Learning to count by 3 is fun and challenging! It helps kids develop their number sense and see patterns in numbers.
We love these skip counting by 3 songs
Our kid’s favorite way to learn skip counting is with songs. He loves singing, and memorized how to count by 3s without really thinking about it. These are our favorite ones.
Practice counting with objects
One of the easiest hands on way to learn skip counting is by grouping objects together and counting.
We did this with pasta. Is it just my kid who loves anything pasta related?
Print and cut out this count by three puzzle
This is a fun and simple counting puzzle that you can print out at home. I love making these types of learning activities for my kid because they work on several skills at once:
- Practice skip counting by three
- Scissors skills
- Glue fine motor skills
Worksheets to practice skip counting by threes
I made these three worksheets for my kid to practice skip counting by threes. Even though the songs helped him memorize how to do it, these worksheets helped him visualise the numbers.
You can download both worksheets together below.
Practice skip counting by threes to sixty
This worksheet lets kids fill in the missing numbers as they count by three. We completed this worksheet as we listened to the songs above.
Fill in every third number in this hundreds chart
The cool thing about filling in the hundreds chart is seeing the patterns form. You can see:
- The diagonal pattern form across the chart
- The need to skip 2 numbers between writing each number
Make a count by 3 number line
I forgot to take a picture of our count by three number line, but we like to make number lines when learning skip counting.
There’s nothing complicated about it, but it’s a fun and easy activity.
Making number lines lets kids practice:
- Scissors skills
- Taping fine motor control
- Writing numbers
Simply cut up wide strips of construction paper, tape them together to make a long strip. Then, on the strip, simply write the numbers across counting by threes.
Once the number lines are done, they often become belts, paths, ramps during playtime!
Here is a tricky questions to take it further
If you have 24 of the same item, how many groups of three does that make?
To find out, gather 24 small items (duplo pieces, sticks, marbles). Put them into groups of 3. How many groups do you have?
If you have 15 of the same items, how many groups of three does that make?
To find out, gather 15 small items (duplo pieces, sticks, marbles). Put them into groups of 3. How many groups do you have?


Counting fractions in a range
Consider the fraction, n/d, where n and d are positive integers. If n < d and HCF(n,d)=1, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, we get:
It can be seen that there are 3 fractions between 1/3 and 1/2.
How many fractions lie between 1/3 and 1/2 in the sorted set of reduced proper fractions for d ≤ 12,000?
考虑形如n/d的分数，其中n和d均为正整数。如果n < d且其最大公约数为1，则该分数称为最简真分数。
如果我们将d ≤ 8的最简真分数构成的集合按大小升序列出，我们得到：
将d ≤ 12,000的最简真分数构成的集合排序后，在1/3和1/2之间有多少个分数？


The students know how to do the math, they just don't understand what the question is asking.
This phrases is true for many teachers throughout their math instruction. Reading and writing in mathematics are of particular interest to educators because these processes are essential to both problem solving and concept development in mathematics. In literacy-rich mathematics classrooms, language processes support students while they are learning new content and helps them to demonstrate what they have learned. Throughout this workshop we will:
Discuss/share different math strategies and/or books dealing with math that promote a literacy rich environment
Engage in math activities that foster a literacy rich environment
Reflect on the Common Core Learning Standards for Literacy and Math and make connections between the Eight Mathematical Practices
By the end of the online course, participants can expect to gain useful and practical ideas for embedding literacy in their Mathematics class. A useful text to support our learning is:
Math and Literature By: Marilyn Burns
- Teacher: Stacy Conti


When working with PIC microcontrollers, I find myself needing numbers formatted in Hex and binary interchangeably: hex for instructions an internal math, binary for port manipulation. Sometimes, if I'm deep in thought about a particular piece of code, I can't shift gears quickly to simple math manipulation. Luckily, I learned some VB6 in college and applied some of that knowledge to this problem.
The result...a simple number converter. Enter your number in any of the 4 ways: Binary, Hexadecimal, Decimal, or ASCII; press "Compute", and it will fill in the other 3 blocks with that value formatted appropriately.


Students have the opportunity to learn more about STEM careers and get hands-on experiences with engineers at John Hopkins University through the Engineering Innovation program sponsored at Clark State Community College.
“It is something I certainly would have done to see if STEM was the right career choice for me,” Kanesha Hall, manager of STEM programming at Clark State said. “In a world where education can cost so much, this is an opportunity to take advantage and see what field of engineering or what career in STEM students may be interested in.”
Hall spoke with students about the Engineering Innovation program as part of the STEMM Idol Speaker Series earlier this year. Students currently in grades 10-12 are encouraged to apply for the program if they meet the following qualifications:
- Grades of As and Bs in math and science classes
- Successfully completed Algebra II
- Successfully completed a lab science course (physics, chemistry and/or biology)
- Successfully completed a course where the trigonometric functions are taught. To meet the trigonometry prerequisite students, must be able to use the trigonometric functions and be comfortable applying the laws of sines and cosines to non-right angle triangles.
“If chosen, students will spend part of their summer taking courses and travel to John Hopkins University in Baltimore,” Hall said.
The course runs from June 18 through July 20. In addition to the putting the Engineering Innovation program on their resume, students will earn 240 internship hours.
Learn more about the program, and application details, here.
Posted February 5, 2018


In this article we’ll show you how to calculate the factorial of 100 with a really quick, step-by-step explanation of how it’s calculated.
First off, what exactly is a factorial? The factorial is the result of multiplying all whole numbers in a chosen number (in this case 100) all the way down to 1.
You will generally see factorials expressed with an exclamation mark after the number, like so:
what is the factorial of 100
100 x 99 x 98 x 97 x 96 x … = 9.3326215443944E+157
In this case, the number of whole numbers in 100 is more than five. You can see how this can quickly get out of hand with larger numbers.
Factorials are used in math quite a lot when calculating the number of possible combinations or permeatations of something. If you think about shuffling a deck of 52 cards, you can use factorials to calculate how many possible orders there are.
When describing a factorial you would usually say 100! like “100 factorial”, “100 shriek”, or “100 bang”. Personally, I prefer shriek!
Hopefully this article has helped you in your quest to calculate the factorial of 100. Feel free to share with your friends, family, teachers, and anybody who might be interested in factorials of numbers (which is surely everyone!).
the factorial of hundred


Birmingham students summer academic camp
By Lisa Brody
Elementary and middle school students enrolled in Birmingham Public Schools will have the opportunity to maintain and increase their academic skills in mathematics and literacy this summer with a free four-week summer learning opportunity sponsored by the school district.
All sessions of the program are in-person, and taught by certified teachers. Students must commit to attending all sessions. The classes are designed to be small and interactive, with each class capped at 12 students. Safety protocols will be followed, with masks worn by students and staff, and distrrict social distancing protocols will be followed and maintained. Students are encouraged to bring their own snack each day.
The program includes remediation of essential skills from the student's current grade level. Each day will include interactive learning opportunities that target students’ specific learning needs in math and literacy, with engaging lessons each day.
For Birmingham Schools students currently enrolled in grades K-4, the summer bridge program will meet Mondays through Thursdays, July 19 - August 12, from 9 a.m. to 12 p.m., at a Birmingham elementary school to be determined. Transportation is not included.
For district students currently enrolled in grades 5-8, the summer bridge program will meet Mondays through Thursday, June 21 - July 15, with the exception of Monday, July 5, from 8:30 a.m. to 12 p.m., at a Birmingham middle school to be determined.
Registration for the program, open to students currently attending virtually and in-person, opens on the district's website Wednesday, April 14, and will close Friday, April 30. Limited space is available. There will be a wait-list if necessary.


I’ve posted several studies attempting to predict golfer performance. This attempted to find the importance of the previous week when predicting the following week. The study was not particularly sophisticated (simple linear regression), but the results indicated that the previous week’s performance should be valued at around 10% of the projection for the golfer the following week (90% would be the two-year performance). This other study attempted to predict golfer performance for an entire season using prior season data. That study found that no matter how many years are used or whether those years are weighted for recency, the resulting correlation is ~70%. Doing better than that for full-season prediction would indicate an additional level of sophistication beyond aggregating prior seasons or weighted data for recency.
This post, however, concerns predicting individual tournament performance using my Bayesian rankings. These rankings are generated each week by combining prior performance and sample performance using the equation ((prior mean/prior variance)+(observed mean/observed variance))/((1/prior variance)+(1/observed variance)). In this way, each golfer’s prediction for a week is updated when new information is encountered. The prior mean for a week is the Bayesian mean generated the prior week. My rankings also slowly regress to a golfer’s two-year performance if they are inactive for a period of weeks. For each week, the prior mean is calculated using the equation (((Divisor – (Weeks since competed)) / Divisor) * (Prior Mean)) + ((1 – ((Divisor – (Weeks since competed)) / Divisor)) * (Two-year Z-Score)). I use 50 as the Divisor, which weights two-year performance at 2% for 1 week off, 27% for 5 weeks off, and 69% for 10 weeks off.
To measure how predictive these rankings were, I gathered data for all golfers who had accumulated 100 rounds on the PGA, European, Web.com, or Challenge Tour between 1-2010 and 7-2013. My sample was 643 golfers. I then examined performance in all tournaments between the 3-28-2013 and 8-8-2013. My sample was 6246 tournaments played. I then generated Bayesian rankings predicting performance before each of these tournaments played. The mean of my predictions was +0.08, indicating I expected the sample to be slightly worse than PGA average. I then compared each prediction to the golfer’s actual performance.
The table below shows the performance of Bayesian and pure Two-year predictions by including all predictions within +/- 0.05 from the displayed prediction (ie, -0.50 includes all predictions between -0.45 and -0.55). The accompanying graph shows the same information with best-fit lines.
Obviously, the Bayesian and Two-year predictions perform similarly. To test which is better I tested the mean square error. This shows how closely the prediction matched actual performance. I also included “dumb” predictions which simply predict all rounds will perform to the mean of all predictions (+0.08 for Bayesian, +0.055 for Two-year). The “dumb” predictions are the baseline for judging any predictions. If a prediction can’t beat it, it’s worthless.
The mean square error for the Bayesian predictions was 0.381 and 0.446 for the “dumb” predictions. The mean square error for the Two-year predictions was 0.389 and 0.452 for the “dumb” predictions. So both sets of predictions provide value over the “dumb” predictions, but both perform fairly similarly when compared to the “dumb” predictions (-0.065 for Bayesian and -0.063 for Two-year).
This study indicates two things; first, using Bayesian methods to predict golfer performance doesn’t substantially improve accuracy relative to unweighted aggregation of the last two years of performance, and second, that predicting golfer performance in individual tournaments is very difficult. A mean square error of 0.38 indicates an average miss of 3.5 strokes for golfers playing four rounds and 2.5 strokes for golfers playing two rounds.


