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Table of Contents
A logic-based combinatorial number-placement problem is called sudoku. The goal of traditional Sudoku is to fill a 9 x 9 grid with numbers so that every column, row, and each of the nine 3 x 3 subgrids that make up the grid (also known as “boxes,” “blocks,” or “regions”) contain all of the digits from 1 to 9. The puzzle creator offers a partially finished grid with a single solution for a well-posed puzzle.
Who invented Sudoku – the ultimate Puzzle Game
The Sudoku puzzle game appeared for the first time in French newspapers in the 19th century. Further, it has been published in puzzle books since 1979 under the name Number Place. The contemporary Sudoku, which means “single number,” was first published in 1986 by the Japanese puzzle publisher Nikoli. Since then, it has grown significantly popular. Thanks to Wayne Gould, who created a computer program to quickly create original puzzles, it first appeared in a U.S. newspaper and later in The Times (London) in 2004.
The modern Sudoku game was probably created under the pseudonym Howard Garns, a 74-year-old retired architect and independent puzzle maker from Connersville, Indiana. It was initially released by Dell Magazines in 1979 as Number Place (the earliest known example of modern Sudoku). In editions of Dell Pencil Puzzles and Word Games that featured Number Place, Garns’ name was always listed among the contributors; however, it was never present in those that did not. He passed away in 1989 before he could witness the success of his invention on a global scale. It’s unknown if Garns was familiar with any of the French publications mentioned above.
Rules of the ultimate Puzzle Game
- You can use only single digit numbers from 1 to 9.
- In a single row or column, no number should be repeated.
- Each number should only be used once in each row, column, and grid.
- Employ pencilling and cross-hatching methods.
- Use the strategy of elimination to guess numbers.
How to play Sudoku
A 9 by 9 grid is used to play Sudoku. There are nine “squares” between the rows and columns (made up of 3 x 3 spaces). You must fill out all nine slots in each row, column, and square with the numbers 1 through 9, without repeating any numbers.
Different types of Sudoku
A 6 by 6 form with 3 by 2 sections is referred to as “Mini Sudoku” and may be seen elsewhere, including in the US daily USA Today.
The puzzle only employs the digits 1 through 6, but the goal is the same as in conventional Sudoku.
A variant of this puzzle known as “The Junior Sudoku” has been published in several newspapers, including some issues of The Daily Mail, for younger puzzle solvers.
Hyper Sudoku or Windoku
The traditional 9 x 9 grid with 3 x 3 sections is used in Hyper Sudoku or Windoku, but there are four additional inside 3 x 3 regions where the numbers 1 through 9 must occur precisely once. Peter Ritmeester created it, and in October 2005 he published it for the first time in the Dutch newspaper NRC Handelsblad. Since April 2007, he has been publishing it daily in The International New York Times (International Herald Tribune). In Will Shortz’s Favorite Sudoku Variations, the term “Hyper Sudoku” first appeared (February 2006). Since the grid looks like a window with glazing bars when the four inside sections are darkened, it is also known as Windoku.
In Twin Sudoku, a 33-box is shared by two conventional grids. There are many different kinds of overlapping grids that might exist. The digits in the overlapping portion are shared by each half, but the rules for each individual grid are the same as in regular Sudoku. In some compositions, neither grid can be solved independently; the whole solution can only be reached once each grid has at least partially been solved.
Working of the Puzzle Game
There are no repeated values in any of the nine blocks (or boxes of 33 cells) in a finished Sudoku grid, which makes it a particular kind of Latin square. It was established that a first-order formula that omits blocks is valid for Sudoku if and only if it is valid for Latin squares, establishing the link between the two theories.
It is well known that solving Sudoku puzzles on n2n2 grids with n blocks is an NP-complete issue. Most 9-by-9 problems can be solved quickly by a variety of computer techniques, such as backtracking and dancing links, but when n rises, combinatorial explosion restricts the qualities of Sudokus that can be built, examined, and solved. A graph coloring issue may be thought of as a Sudoku puzzle. With a partial 9-coloring, the goal is to create a 9-coloring of the provided graph.
Effect of the game on the brain
Playing Sudoku not only helps young people focus and remember things, but it can also help senior people from developing Alzheimer’s disease, one of the most prevalent kinds of dementia. The thrill of creating the game may both increase your happiness quotient and keep you from getting bored.
The game demands strategic thinking and innovative problem-solving. You must restart your game from scratch if you stop playing in the middle. Although it may be annoying, it really helps you improve your ability to concentrate and refocus.
Playing Sudoku requires both logic and memory. Also, when you memorize the numbers and use reasoning to fill in the next blank, it improves your memory.
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Maki Kaji, the Japanese puzzle publisher Nikoli, is known as the father of the puzzle game of Sudoku.
It was originally called “Suji-wa-Dokushin-ni-Kagiru,” which translates to, “Numbers should be single, a bachelor.”
The puzzle game requires attention of the player to analyze the grids and fill in the numbers. Therefore, it helps to keep the brain healthy and is evident in delaying the onset of dementia.
It uses a Latin square of order n is an n × n array in which each row and column contains every n symbols exactly once. The numeral n is the order of the Latin square. Sudoku is a Latin square of order 9. However, it additionally includes the condition that every block must also contain the numbers 1 through 9.
According to the Guinness Book of World Records, the fastest time to solve a “Very Easy” Sudoku puzzle was 1 minute 23.93 seconds on May 20, 2006. The record was set by Thomas Snyder, an American Sudoku champion.


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“Why do we have to learn this?”
“Why do we have to show our work?”
“When are we ever going to use this in real life?”
Math teachers endure a barrage of these questions on a daily basis. And sometimes, students can seem justified in their protests. (When is the last time you solved for the derivative? Or figured out the cosine of an angle?)
More often, however, students are not challenged to apply their skills in a real world environment. Skills that, if nothing else, are essential for success in the classroom and on standardized tests such as the SAT and the ACT.
From my extensive experience as a test prep tutor, I have noticed certain areas of math that students should have already mastered back in middle school, but still cannot completely grasp as juniors and seniors. Here are some of those areas, and some tips on how you can have your sons and daughters practice in “real life.”
Leaving the Tip: Parents who make their children figure out how much tip to leave on the bill at a restaurant usually teach easy shortcuts: Find 10 percent then double it for a 20 percent tip, or find 10 percent, then 5 percent for a 15 percent tip. And while these tricks usually work, children do not understand why they work. Ask them to leave an 18 percent tip and watch them falter helplessly. Challenge your kids to leave 17 percent and 22 percent tips. Anything that is an unordinary number to force them to use actual math skills.
Additionally, have them solve for the total bill amount. High school students struggle with the idea of increasing a value by a certain percentage. Present the restaurant bill in this light: If you leave a 21 percent tip, the final amount is a 21 percent increase from the total charged for food and taxes.
Shopping for discounts: While tips represent a percent increase, shopping for sales requires students to solve for a percent decrease. Take your kids shopping, with one rule: For any item on sale, if they cannot solve for the discounted price, they are not allowed to buy it. If the price tag shows the already discounted price, do not let them peek—just tell them the original price and the discount percentage. And even when sales are over, keep the game going. Make up discounted rates and if your kids cannot solve the problems you create for them, they cannot purchase that item.
Sports statistics: Let’s make the assumption that your son or daughter either plays sports or watches sports. If they play a sport, pick two or three statistical categories and have them figure out their average per game over the span of a season. The quantity and quality of the statistic does not matter. Whether they are solving for large numbers (points per game in basketball), small numbers (goals per game in soccer), good statistics (race times in track or swimming), or bad statistics (double faults in tennis), the process of problem-solving is the same. And if they religiously watch a certain team play, have them act as the team statistician. Maybe they can figure out the Phillies average runs scored in a game, or the Eagles average points given up per game.
Class Grades: Every student waits for their report card to learn their GPA, but why should they have to wait that long? Have them figure out their GPA throughout the semester. If a student knows their average grade in each class, have them calculate their GPA for that quarter or trimester. Then, as the year goes on, they can figure out their GPA in each class to date, and combine those calculations together to get their cumulative GPA.
And most significantly, later on in a quarter, challenge them to figure out what test score they will need to achieve or maintain the grade they want in a class. If they know their test average in world history is an 87, but they want at least an A- for the quarter, have them figure out the minimum score they need on the last exam to achieve their goal.