Sir Isaac Newton was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is perceived and considered by a substantial number of scholars and the general public as one of the most influential men in history. His Philosophie Naturalis Principia Mathematica, published in 1687, is by itself considered to be among the most influential books in the history of science, laying the groundwork for most of classical mechanics. In this work, Newton described universal gravitation and the three laws of motion which dominated the scientific view of the physical universe for the next three centuries. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation, thus removing the last doubts about heliocentrism and advancing the scientific revolution.
In mechanics, Newton enunciated the principles of conservation of both momentum and angular momentum. In optics, he built the first practical reflecting telescope and developed a theory of colour based on the observation that a prism decomposes white light into the many colours that form the visible spectrum. He also formulated an empirical law of cooling and studied the speed of sound.
In mathematics, Newton shares the credit with Gottfried Leibniz for the development of the differential and integral calculus. He also demonstrated the generalised binomial theorem, developed the so-called "Newton's method" for approximating the zeroes of a function, and contributed to the study of power series.
Newton remains influential to scientists, as demonstrated by a 2005 survey of scientists and the general public in Britain's Royal Society asking who had the greater effect on the history of science, Newton or Albert Einstein. Newton was deemed to have made the greater overall contribution to science, although the two men were closer when it came to contributions to humanity.
Newton was also highly religious, though an unorthodox Christian, writing more on Biblical hermeneutics than the natural science he is remembered for today.


We then played a math dice game. Specially made paper dice had the numbers 4 through 9. Two dice are rolled. One of them is multiplied by ten and the other is subtracted from that number. Not as easy as you might expect for 3rd graders. Players take turn rolling the dice. The goal for each player is to come up with one number in each of the decades from the 30's through the 80's. The first one to get numbers all six decades win. My young student and I ended the game in a fiercely competitive tie, 5-5. Neither of us could roll an 8, which is necessary to get a number landing the decade of the 70s.
A newspaper article appeared earlier in the week about guy who who was an active math tutor at an elementary school "Retired salesman gives new spiel at Plymouth school: After a career in sales, 85-year-old Andy Anderson volunteers each week at a Plymouth school." I wonder if he's ever been compared to a movie figure? Possibly George Burns? No, probably not. One of his most famous appearances was in "Oh God", and that was filmed way back in the 70s.


Genre: Self-help Book, Non-Fiction
Author: Jordan Ellenberg
Book: How Not to Be Wrong: The Power of Mathematical Thinking (Buy the Book)
How Not to Be Wrong is a consumer-friendly briefing of a few mathematical concepts and their importance to everyday life. Throughout this work, Ellenberg doesn’t talk about complicated formulas or equations, but instead, the fundamentals of mathematical thinking.
He argues that math isn’t just indefinite lists of computations, but that the seemingly endless computations we do in school are like training for life’s challenges – he likens this to an athlete practicing endless training drills, though he never uses them directly in the games. Once you know math, it’s like wearing a pair of x-day glasses that allow you to identify underlying structures and patterns in everyday life.
Knowing math provides you with a tool to gain a deeper understanding of the world around you. You can use mathematical thinking to get problems of commerce, politics and even theology right. When students ask:
“when am I ever going to use math in real life?”
they should be referred to this book.
Ellenberg breaks his book into 5 main sections corresponding to 5 broad themes in mathematics and its applications: linearity, inference, expectation, regression and existence.
In the first section, he discusses an assumption that many people make: that relationships between inputs and outputs are always linear. Through a few examples, he describes the pitfalls of this often-incorrect assumption. More of a good thing is not always better, and the scale at which you observe relationships changes their linearity.
In the second section, he discusses inference – the process of drawing conclusions about a broader population by observing some smaller set of samples. While many extremely powerful statistical tools rely on inference, they require a deep understanding of inference as misuse frequently leads to extremely incorrect conclusions.
In the third section, he introduces the concept of expected value – not the value that you expect if you engage in one uncertain event but the average value that you would expect given a large number of chances. Additionally, it is important not only to consider the most likely outcome but incorporate all possible outcomes into your analysis. Lastly, given a large number of chances, improbable things become probable.
In the fourth section, he discusses regression to the mean. Regression to the mean occurs in many fields including business, sports, genetics and others. Often times, the exceptional performance of the best players and the worst players is a function of random fluctuations. Over time, these effects wear out, and their performance tends to more closely resemble the broader population.
In the fifth section, he discusses existence – a combination of seemingly paradoxical realities and a few abstract mathematical concepts such as set theory and non-traditional coordinate systems. Most notably, he explores the difficulty in fairly understanding public opinion and the conclusion that in many circumstances it simply does not exist.
Ellenberg closes the book by summarizing the key findings. Chief among these is the notion that there is structure in the world, and math allows us not only to understand that structure but make the best possible decisions based on our observations.
Although the book is titled How Not to be Wrong, Ellenberg does not assert that knowing math guarantees you will always achieve the most desirable outcome – math gives us a principled method of being uncertain. Mathematical certainty is not the same as the everyday convictions we have become accustomed to thinking of as certainties.
Overall, How Not to be Wrong is an engaging read that helps the reader understand important concepts. Ellenberg’s abundant use of relatable and interesting stories makes the concepts memorable, and the structure of the book makes the content digestible.
The mathematical concepts explored here are relatively basic, though a reader should have a basic understanding of geometry, algebra, probability and data visualizing to gain the most utility from this book.
Abraham Wald was an Austrian mathematician during World War II. He was a part of the high powered “Statistical Research Group” which used math to make wartime recommendations to the military. One of his most famous recommendations was the case of the “missing bullet holes”. When asked to determine where to reinforce the armor on fighter jets, he inspected the placement of bullet holes on fighter jets that returned from combat.
Upon inspection, he counterintuitively recommended reinforcing the armor where there were few bullet holes, while the generals had instead expected him to tell them how much more armor was needed in the places with many bullet holes. While there were numerous equations involved in computing the optimum armor distribution, the underlying thinking was clear: Wald assumed planes were hit with relatively even distribution, so the areas of the planes with few holes weren’t getting hit less, but instead, hits to those areas were lethal.
Therefore, armor should be reinforced where the holes were scarce, not where the holes were plentiful. In this case, mathematical thinking is
about asking yourself what assumptions you are making and what the implications of those assumptions might be. The Air Force generals incorrectly assumed that the planes coming back were representative of all planes in the war, but Wald knew that it was a biased sample as these were only the planes that survived. He was able to look past the context of planes, bullets and war to see the underlying structure of the problem: a phenomenon called survivorship bias.
Wald’s recognition of the generals’ incorrect assumption may not seem to resemble the math we are accustomed to. It does not include complex equations, trigonometric identities or calculus. It probably just seems like common sense, and that is the exact point of this story: all math starts from a point of common sense. Math is an extension of common sense by other means. In Ellenberg’s words:
“Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.”
Math offers rigorous structure to common sense that allows you to accurately reach beyond the obvious. However, it is important to maintain your common sense as you dive deep into the abstract reaches of math – it is this “interplay between abstract thinking and our intuitions about quantity, time, space, motion, behavior and uncertainty” that allows math to work.
When considering the various types of mathematical facts, Ellenberg segments them into four categories based on a 2 x 2 matrix (left). On one axis, facts are categorized by the complexity of the computations involved: some are simple, while others are more complicated.
On the second axis, facts are categorized by how meaningful the contribution is: some are shallow, while others are profound. Contemporary professional mathematicians attempt to spend most of their time in the complicated-profound quadrant; however, there are many interesting observations in the simple- profound quadrant. How Not to Be Wrong focuses on the simple- profound quadrant.
Ellenberg breaks his book into 5 main sections corresponding to 5 broad themes in mathematics: linearity, inference, expectation, regression and existence. In each section, he offers digestible examples of the usefulness of each concept as well as common pitfalls and applications for these concepts. In each section, he offers digestible examples of the usefulness of each concept as well as common pitfalls and applications for these concepts.
PART 1: LINEARITY
In this section, Ellenberg describes numerous cases in which challenging the assumption of linearity can lead to interesting findings. Sometimes common phenomena are assumed to be linear but are, in fact, non-linear. In other cases, phenomena are assumed to be non-linear but can, in fact, be considered linear when viewed on a different scale. Understanding when to apply linear and non-linear thinking leads to useful insights.
Some things are non-linear
There is a common fallacy: “if a thing is good, more of it must be better.” This fallacy is an example of assuming that a non-linear phenomenon is in fact linear. Ellenberg debunks this fallacy using a blog entry authored by the Cato Institute that condemned taxation and government provided social benefits based on this principle. In the post titled: “Why is Obama Trying to Make America More Like Sweden when Swedes Are Trying to Be Less Like Sweden?” Daniel J. Mitchel argues that Swedishness (in this context defined by the amount of government funded social welfare) is obviously bad, as even the Swedes have recognized that being less Swedish would lead to higher prosperity. His vision was something like the curve on the left. In reality, the issue is not linear and the reality looks something like the middle curve.
This is an example application of the Laffer Curve – a simple economic model developed by Arthur Laffer. The Laffer Curve looks something like the curve on the right. At a 0% tax rate, there would obviously be no tax revenue. Likewise, at a 100% tax rate, people would not have any incentive to work for pay, and there would again be no tax revenue.
The optimal tax rate lies somewhere in the middle. While the exact tax rate or level of “Swedishness” that will maximize tax revenue or national prosperity is context-specific and still up for debate, simple statements like “more tax is always good” or “everyone needs to be less Swedish, just look at the Swedes” are not accurate.
Sometimes non-linear things can actually be linear
Though curves may always appear to either be linear or non-linear, it is important to realize that the distinction between linear and non-linear is not always clear depending on the scale at which the curve is viewed. At small scales, non-linear curves can be approximated by small linear segments.
Archimedes drew on this concept when he developed the method of exhaustion to calculate the area of a circle. At the time, there was no way to calculate the area of a circle because of its non-linear curvature. However, by drawing polygons using straight lines and known angles (areas which could be computed), he could estimate the area of the circle.
The method goes something like this. Start with a circle with a radius of 1 and draw a square inside of it (the inscribed square) where all 4 corners touch the circle at one point. This square can be divided into 4 equal right triangles, and by the Pythagorean Theorem, the length of the sides of the square can be computed to be equal to the square root of 2.
From there, the area of the square can be computed to be 2. Next, another square is drawn outside of the circle (the circumscribed square) and this one also touches the circle at only 4 points. The length of each side of this square is 2, and the area of this square is therefore 4. From this, we know the area of the circle is more than 2 but less than 4. This method can be extended to an octagon, then a 16-gon, and so on and so forth 15 times.
At this point, the resultant 65536-gon looks so similar to the circle that it doesn’t matter if it’s a circle or a polygon, the areas will be so close that the difference doesn’t matter for any practical purpose. As this process goes on to infinity, the two become the same.
Numerous other interesting phenomena arise as we delve into the infinite (or practically speaking, small enough that the calculable mirrors the behavior of the infinite). This similarity between the infinite and small enough or numerous enough to mimic the infinite is the basis of calculus.
PART 2: INFERENCE
Inference is the basis of many statistical methods. Simply stated, it is the assertion that conclusions about full sets of data (or populations) can be drawn by observing a randomly selected subset of that data (samples). Inference is a very powerful statistical tool that has led to many discoveries in traditional fields such as science, engineering, finance and medicine, but also in surprising places such as theology.
However, like other mathematical tools, when used incorrectly, statistical tests rooted in inference will reject true conclusions and fail to reject untrue hypotheses. While using inference as a basis for decision making (whether formally through statistics or informally through crude observation) it is important to understand that improbable things happen all the time.
Given enough chances, improbable becomes probable
Ellenberg uses a few stories to illustrate that given enough chances, improbable things happen often. The parable of the Baltimore stockbroker goes something like this. One day, you receive a stock tip from a broker that says a certain stock will rise in the next week. A week passes and that stock does in fact rise.
The next week you receive another tip that a different stock will fall, and a week later, sure enough, the stock falls. 10 weeks pass, and you receive 10 tips that correctly predict the movement of 10 stocks. By the 11th week, you are so impressed by his stock-picking prowess that you invest your money. However, these 10 picks were not evidence of a prescient stockbroker, you were the victim of a scam.
If you look from the stock broker’s point of view it becomes much clearer how he actually picked correctly 10 times in a row. From the stockbroker’s point of view, the story goes something like this. Suppose any given week there is a 50% chance that a given stock rises. Then, over the course of 10 weeks, there is a 1/1024 chance that 10 random picks would be correct. The first week, he sends his picks to 10,240 people – 5,120 get a letter that says the stock will rise, and 5,120 get a letter that says the stock will fall.
Given the 50% chance that the stock will rise, 5,120 people will have received a correct tip. When the next week rolls around, he sends his newsletter to the 5,120 who received the correct pick, this time sending 2,560 tips that say rise, and 2,560 tips that say fall. He repeats this for 10 weeks and by the 11th week, there are 10 people who received 10 correct stock picks in a row. The important thing to consider here is:
how many chances did the stockbroker have to get these picks correct?
On the receiving end of his newsletters, it is impossible to know. Therefore, it is unwise to believe his performance was the result of some special financial foresight.
This idea is not limited to the realm of parables. Ellenberg describes numerous real world examples of this phenomenon. Throughout the 20th century, some mathematically inclined Rabbi decided to search the Torah for hidden messages. They employed a technique called “equidistant letter sequence” or ELS to interpret the text. ELS goes something like this. Suppose you have a text that says: DON YOUR BRACES ASKEW. A 5 letter ELS would reveal the hidden word: DUCK, by extracting every fifth letter.
Upon inspection, the Rabbi found numerous hidden messages ranging from the word TORAH to the death dates of famous Rabbi. However, it is important to realize that just like the Baltimore stockbroker, the number of chances matter. The Torah has 304,805 letters in the text. Upon further investigation of the methods and similar studies revealing hidden messages in other texts such as War and Peace, the Rabbis’ findings were ultimately rejected.
The law of large numbers
In short, the law of large numbers roughly states that it becomes increasingly more unlikely to see inaccurate results as the number of experiments increases. Coin flips are a great example of the law of large numbers. If someone were to flip 10 coins and saw 8 heads and 2 tails, it does not mean the coin is unbalanced.
However, it is (for practical purposes) impossible to flip 10,000 balanced coins and observe 80% heads due to the law of large numbers. In fact, Abraham de Moivre calculated that it is more unlikely to flip 54% heads in 1,000 tries than it is to flip 60% heads in 100 tries.
Carelessly applied statistics leads to incorrect conclusions
When trying to draw inferences about systems using mathematics, we are often presented with yes or no questions. However, when analyzing data to validate your hypothesis, the data must not only be consistent with your theory but also inconsistent with the negation of your theory. For example, if I believe I have telekinetic powers to make the sun rise, the sun rising when I tell it to does not prove my hypothesis. The sun must also fail to rise if I don’t use my telekinetic powers.
The second statement – that the sun must fail to rise if I don’t use my telekinetic powers – is the null hypothesis. Null hypothesis testing basically says: 1) suppose your null hypothesis is true and let p be the probability under that hypothesis that you observe the results that you have 2) if p (also called the p value) is small, then your results are statistically significant (ie, it is sufficiently unlikely that you would observe the results that you have if your null hypothesis is true), if p is large, then the null hypothesis cannot be ruled out.
The null hypothesis test or p value test has become a standard among medical, scientific and social science publications, but it is important to understand the limitations of this approach. The generally accepted convention is to select a p value of 0.05.
In other words, there is a 1 in 20 chance that a researcher would observe these results if the null hypothesis is true. In most cases, researchers have sufficient reason to believe their hypothesis before testing it, so the 1 in 20 odds are enough. However, we’ve seen with the Baltimore stock broker that improbable things happen all the time given enough chances.
There are many ways the p value test can be misused. First, experiments without sufficient theoretical framework can be misinterpreted. Ellenberg gives two examples of this. In 2009, a group of researchers published a satirical paper (though the data and analysis were real) in which they showed statistically significant evidence that a dead salmon could read emotions.
When various pictures where shown to the salmon, certain voxels (brain waves observed through fMRI scans) lit up. Of course, these correlations were nothing more than random noise – with the vast amount of data it was highly likely that some of the voxels correlated with the timing of the pictures that were shown.
In a more broadly published field of study, Haruspex (people who sacrifice sheep then interpret the entrails to predict the future) execute numerous experiments and publish the ones that work. Of course, given what we already know about the p value, we expect that 1 in 20 experiments will show statistically significant results. However, hundreds of these papers are published in The International Journal of Haruspicy.
These two stories do not mean that all papers or scientific conclusions based on p value testing should not be trusted, just that scientists and researchers should be careful to present sufficient theoretical framework to form a causal argument and employ additional tools such as multiple comparisons analysis to validate surprising results.
A second way in which p value tests can be misused is when they are underpowered. Small results may not show up on p value tests. An example of this is the “hot hand” myth in basketball. The “hot hand” says that sometimes players display short bursts of greatness in which they seemingly cannot miss a shot.
After many rigorous statistical analyses sought to prove it did not exist – and indeed they could not detect it – a group of computer scientists showed that the statistical tests were simply not high power enough to detect the effect, even if it was real. They did so by simulating players’ seasons but explicitly adding in sequences where the players possessed the hot hand. But even in these sequences, the statistical tests were unable to show a result.
The phrase “statistically significant” is also misleading. Really, this phrase simply means an effect exists, not necessarily that the effect is important, large or meaningful. A third way, therefore, that statistical tests may be misused is overstating the meaningfulness of statistically significant results. An example of this occurred in 1995 when the UK Committee on Safety of Medicines released a “Dear Doctor” indicating that a certain type of birth control doubled the risk of thrombosis.