More to come at a later date, but this is a good place to start! The more students use these math skills beyond class and homework, the more fluent they will become in problem solving for percentages and averages.
Read More Education, Test and Prep Insights from Eric Karlan
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I love "destroy target player", but it feels tacked on on a card that doesn't destroy anything else. I'm not sure if there was a way of doing that better.
I notice, the cards overall seem a bit less out there than previous un sets. They're still un-rules-y, but there's less art exploding all over the text box, and less disregard for normal templating (like the augment creature which can augment at instant speed, they spell it out in rules text, not just change the reminder text). Even if the rules still need a similar amount of careful FAQ-ing :)
Ah! They introduced transcendental numbers (pi damage, although you can round to 2 s.f. if it's easier to count).
And Rosewater makes a ruling about infinity life -- it doesn't work like an ordinal, or a cardinal, which I think would be more interesting, it's just infinity life minus infinity damage is still infinity, infinity plus infinity is still infinity
Did he make a ruling about what happens if you have minus infinity life and then gain infinity life?
Not specifically, but I can't imagine minus infinity gaining infinity is different to infinity losing infinity, i.e. both stay the same.
The choice is nothing but arbitrary, it seems to me.
You mean, to use that version of arithmetic, or what the particular results are?
It seems to follow "when you gain infinity life, lose infinity life, do anything else infinite, the result is the same as doing it for 1, but repeated an infinite number of times" rules, which is (I think?) unambiguous, but sometimes unsatisfying.
If you have negative infinity life and get dealt positive infinity damage, the answer is you end up with zero life. Infinity plus its negative is zero. I don't think Magic would ignore such a fundamental rule in mathematics.
I'm not entirely convinced.
Rosewater says, if you have infinity life and take infinity damage, you're still on infinity. Is negative infinity different? I don't know if the way Rosewater chose right, but that's what he says in the FAQ.
There's not a single way of dealing with infinity which is appropriate in all circumstances. If you treat infinity as a number, and say "inf - inf = 0" then you break equations like "x + 1 - x = 1" when x is infinity. If you say "inf - inf = inf" then you break "x - x = 0". Other compromises break other assumptions.
If you allow infinity at all, you get some weirdness. E.g. floating point arithmetic on computers, you get NaN ("Not a number") when you try to calculate INF - INF, -INF + INF, INF/INF or 0/0. The reason being, those are operations you might sometimes expect to be reversible, and sometimes not.
That doesn't work in magic, because your life might be INF but can't really be NaN. So they picked one. I'd prefer an implementation that allowed you do take infinite life to zero SOMEHOW (I have some in mind but haven't played them) but I understand that wouldn't be a good choice for most players who aren't mathematicians.
So they had to pick some interpretation. I think allowing inf - inf to be 0 might be a more fun choice for playing magic. But I don't think it's more mathematically correct (you can justify inf - inf being any other finite number or inf, equally well).
If you assume cardinality applies, you can make infinite tokens somehow, devour them with Thromok, and deal infinity2 damage. That is undeniably larger than infinity
Hah! Right, Thromok is very useful for this, I hadn't realised that.
But I'm not sure that works for dealing damage to someone with infinite life. If you define a notion of subtraction like "subtract 1, that many times", then subtracting 1 from ω leaves ω, so however many infinite times you do that, ω, ω+1, ω2, all the way up until 2^ω, it still leaves ω.
Whereas if you define subtraction to consider both ordinals at once, then ω - ω (or ω - ω+1) would get to zero anyway.
Is there a case where ω2 works better?
I've looked for magic rules which would naturally generate an uncountable infinity, but I don't think I found any.
Mary O'Kill's switch and combined creatures are two things I'm going to be exploring in the future. Its something many people dabbled into (the latter more than the former) and it's often been more of a matter of choosing one option over the other and with a semi-canon reference mechanic a lot of this is just easier to handwave.
I'm quite happy about where Very Cryptic Command is going. And alternate illustrations are neat, too.
Add your comments:


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The tutorial explains how to do sum in Excel by using the AutoSum feature, and how to make your own SUM formula to total a column, row or selected range. You will also learn how to sum only visible cells, calculate running total, sum across sheets, and find out why your Excel Sum formula is not working.