This “statistically significant” effect turned out not to be very meaningful or important as the risk of thrombosis was already very small to begin with. This letter led to a dramatic reduction in the number of women taking birth control, naturally leading to a spike in the number of unplanned pregnancies (26,000) and abortions (13,600). When the dust settled, the researchers who released the study said the additional risk presented by the new birth control may have, at most, taken the life of one woman.
The stories of the British birth control scare, the “hot hand” myth, and dead fish reading minds all show the importance of using statistical tools correctly. Particularly, it is important to 1) choose an appropriate test 2) understand what you can and cannot test for and 3) understand the context and other factors that may be influencing your outcomes.
PART 3: EXPECTATION
Expected value is a commonly used metric when dealing in uncertainty. It is essentially the average value that you would expect to see over a large number of occurrences.
If gambling is exciting, you’re doing it wrong
When deciding whether or not to play the lottery, you should consider the expected value of the lottery ticket. Expected value can be calculated by multiplying each potential outcome by the probability of that outcome, and summing up these numbers for all of the potential outcomes.
For example, suppose you have a lottery ticket with a 1/10,000,000 chance of winning the $6 million dollar jackpot, and a 9,999,999/10,000,000 chance of being worthless. The expected value of that ticket would be 1/10,000,000 × $6,000,000 + 9,999,999/10,000,000 × $0 = $0.60. That does not mean you expect the single ticket to be worth $0.60, but instead, if you purchased a large number of tickets, say 10 million, you would expect the average value of those 10 million tickets to be $0.60.
Normally, lottery tickets are priced such that the expected value of the ticket is less than the purchase price of the ticket in order to ensure the state makes money. Due to this expected value calculation, the lottery is usually not a good bet.
In 2004, Massachusetts decided to change their lottery system in order to increase participation. Instead of the traditional lottery where the jackpot continued to accumulate week to week until someone correctly guesses the 6 ball combination to win the jackpot, Massachusetts introduced a new game called Cash Windfall.
In Cash Windfall, the jackpot would continue to grow week to week as long as no one guessed the correct 6 ball combination; however, the difference came in the weeks that the jackpot reached a designated size: the massive jackpot triggered a roll down. During a roll down, it was not necessary to guess the 6 ball combination in order to win the jackpot. Players who correctly guessed 5 of 6, 4 of 6, and even 3 of 6 numbers would split the jackpot in various proportions, therefore raising the chances of success and increasing participation in the lottery.
A group of MIT students studying the lottery for class discovered an interesting phenomenon – during the rolldown weeks, the expected value of a lottery ticket exceeded the purchase price of a lottery ticket. Massachusetts had inadvertently designed a game that was, in fact, a good bet. Based on the expected values of each winning combination, the expected value of a $2 ticket was $5.54 as illustrated below.
|Chance of Winning
|Expected number of winners
|Roll Down Allocation
|Roll– Down per Prize
|Expected Value Calculation
|Match 5 of 6
|1 in 39,000
|1/39,000 × $50,000
|Match 4 of 6
|1 in 800
|1/800 × $2,385
|Match 3 of 6
|1 in 47
|1/47 × $60
|Total Expected Value:
|Match 5 + Match 4 + Match 3
The MIT students, along with a few other betting clubs, placed massive bets – and won big – during the roll down weeks for the duration of the Cash Windfall game. The lesson here is that while expected value is not the value you can expect while placing one bet, by invoking the law of large numbers, expected value can be realized as the average value of a bet when a sufficiently large number of bets are placed.
Risk/waste reduction is not always beneficial
There is a Laffer style tradeoff between reducing risk or eliminating waste, and the cost of reducing risk or eliminating waste. This means that if you spend too little effort reducing risk, the risk will be large and will hurt you. Conversely, however, spending too much effort reducing risk, the effort spent reducing the risk will cost you more than taking the risk would have.
“If you never miss a plane, you’re spending too much time at the airport.” To most people, this quote seems counterintuitive: if I miss my plane, won’t I spend more time at the airport? The answer to this question is maybe. In order to compute the tradeoff, economists use the standard measurement of utils to measure the utility of commodities such as time or money.
This is necessary as the value of the time spent getting to the airport early, may not be as valuable as the time lost at the airport after missing a flight. For example, if missing a flight means missing an important business meeting or a friend’s wedding, the missed flight costs much more than getting to the airport early.
The computation to understand the tradeoff can be carried out as follows. Suppose each hour you spend at the airport before the flight costs you 1 util, but missing a flight will cost you 6. Given three options: getting to the airport 2, 1.5 or 1 hours before your flight, your chance of missing the flight is 2%, 5%, and 15%, respectively.
|-2 x 2% x (-6) = -2.12 utils
|-1.5 x 5% x (-6) = -1.8 utils
|-1 x 15% x (-6) = 1.9 utils
The expected value computations for each option are carried out in the table on the left. Based on these calculations, if your goal is to maximize utility (as it is for most people), option 2 yields the smallest loss of utility and is, therefore, the best option although it comes with a non-trivial chance of missing your plane. Of course, when applying expected value calculations, the law of large numbers is important to consider.
Therefore, someone who flies infrequently who has not missed a flight is not necessarily spending too much time at the airport. From a practical standpoint, it is also important to consider the fact that you don’t always need to arrive at the airport with the same margin of error. If for some important events the cost of missing a flight is high, then the optimal point shifts to the left. For other, more flexible travel, the optimal point may shift to the right.
Reason to believe is not the same as evidence to believe
Ellenberg and other mathematicians have a curious obsession with trying to prove or disprove the existence of God using math. Blaise Pascal, a 17th century French mathematician, took a different approach when applying math to his faith. Though his thoughts were published after his death as the Pensées, he is most famous for thought number 233.
“God is, or He is not. But to which side shall we incline? Reason can decide nothing here.”
For this reason (and his mathematical inclination toward understanding uncertainty), Pascal viewed the decision to believe in the God of Christianity as a wager. In order to decide if to believe or not, he carried out a straightforward expected value computation: expected value of believing God is real = chance that God is real × infinite joy + chance that God is not real × some finite cost of a life of piety.
The expected value of not believing in God = chance that God is real × infinite damnation + chance that God is not real × some finite enjoyment of life on earth. According to this formula, as long as there is any non-zero chance that God is real, belief in God yields an infinitely positive expected value, while the belief that God is not real yields an infinitely negative expected value.
However, the flaw in this argument is that it does not consider all possibilities. For example, if there exists a possibility of a god who damns Christians eternally, his calculation is no longer valid as we have no principled way of knowing the relative odds of the existence of either option.
PART 4: REGRESSION
Regression to the mean
In many fields of study including sports, business, genetics, weather and many more, a mysterious force often times pushes values toward some average function. This can often be explained by the concept of regression to the mean. In the 1920s and 1930s, Horace Secritst – a renowned statistics professor at Northwestern University – published an extensive review of the performance of hundreds of businesses entitled The Triumph of Mediocrity.
In this work, he detailed how regardless of industry, the top performing stores performed more poorly (relative to the competition) by the end of his study, and the bottom performers tended to increase performance (relative to the competition). In all, stores among the top sextile and bottom sextile in every industry tended toward the mean performance of the industry. This held true in about every performance statistic he measured. His conclusion was that competition drove businesses toward mediocrity.
A British Eugenicist, Francis Galton, challenged this view by observing heritable traits such as height. One of his most notable findings was that while tall parents tended to have tall children, the children were not likely to be as tall as their parents. He demonstrated this by plotting the height of sons and fathers and on an x-y scatter plot and saw that the heights tended toward the average height. Regression to the mean is based on the fact that performance is often times a function both of random fluctuations or luck, and controllable factors or skill.
However, the controllable factors do not often contribute as much as it is thought to contribute and over time as the luck runs out, performance regresses to the mean value. Ellenberg goes on to demonstrate the presence of these effects in sports (the phrase “on pace for” is misleading as exceptional players early in the season will often regress to the mean) and weather patterns.
Limitations on correlation
Correlations are not transitive. If A and B are positively correlated, and B and C are positively correlated, A and C are not necessarily positively correlated. On first glance, this may seem both obvious and paradoxical. In order to understand it a bit better, Ellenberg considers the case of mutual fund investors. Suppose Laura has a fund split evenly between Facebook and Google. Tim’s fund is half GM and half Honda. Sarah has half Honda and half Facebook in her fund.
It is not unreasonable to expect Laura and Sarah’s funds to be positively correlated. The same could be said for Sarah and Tim. However, does that mean you can expect Laura and Tim’s portfolios to be positively correlated? No, you cannot. The non-transitivity of correlation is a problem that appears in many areas of life including medical research, politics, genetics, mutual funds and many other fields.
This makes medical research very difficult. For example, levels of high- density lipoprotein (HDL) are positively correlated with a lower risk of heart attack. However, that does necessarily mean that raising HDL by any means is also positively correlated with a lower risk of heart attack. It could also be the case, that some non-obvious factor is impacting both HDL levels and heart attack risk – this is analogous to the Honda stock affecting both Tim and Sarah’s portfolios.
If a drug intended to lower the risk of heart attack does so by raising HDL levels by means of the mystery factor then it may also reduce the risk of heart attack, but if it raises HDL levels by some other factor, then it is just shooting in the dark. This is one of the principal challenges in medical research. Understanding the difference between correlation and causation is crucial in understanding the relationships among various correlated variables.
This is why the phrase: “correlation does not imply causation” is so popular. But ultimately, these relationships can only lead researchers in the right direction and they have to directly test their hypotheses. If A and B are correlated, and B and C are correlated, test to determine whether A and C are correlated.
This same phenomenon can be observed in the world of politics. Rich states vote for Democrats, rich people vote for Republicans. This seems paradoxical: don’t rich people come from rich states? In the general case, the non-transitivity of correlation can be explained by Karl Pearson’s definition. While the formula itself is complex, it is helpful to think in geometric terms. Your set of variables can be thought of as an n-dimensional space, in which each set of variables corresponds to a vector. Under this geometry, the correlation between two points can be thought of as the cosign of the angle between two vectors. While you may not remember exactly how cosign operates, the only important things to know about cosign are that cos(0) = 1, cos(90) = 0, and cos (180) = -1. In other words, an acute angle (0 ≤ 𝜃𝜃 < 90) between vectors means there is positive correlation, a right angle (𝜃𝜃 = 90) between vectors means there is no correlation, and an obtuse angle (90 < 𝜃𝜃 ≤180) between vectors means they are negatively correlated. Once this definition of correlation is understood, it is easy to see how correlation is not transitive. In the graphic to the right, vector 1 and 2 are positively correlated (acute angle), vector 2 and 3 are positively correlated (acute angle), but vectors 1 and 3 are not positively correlated (right or obtuse angle).
PART 5: EXISTENCE
Majority rules systems only tend to work well when choosing between two options. For example, a 2011 Pew Research poll showed that while 77% of Americans wanted smaller government (that is, to cut government spending) when categorizing the government spending into 13 categories, the majority of respondents wanted to increase spending in 11 of 13 categories.
Only foreign aid and unemployment insurance – collectively less than 5% of spending – did not gain majority support. Some critics of the American public conclude that Americans just want a free lunch. A February 2011 Harris poll summarized this as “Many people seem to want to cut down the forest but keep the trees.”
However, the issue is a bit more nuanced. Consider a simplified example in which a third of the electorate wants to raise taxes without cutting spending, a third wants to reduce spending by cutting Medicare, and another third wants to reduce spending by cutting defense spending. If you ask: “should we cut defense spending?” a majority of the country says no. Similarly, a majority disapproves of raising taxes and cutting Medicare.
So, while a majority of the country wants to cut spending, the majority is also opposed to cutting spending in any single category. The same divergence from a clear majority choice can occur in presidential elections. Consider the 1992 election where Clinton won 43% of the vote, Bush won 38% and Perot won 19%. A majority of the country voted against each candidate, but it still seems fair enough that Clinton won.
However, consider a possible circumstance: the 19% of Perot voters were broken down into 13% who preferred Bush over Clinton, and 6% who preferred Clinton over Bush. Now, if you were to poll the electorate 51% preferred Bush over Clinton. The presence of a third candidate, though no majority preferred him, may have changed the outcome of the vote. Here is a case where public opinion is tough to decipher, and a fair conclusion might be that public opinion does not exist in this case.
In order to address the paradox above, some democracies implement voting systems where voters can submit the first, second, third, etc. choice. This may seem like a fair system; however, circumstances arise that make it impossible to decipher public opinion.
Burlington, Vermont, one of the few US municipalities that employs instant run off voting (a voting system where the votes are counted twice, taking into consideration voters’ second preference), a different candidate is preferred based on how the votes are counted. Under the traditional American voting method, the Republican candidate would have won. Under the instant run off voting system, the Democratic candidate won.
If you consider all the preferences simultaneously, a majority of the voters preferred the Independent candidate over both the Democratic candidate and the Republican candidate; however, he did not receive enough first place votes to make it to the run off. Consider a simplified ballot count from the Burlington election. Under this count, a majority of voters prefer Montroll to Kiss; however, a different majority prefers Wright to Montroll, while a third majority prefers Kiss to Wright.
Who is the fair winner? Circles like this are called Condorcet’s paradox, named after the 18th century French Enlightenment philosopher Marquis de Condorcet. The conclusion here is that sometimes public opinion is impossible to decipher.
HOW TO BE RIGHT
How Not to Be Wrong: The Power of Mathematical Thinking paints math in a way that many (even most) people probably don’t think of it. Many of the computations that support the concepts that Ellenberg talks about are complex and out of reach to many people; however, his examples bring to light the usefulness and applicability of the underlying principles in everyday scenarios when we aren’t the ones performing the computations.
Given the title, it is only fitting that Ellenberg concludes with a summary of how to use the mathematical principals described in the text to always be right. However, the answer is not what many readers want to hear.
“Math gives us a way of being unsure in a principled way: not just throwing up our hands and saying ‘huh,’ but rather making a firm assertion: ‘I’m not sure, this is why I’m not sure, and this is roughly how not-sure I am.’ Or even more: ‘I’m unsure, and you should be too.’”
In other words, it is not possible to always know everything about nature, society or the outcome of the lottery; however, mathematics gives us a framework to uncover hidden structures, deal in the uncertain and make best guesses about everyday life that we can’t directly observe.
While mathematicians traffic in logic and rationality, purely deductive thinking is a dangerous proposition. F. Scott Fitzgerald once said: “The test of a first rate intelligence is the ability to hold two opposed ideas in the mind at the same time and still retain the ability to function.” As humans, we are not entirely rational beings, and this is a good thing.
A common piece of folk advice among mathematicians is that when you are working on developing a theorem you should spend some time trying to prove it correct, but also spend a significant amount of effort trying to disprove the very thing you are trying to prove.
This act of proving by day and disproving by night is valuable not just in mathematics, but also in other areas of your life including social, political, scientific and philosophical beliefs. If you can’t talk yourself out of your existing beliefs, you’ll understand a lot more about why you believe what you believe.
Ellenberg concludes his book with the following words, which do not need any further paraphrasing, summarizing or explanation:
“The lessons of mathematics are simple ones and there are no numbers in them: that there is structure in the world; that we can hope to understand some of it and not just gape at what our senses present to us; that our intuition is stronger with a formal exoskeleton than without one. And that mathematical certainty is one thing, the softer convictions we find attached to us in everyday life is another, and we should keep track of the differences if we can.
Every time you observe that more of a good thing is not always better; or you remember that improbable things happen a lot, given enough chances, and resist the Baltimore stockbroker; or you make a decision based not just on the most likely future, but on the cloud of all possible futures, with attention to which ones are likely and which ones are not; or you let go of the idea that beliefs of groups should not be subject to the same rules as beliefs of individuals; or, simply, you find that cognitive sweet spot where you can let your intuitionrun wild on the network of tracks formal reasoning makes for it; without writing down an equation or drawing a graph, you are doing mathematics, the extension of common sense by other means. When are you going to use it? You’ve been using mathematics since you were born and you’ll probably never stop. Use it well.”
Britt always taught us Titans that Wisdom is Cheap, and principal can find treasure troves of the good stuff in books. We hope only will also express their thanks to the Titans if the book review brought wisdom into their lives.


This calculator can be used to calculate the Field Of View of both fisheye and normal lenses and the number of images needed for a panorama using a given overlap. You can also find out how much overlap you get with a given number of images. By playing with the number you can optimize your shooting strategy.
The calculation for fisheye lenses is theoretical. If the image is larger than the image circle of the lens (i.e. circular fisheye) the calculator will not warn you for the fact that the image area will not be covered by the image circle of the lens.


For arbitrary precision mathematics PHP offers the Binary Calculator which
supports numbers of any size and precision, represented as strings.
Since PHP 4.0.4, libbcmath is bundled with PHP. You don't need any
external libraries for this extension.
In PHP 4, these functions are only available if PHP was configured with
In PHP 3, these functions are only available if PHP was not configured with
The behaviour of these functions is affected by settings in php.ini.
Table 1. BC math configuration options
For further details and definition of the PHP_INI_* constants see
Here is a short explanation of the configuration directives.
Number of decimal digits for all bcmath functions.
This extension has no resource types defined.
This extension has no constants defined.


Prof. Mauro Giudici
Università degli Studi di Milano, Dipartimento di Scienze della Terra “A. Desio” mauro.giudici at unimi.it
Dr. Romina Gaburro
University of Limerick, Department of Mathematics and Statistics romina.gaburro at ul.ie
Prof. Alessandro Comunian
Università degli Studi di Milano, Dipartimento di Scienze della Terra “A. Desio” alessandro.comunian at unimi.it
at = @
Inverse problems appear in several fields of science and technology, including geophysics, bioimaging, material science, etc. and they require specific mathematical techniques to handle ill-posed problems. While fundamental results on uniqueness and stability of inverse problems have been obtained by mathematicians, other scientific communities seldom profit from these advances. Instead, practical approaches to inverse problems in geophysics are often based on modifications and generalizations of the least-squares method and possibly on stochastic approaches, e.g., when set in a Bayesian framework. A significant step forward for scientific research in geophysics requires that the theory and application of inverse problems “meet each other” in an open forum where both early-stage researchers and senior scientists are brought together with the aim of sharing their diverse scientific background and expertise in the fields of inverse problems, imaging and Earth exploration.


Kazakh student bags bronze at int’l Olympiad
KOSTANAY. KAZINFORM Two pupils of Kostanay region Arseniy Samoilenko and Zhan Dautov took part in the Tuymaada international school Olympiad in mathematics, physics, and computer science. Arseniy Samoilenko won the bronze medal in physics, Kazinform reports.
The Tuymaada international school Olympiad in mathematics, physics, computer science was held online between July 25-August 3. It brought together more than 250 schoolchildren from 7 countries of the world, namely, Kazakhstan, Russia, Indonesia, Romania, Singapore, Bulgaria and Iran. 74 pupils represented Kazakhstan at large.
Notably, Arseniy Samoilenko won a grand prix of the Zhas Orken 2021 republican prize in the Young Groundbreaker nomination.


Deep Learning Course
Researcher at Vernacular.ai
SpeechLab, NTU Singapore
I am currently working as a Researcher at Vernacular.ai, Bangalore.
I did my dual degree in B.E. EEE & M.Sc. Mathematics at BITS Pilani Goa Campus, India.
Earlier worked with Prof.Chng Eng Siong at Speech and Language Laboratory, NTU, Singapore.
Last Updated: 11-Jul-2020
Deep Reinforcement Learning
Natural Language Processing
Unsupervised Representation Learning
MSc in Mathematics, 2021
BE in EEE, 2021
Off Policy Monte Carlo Prediction with Importance sampling
Unsupervised Learning 101
Convolutional Feature Extraction and Neural Arithmetic Logic Units for Stock Prediction


I am co-organizing a special session on Computational Microelectronics at the 10th Large-Scale Scientific Computations Conference (LSSC) 2015, being held from June 8 – 12, 2015 in Sozopol, Bulgaria. The conference is organised in cooperation with SIAM. The general area of the conference is on (but not limited to):
- Hierarchical, adaptive, domain decomposition and local refinement methods;
- Robust preconditioning algorithms;
- Monte Carlo methods and algorithms;
- Numerical linear algebra;
- Control systems;
- Parallel algorithms and performance analysis.
- Large-scale computations of environmental, biomedical and engineering problems;
Within the special session Computational Microelectronics, we focus on topics ranging from stochastic to deterministic approaches in the area of TCAD. Mandatory abstract submission deadline (1 page) is due January 15, 2015. Full paper submission (8 pages, Springer LNCS format) is due March 15, 2015. The proceedings will be published in a dedicated Springer LNCS volume.


The game simulates real coin flip and guess the result, which is completely built by smart contracts based on blockchain, so no one can cheat on the result, it's pure fairness. If you are familiar with smart contract technology, you could also play by directly accessing our contract address.
The player must select at least one and no more than five coins to play, choose head or tail on each coin, and the results of each coin are revealed in sequence; the player must correctly guess which side of each coin is up to win. Various coin counts correspond to different odds, with the highest having up to 32 odds.
For example, a player selects two coins to play and chooses head for the first coin and tail for the second. So this is the list of all possible flipping results.
Case 3 appears to have the same result as the player chose, one for head and one for tail, but the order is incorrect, so the game is still lost.
When a player wins, the system deducts 3.5% of the reward as a maintenance fee.
Step.1 Click the transaction in My Flips
Step.2 Select the logs tab and scroll down
Step.3 Get the random number from Chain Link VRF
Step.4 Compare the choice with random number
Coin Flip stores the choices in binary, 0 for Head and 1 for Tail. For example, if you choose two coins to flip, the first is Head and the second is Tail, your choice will be converted to a number, which is 2 in DEC and 00010 in BINARY, so the formula for comparison with a random number is:
So it is:


I am a Machine Learning Researcher at Morgan Stanley. My aim is to advance probabilistic methods for more reliable AI across supervised, generative, and time-series domains. I got my Ph.D. in applied math at Purdue University in Dec 2021, where I was advised by Prof. Guang Lin and Faming Liang.
Current interests: Monte Carlo, diffusion models, neural SDE and filtering (with applications in finance), Thompson sampling.
Dec, 2023. Glad to participate Transport, Diffusion, and Sampling Workshop
Nov, 2023. Talk at Financial Mathematics Seminar at FSU
Apr, 2023. NewBeginnings: One ICML on diffusion models and Schrodinger bridge is accepted!
Feb, 2022. Talk at Machine Learning Seminar at HKU
Oct, 2021. I have defended my thesis!
Oct, 2020. Talk at Machine Learning + X Seminar at Brown University
Reviewers: ICML, NeurIPS, ICLR, AISTAT, AAAI, IJCAI, Machine Learning Journal.
La pensée n’est qu’un écliar au milieu d’une longue nuit. Mais c’est cet éclair qui est tout.


International Journal of Education. 2020; 2: (1) ; 10.12208/j.ije.20200008.
Numbers play a vital role in people's lives. Counting can be used not only to understand things around you, but also to express and communicate. It is also an effective tool for solving problems in life, and it is also the foundation for students' math learning. The "Mathematics Curriculum Standards" clearly puts "number sense" as the content of mathematics learning. Among the ten core concepts of the new curriculum, "number sense" is placed in the first place. It can be seen that the concept of "number sense" is understood and allowed The new curriculum emphasizes and values students' sense of number in the process of mathematics learning.


All are harder questions from past jmcs within the 2004-2012 papers (q16-25 on each intermediate maths olympiad question compilation on the topic of. Junior mathematical olympiad past papers 2007-2010 ukmt maths challenge — millburn academy ukmt maths challenge past intermediate standard papers. Past papers/sample questions (mathematical olympiad) [email protected] (science olympiads) [email protected] (mathematical olympiads) location. Collection of three books, each catering to three separate levels elementary, intermediate and advanced developed for high ability pupils to acquire mathematical.
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Intermediate mathematical olympiad maclaurin past papers 2008-2010 250 junior mathematical olympiad past papers 2011-2014. Resource books, practice questions and solutions aimed at teaching maths problem solving to primary years 5 and 6 and secondary years 7 and 8. Sa maths olympiad 2018 dates: closing how to gain access to pass papers international mathematical olympiad. Secondary papers (junior, intermediate and questions and full solutions from past amc papers and annual record of the australian mathematical olympiad. Intermediate 2 maths past papers 2003 2 per a4 in the intermediate mathematical olympiad papers: cayley, hamilton and print past papers free of charge. It was decided that to widen usage, there would be no (direct) indication as to which paper (ie junior, intermediate or senior) a question is from.


Community College Students and Achievement in Mathematics: Predictors of Success
A large majority of college degree programs and majors require students to take courses in the mathematical sciences consequently enrollments in mathematics courses are among the highest of any subject. Equally high are the withdrawal and failure rates in math courses compared to most other courses. Failure in mathematics can restrict students’ choices of majors and later career paths. Presently, relatively little is known about factors related to low achievement in mathematics. Research has shown that gaps in achievement levels between males and females and between white, African-American, and Hispanic students are narrowing yet still persist. The aim of this dissertation is to determine the factors that predict students’ achievement in mathematics. Specifically, this study investigated the relationship between cognitive and non-cognitive variables and achievement in urban community college mathematics.
Data from the Transfer and Retention of Urban Community College Students (TRUCCS) project was subjected to secondary analyses. The TRUCCS project, funded by the Field Initiated Studies (OERI R305T000154) and the Lumina Foundation, was a five-year longitudinal study of approximately 5,000 community college students in urban Los Angeles. Data were collected from students across nine campuses comprising the Los Angeles Community College District (LACCD). The TRUCCS project sought to advance new definitions of achievement more consistent and applicable to urban community college students and this study adds to that new knowledge.
The results of this study showed that cognitive items that included English placement scores, the highest math course taken, and college GPA significantly predicted achievement in mathematics for urban community college students whereas having taken remedial math courses and being employed were negative predictors. Always completing homework assignments was the only non-cognitive item to significantly predict math achievement (overall math and transfer level math). These results add to the small body of research that has shown cognitive and non-cognitive variables to be positively associated with performance in various four-year college math and science courses. This is especially noteworthy as a literature search failed to produce any studies that simultaneously examined cognitive and non-cognitive variables as predictors of achievement in mathematics for students enrolled in community colleges. While non-cognitive variables such as academic self-concept and achievement-related expectancies were shown to be predictors of achievement in various four-year college courses, this study showed otherwise for urban community college students.


At Monkey Puzzle Sidcup we take full advantage of the specialisms and the expertise of our staff, Davinder one of the Directors is a qualified science teacher, she plans and delivers science lessons and STEM sessions that allow the children to build on skills gradually. Not only do these sessions cover the STEM areas but can also be cross linked to the 7 areas of the EYFS curriculum.
Science, Technology, Engineering and Mathematics
STEM education stands for Science, Technology, Engineering and Mathematics. STEM education has been at the forefront of all learning at all different ages and stages. The world is dependant on these keys areas of learning; our general wellbeing and development is based on our knowledge of these areas.
The pedagogical principles underpinning engineering education emphasise innovation, invention, investigation, creativity, teamwork, enterprise, perseverance and responsibility and extend the skills, attributes and capabilities of learners through the application of:
- high levels of scientific and mathematical understanding
- solution-oriented approaches
- a range of problem-solving techniques
- systems approaches
- modelling and simulations.
Making the links between curriculum areas explicit to learners illustrates the interdisciplinary nature of STEM careers.
These photos can show how the activities capture the children’s attention, we are inspiring and challenging our children. Today’s curious explores make tomorrow’s STEMHEROES!


wax | wendy
I like shiny things. And when I find them, I put them here.
Monday, March 14, 2011
For Pi Day, a musical interpretation to 31 decimal places by
Michael John Blake
I wonder what would happen if you keyed this into an FTL drive?
wax | wendy: Happy Pi(e) Day! 2010
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In its turn, Φimp can be written as Φimp = C impΦ where Φ is the fluid flow and C imp the incoming number concentration of impurities. Gathering together the previous results in this letter, we get (5) with the z e (n) and ρ e (z e ) dependences given by Equations 1 and 3. Equations for Φ(t)and ∂C imp (x,t)/∂x In order to solve the filtration dynamics (i.e., to obtain n(x t) and C imp(x t)), it is necessary to supplement Equation 5 with formulas for Φ(t) and C imp(x t). Regarding the fluid flow, we apply the Poiseuille
law for incompressible fluids of viscosity η in a cylindrical channel of length L and radius r e (x t):  (6) In this equation, P is the pressure difference between both ends of the finite-length channel, which we selleck chemicals take constant with time. Note that Φ becomes zero when at some x, the n value becomes n clog ≡ r 0/r 1, i.e., r e becomes zero at that location and the channel becomes fully closed by impurities. Note also that Equation 6 reduces in the particular case r 0 ≫ r 1 n(x,t) (which is common in experiments) to . We construct now the supplementary equation for C imp(x,t). For that, we again consider the differential channel slice going from x up to x + d x. The number of GDC 0032 impurities that become trapped in its walls
during an interval d t is (2Π r 0 d x)(∂n/∂t)d t (the factor 2Π r 0 d x is again due to the areal normalization in the definition of n). The numbers of impurities entering and exiting the slice in the liquid flow are Φ(t)C imp(x,t)d t and Φ(t)C imp(x + d x,t)d t respectively. Mass conservation balance therefore gives (7) Notice that Equations 5 to 7 are coupled to each other. In fact, they form now a closed set that can be numerically integrated by providing the specific values for the characteristics of Bumetanide the filter, for any given pressure difference P and incoming impurity
concentration C imp(0,t). In what follows, for simplicity, we will always consider for the latter a constant value C 0. The computation to numerically integrate Equations 5 to 7 is relatively lightweight (e.g., calculating our Figure 2 took about 15 min in a current personal computer that considered 2 × 104 finite-element x-slices). Figure 2 Time dependence. (a) Results, obtained by integrating Equations 5 to 7, for the time dependence of the areal density of trapped impurities (continuous lines) at the entrance of the channel n(x = 0,t) and at its exit point n(x = L,t), and also the global average areal density of trapped impurities . The areal density axis is normalized by the saturation value n sat. The time axis is normalized by the half-saturation time, defined by . The parameter values used are as follows (see main text for details): ρ 0 = 13 nm, ρ 1 = 0.11, λ D = 5.1 nm, , r 0 = 500 nm, , Ω0 = 0, Ω1 z 0 = 1.2 × 105/m, L = 7.


Quadratic equation solver we can help you solve an equation of the form ax 2 bx c 0 just enter the values of a b and c below is it quadratic only if it can be put in the form ax 2 bx c 0 and a is not zero the name comes from quad meaning square as the variable is squared in other words x 2 these are all quadratic equations in disguise. Shows you the step by step solutions using the quadratic formula this calculator will solve your problems.
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Fractions can be divided into several groups. The most commonly used fractions are those with values between zero and one. These fractions are called proper fractions. Fractions equal to one are called unit fractions (for example, 4/4). Fractions with values greater than one are called improper fractions. Improper fractions can also be written as mixed numbers, as a whole number and a proper fraction.
A common way to determine if a fraction is proper or improper is to consider whether the numerator or denominator is the larger number. A proper fraction will be "larger" on the bottom -- an improper fraction will be "larger" on the top.


Card counting at the casino? How does it work?
Casino card counting like Winboss Casino Online, especially in the game of blackjack, is a technique by which players aim to determine the proportion of high-value cards (such as tens and aces) to low-value cards (such as 2-6) that remain in the deck. The idea behind this technique is that when the deck has a higher proportion of high-value cards, the player has a higher probability of getting a blackjack (21 points from the first two cards) and the dealer has a higher probability of "drown" (ie exceed 21 points).
Card counting in blackjack, a method that has become famous for gaining an edge over the casino, has a fascinating history, interweaving mathematics with tales of wit and ingenuity. This technique evolved with the development of probability theory and gambling knowledge, becoming more sophisticated and well-known over time.
Card counting began to gain attention in the 1950s, but its roots stretch back to the beginnings of probability theory in the 21th century. Blackjack, originally known as "vingt-et-un" (XNUMX in French), was a popular game in France and Spain and came to North America with European settlers. However, it was only in the XNUMXth century, with the development of modern mathematics, that systematic methods of "beating" the game began to appear.
The turning point in the history of book counting was the publication of the work "Beat the Dealer” by Edward O. Thorp in 1962. Thorp, a mathematician and professor at MIT, is considered the father of modern book counting. He used early computers to model and analyze the game of blackjack, developing the first card counting system based on sound mathematical principles, known as the "Ten Count System". His book provided the first scientific proof that the game of blackjack could be beaten by a strategy based on statistics, which caused a wave of interest and excitement among gamblers.
Casinos have started to be aware of this technique
After the publication of Thorp's work, book counting became a phenomenon. Casinos have become aware of this technique and have implemented various measures to counter it, such as using multiple decks or shuffling cards frequently. At the same time, players began to develop and refine various card counting systems, making them more efficient and harder to detect. The appearance of books such as "Playing Blackjack as a Business" by Lawrence Revere and "The World's Greatest Blackjack Book" by Lance Humble and Carl Cooper helped make these techniques popular and accessible.
The 80s and 90s brought a new level of sophistication to card counting with the formation of blackjack teams. The most notorious example is the blackjack team at MIT, which was the subject of the book "Bringing Down the House" by Ben Mezrich and the movie "21". These teams used a combination of advanced techniques, including specialized roles for team members, complex signaling systems, and large capital to maximize efficiency and reduce individual risk.
The team members were divided into "counters", who stayed at the low-stakes tables and counted the cards, and "big players", who entered the game with high stakes when the conditions were favorable. This team approach brought considerable success and led to significant winnings, but it also attracted more attention from casinos, who began using advanced surveillance technologies and other measures to identify and block teams of card counters .
The technique of counting cards in blackjack is a fascinating and complex subject, with several approaches and systems developed over time. These systems vary in complexity and effectiveness and are tailored to meet different playing conditions and casino strategies.
The Hi-Lo system is probably the most widely known and used card counting system. Developed by Harvey Dubner in the 60s, it is a "level one" counting system, meaning that each card is assigned a value of +1, -1, or 0. In the Hi-Lo system, cards 2-6 are + 1, cards 7-9 are 0, and cards 10, J, Q, K, A are -1. The player keeps count as the cards are played and uses this number to estimate when the deck is favorable (high and positive number) and when it is not (low or negative number). The advantage of the Hi-Lo system is its simplicity and efficiency, making it a good system for both beginners and experienced players.
The KO system, developed by Ken Fuchs and Olaf Vancura, is a variation of the Hi-Lo system that does not require the current number to be converted to a "true number". This makes it easier to use, especially for novice players. In the KO system, cards 2-7 are valued at +1, cards 8 and 9 are 0, and high value cards are -1. The key difference is that the KO system starts with an "advance" number based on the number of packs used, and the current number keeps going above zero, eliminating the need to calculate a true number.
The Omega II system, created by Bryce Carlson, is a "level two" numbering system, meaning that the values assigned to cards are more varied. In this system, cards 2, 3, and 7 are valued at +1, cards 4, 5, and 6 are +2, cards 8 and A are 0, cards 9 are -1, and cards 10 and faces are -2. This system is more complex and requires more concentration and precision, but it provides a more accurate estimate of the remaining composition of the deck.
The Zen Count system, developed by Arnold Snyder, is another level two system that offers a good balance between the efficiency of the Hi-Lo system and the accuracy of more advanced systems. In Zen Count, cards 2, 3, and 7 are valued at +1, cards 4, 5, and 6 are +2, cards 8 and 9 are 0, 10 cards and faces are -2, and aces are -1. This system is popular because it improves the accuracy in estimating the player's advantage, especially in insurance decisions in the game.
The Red 7 system, created by Arnold Snyder, is an interesting variant of the Hi-Lo system that introduces a differentiation between red and other colored cards. In this system, cards 2-6 red and 7 are valued at +1, cards 2-6 black are 0, cards 8 and 9 are 0, and high value cards and aces are -1. The peculiarity of this system is that it takes into account the suit of the 7 cards, which makes it a little more complex, but also provides an additional advantage in certain game situations. The Red 7 system is appreciated for its relative simplicity and the added advantage it provides through this color differentiation.
The Canfield Master System, developed by Richard Canfield, is a more advanced, level three counting system. It gives different values to several cards, thus increasing accuracy as well as complexity. In this system, cards 3-7 are valued at +1, cards 2 and 8 are 0, 9 cards are -1, and high value cards and aces are -2. As a level three system, it requires more concentration and practice to use effectively, but can provide a greater advantage to experienced players.
The decision to try casino card counting involves a number of considerations and evaluations, both from an efficiency and risk perspective. Card counting can theoretically give the player an advantage in the game of blackjack, but this practice also comes with significant challenges.
In theory, card counting can give the player an advantage over the house. By tracking the ratio of high value cards to low value cards remaining in the deck, a player can adjust bets and play decisions to maximize the chances of winning.
Unlike other casino games that are strictly based on luck, card counting blackjack requires mathematical skills, good memory and quick decision making. For some players, this adds an interesting and challenging dimension to the game.
If practiced correctly and consistently, card counting can, in theory, generate long-term profits. That requires strict adherence to a system, discipline and enough capital to absorb the natural fluctuations of the game.
Counting cards requires intense concentration and advanced mental skills. You must be able to keep an accurate count while playing fast, interact with the dealer and other players, and adapt to game conditions. This can be very demanding and is not for everyone.
Blackjack, even with card counting, remains a game of luck. Even with a statistical advantage, there are losing periods. Players must have a large enough bankroll to withstand these fluctuations and be prepared to lose money.
Casinos are very aware of card counting and use various methods to combat it. These include the use of multiple decks, frequent shuffling of cards, betting limits, watching players through surveillance cameras, and even refusing to allow certain players to play. If you are caught counting cards, you can be asked to leave the casino and even banned.
Card counting in casinos may provide a theoretical advantage in the game of blackjack, but it comes with significant challenges and risks. It requires mathematical skills, an excellent memory, strict discipline and sufficient capital. Players should also be aware of the measures taken by casinos to combat this practice and be prepared for possible legal or professional consequences.


Traditional learning has become more fractured during the pandemic, with reduced in-person learning. It is difficult to stay engaged in an online classroom, something I know from experience. One-on-one tutoring can help keep students motivated and learning on track. Tutoring may clarify concepts from class, or prepare students for science competitions or standardized tests.
What I teach
I tutor physics and math through the introductory college level, and chemistry, biology, astronomy, and economics through the advanced high school (AP+) level. I have experience teaching students from 4th grade through senior year of high school.Contact me
I greatly enjoyed taking part in competitions like science bowl and the physics, math, and chemistry olympiads. I can offer strategies to help you succeed and recommend resources to use, as well as guiding your problem solving.
If you’re like me, you have an interest in science that just isn’t satisfied by what’s taught in school. Having a tutor gives you personalized attention, so you can explore topics that appeal to you, and follow your curiosity wherever it leads. I enjoy explaining scientific concepts and getting students excited about learning.
- Developed, directed and taught at a week-long summer science program (LSSP) for advanced middle school students.
- Gave physics lectures after school at LHS
In the classroom, sometimes concepts are presented in a way that is unclear, or explained only superficially. Working with a tutor allows you to clarify material until you feel confident in your understanding.
- Earned A's and A+'s in AP physics, chemistry, biology, calculus, computer science, and economics classes at LHS
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I scored an 800 on the math section of the SAT, and a 780 on the verbal section. I also scored 800's on the physics, chemistry, biology, and math 2 subject tests.
I scored 5's on all the AP exams I took (calculus BC, chemistry, biology, computer science, and micro/macroeconomics).


MathAlly Graphing Calculatorpor MathAlly
Here are some of our current features:
-Enter values and view results as you would write them
-Swipe up, down, left, or right to quickly switch between keyboard pages.
-Long click on keyboard key to bring up dialog about key.
-Undo and Redo keys to easily fix mistakes.
-Cut, Copy, and Paste.
-User defined functions with f, g, h
-Simplify and Factor algebra expressions.
-Polynomial long division.
-Solve algebra equations for one variable.
-Solve equations with inequalities such as > and <
-Simplify trigonometric expressions using trigonometric identities.
-Graph three equations at once.
-Normal functions such as y=x^2
-Inverse functions such as x=y^2
-Circles such as y^2+x^2=1
-Ellipses, Hyperbola, Conic Sections.
-Add markers to graph to view value at given point.
-View delta and distance readings between markers on graph.
-View roots and intercepts of traces on graph.
-nCr and nPr functions
-Change numeric base between binary, octal, decimal, and hexadecimal
-Bitwise operators AND, OR, XOR, and NOT
-Vector dot product and norm.
Q. Is there are tutorial anywhere explaining how to use the calculator?
A. There are three into tutorials in the app for the calculator, graph equations, and graph screens. Additional tutorials can be found on our website http://www.mathally.com/
Q. How do I get to the keys for pi, e, solve, etc?
A. There are four keyboard pages. Each swipe direction across the keyboard moves you to a different page. The default page is the swipe down page. To get to the page with trig functions, swipe left. To get to the matrix keys, swipe up. To get to the last page, swipe right. No matter what page you are on, the swipe direction to move to a specific page is always the same.
Q. How do I enter = for a graph equation?
A. Swipe your finger from the bottom of the keyboard to the top of the keyboard to change to the page with the = key on it.
Q. How do I copy and paste?
A. Long click on expression to highlight it. Then use blue tabs and arrow keys to change area that is selected. Finally, use the copy, cut, and paste keys on the keyboard (swipe up page) to copy, cut, and paste. In addition, you can tap a previous input or result in the calculator history and it will be pasted in entry.
Q. What do you have planned for future releases?
A. You can keep up to date on the latest news on our blog at http://mathally.blogspot.com/ . This news will include what is coming up in future releases. Also feel free to leave comments and let me know what you think!
This is an active project. I am continuing to work to add more features to make this the best free graphing, symbolic, and scientific calculator for android. If you find a bug or have questions, please email me. For the latest news, visit our blog.
Nombre del paquete
Descargar apk/Código QR
Valoración media: 4.6
10.000 - 50.000


The following article is a structural comparison between two DS puzzles games: Sudoku, I recommend either Brain Age versions, and Picross DS.
Both of these games are designed around a grid where players must mark the squares in specific ways. In Sudoku, the squares are filled with one number each. And in Picross each square is either filled in or left blank.
Each challenge is a puzzle that players must solve by understanding the intricacies of the coding system. The codes in these games are made up of a few rules.
Sudoku (image left): Each row, column, and box must only contain the numbers 1-9 without any duplicates. Play it here.
Picross (image right): Each column/row must contain filled in squares in the groupings specified by the string of numbers posted to the top/left of the grid. Each grouping must be separated from each other mark by at least one empty square. Play it here.
With just a bit of data provided by the coding system for type of puzzle, the player is able to, through the process of deduction, figure out each square on the grid. By filling in part of the grid, the range of possibilities decreases for the remaining squares on the grid. This process continues until the grid is complete.
At this point, I wanted to state that both of these puzzle systems are very solidly designed. In my puzzle game experience, I've found that the simpler the puzzle system, the better the puzzle game. For it's more impressive if one can design many puzzles with a wide range of difficulty using a puzzle system that's very simple. So at this point, I want to consider the complexity and flexibility of the Sudoku and Picross systems.
Each Sudoku square potentially contains 1 of 9 numbers. With more difficult puzzles, a good strategy is to write down the possible numbers of a square in tiny print so you can help yourself narrow down the answer. Every square in Picross is much simpler. To solve Picross puzzle, each square must be marked or left unmarked. Similar to Sudoku, a good strategy in Picross is to mark all the known unmarked squares with an "x." In this way, a blank Picross square is potentially 1 of 2 states.
Though Sudoku has a wider range (more complex) for its input system, the design trade off is a simple presentation of the code data. In Sudoku, each puzzle challenge presents a few numbers up front. These numbers are the code that players use to solve the puzzle (see red numbers in image above-left). Just a few numbers scattered here and there does the trick. The opposite is true for Picross. The more complicated the image and the more difficult the puzzle, the more complex the code is along the top and left side of the puzzle (see image above-right). For these two puzzles games, the relationship between their code complexity and their input complexity is an inverse relationship that reflects the nature of puzzle game systems that are designed around deduction. In other words, to create a decent challenge, there must be enough possible answers, solutions, moves, or inputs at the outset to conceal or distract the only solution so that players have to slowly work at the puzzle using what little they can uncover to start a puzzle solving chain reaction.
Sudoku is a game that works around the number 9. The grid is 9x9. Each of the 9 rows, columns, and 3x3 grids must only contain the numbers 1-9 without repeating a number. Each puzzle challenges players to fill in the right numbers. But after the puzzle is solved, players are left with a grid of numbers. This ultimately describes all Sudoku games in the Brain Age DS games. Sure there are many variations on Sudoku out there. But these variants require additional rules. The system that I have in my Brain is is what I'm stuck with. In this way, the Sudoku puzzle design is not very flexible.
On the other hand, the few rules of Picross are not bound to a particular grid size. Picross puzzles can be 5x5, 10x10, and 15x15. In addition to the flexible size of Picross puzzles, there is one key difference that wins the puzzle game war between the two games. That difference is forms.
It doesn't matter how pixelated the pictures are or how simple they appear. Every mark you make solving a Picross puzzle, you're actually creating a picture. In a very paint by numbers fashion, with a little methodical persistence, anyone can become an "artist." Going from a blank grid and slowly etching out an image is like pressing your face up against one of those magic 3D images. All of a sudden, you see a shape that wasn't there before. It's like art. It's like magic. And it's these forms that help give every Picross puzzle interesting layers that appeal to multiple senses/types of accomplishment and satisfaction.
If you ever find yourself in a position where you want to make a guess in either game, guessing in Sudoku is like putting down a random answer on a match quiz. Guessing in Picross is a little less random and abstract. If you can visualize the image behind the puzzle that you're solving in Picross, it's possible to make educated guesses in based on the image. Can you fill in the missing squares in the cat above. I bet you can get close.
On top of all of this, the forms also give every player the power to create their own puzzles. Using the DS touch screen, simply draw a picture over a blank grid and the computer system will make sure your artwork is playable. There would be no point to creating your own Sudoku puzzles. They're all just numbers anyway.
In conclusion, Picross is a puzzle game that turns gamers into code crackers, artists, and puzzle makers. The code system is no more complicated than a bit of addition and subtraction, yet the forms are what separate Picross from Sudoku. It's amazing what a little form can do to a video game. I don't wonder what the next level is for these two puzzles systems. What's done in 2D can always go...


GPU Programming Lab 4: Tessellation with Squares
Due Monday, September 26
Last Wednesday, the in-class demo on tessellation shaders used triangles for subdivision. This is fine, but for a cube, perhaps it would be more natural to define the cube using quads, and subdivide into triangles using tessellation.
For this week's lab, modify the tessellation demo so that it uses quads instead of triangles. You'll have to change the cube element list into quads, change the patch side to 4, and make a number of changes on both the tcs and tes. Refer to khronos and figure out what changes you'll need, but it summarizes as dividing squares instead of triangles.
I expect the biggest hangup will be mixing the 4 vertices from the square to find the 3D coordinates of each vertex. Remember that when using quads, only the first two tessellation coordinates are significant, and they indicate coordinates within the square.