If you want a quick sum of certain cells in Excel, you can simply select those cells, and look at the status bar at the bottom right corner of your Excel window:
For something more permanent, use the Excel SUM function. It is very simple and straightforward, so even if you are a beginner in Excel, you will hardly have any difficulty in understanding the following examples.
- How to sum in Excel using simple arithmetic calculations
- Excel SUM function
- AutoSum in Excel
- How to sum a column in Excel
- How to sum rows in Excel
- Excel Total Row to sum data in a table
- How to sum only filtered (visible) cells
- How to calculate running total (cumulative sum)
- How to sum across sheets
- Excel conditional sum
- Excel Sum not working - fixes and solutions
How to sum in Excel using a simple arithmetic calculation
If you need a quick total of several cells, you can use Microsoft Excel as a mini calculator. Just utilize the plus sign operator (+) like in a normal arithmetic operation of addition. For example:
However, if you need to sum a few dozen or a few hundred rows, referencing each cell in a formula does not sound like a good idea. In this case, you can use the Excel SUM function specially designed to add a specified set of numbers.
How to use SUM function in Excel
Excel SUM is a math and trig function that adds values. The syntax of the SUM function is as follows:
SUM(number1, [number2] ,…)
The first argument is required, other numbers are optional, and you can supply up to 255 numbers in a single formula.
In your Excel SUM formula, each argument can be a positive or negative numeric value, range, or cell reference. For example:
=SUM(A1, A2, A5)
The Excel SUM function is useful when you need to add up values from different ranges, or combine numeric values, cell references and ranges. For example:
=SUM(A2:A6, A9, 10)
The below screenshot shows these and a few more SUM formula examples:
In real-life worksheets, the Excel SUM function is often included in bigger formulas as part of more complex calculations.
For example, you can embed SUM in the value_if_true argument of the IF function to add numbers in columns B, C and D if all three cells in the same row contain values, and show a warning message if any of the cells is blank:
=IF(AND($B2<"", $C2<>"", $D2<>""), SUM($B2:$D2), "Value missing")
And here's another example of using an advanced SUM formula in Excel: VLOOKUP and SUM formula to total all matching values.
How to AutoSum in Excel
If you need to sum one range of numbers, whether a column, row or several adjacent columns or rows, you can let Microsoft Excel write an appropriate SUM formula for you.
Simply select a cell next to the numbers you want to add, click AutoSum on the Home tab, in the Editing group, press the Enter key, and you will have a Sum formula inserted automatically:
As you can see in the following screenshot, Excel's AutoSum feature not only enters a Sum formula, but also selects the most likely range of cells that you'd want to total. Nine times out of ten, Excel gets the range right. If not, you can manually correct the range by simply dragging the cursor through the cells to sum, and then hit the Enter key.
Tip. A faster way to do AutoSum in Excel is to use the Sum shortcut Alt + =. Just hold the Alt key, press the Equal Sign key, and then hit Enter to complete an automatically inserted Sum formula.
Apart from calculating total, you can use AutoSum to automatically enter AVERAGE, COUNT, MAX, or MIN functions. For more information, please check out the Excel AutoSum tutorial.
How to sum a column in Excel
To sum numbers in a specific column, you can use either the Excel SUM function or AutoSum feature.
For example, to sum values in column B, say in cells B2 to B8, enter the following Excel SUM formula:
Total an entire column with indefinite number of rows
If a column you want to sum has a variable number of rows (i.e. new cells can be added and existing ones can be deleted at any time), you can sum the entire column by supplying a column reference, without specifying a lower or upper bound. For example:
Important note! In no case you should put your 'Sum of a column' formula in the column you want to total because this would create a circular cell reference (i.e. an endless recursive summation), and your Sum formula would return 0.
Sum column except header or excluding a few first rows
Usually, supplying a column reference to the Excel Sum formula totals the entire column ignoring the header, as demonstrated in the above screenshot. But in some cases, the header of the column you want to total can actually have a number in it. Or, you may want to exclude the first few rows with numbers that are not relevant to the data you want to sum.
Regrettably, Microsoft Excel does not accept a mixed SUM formula with an explicit lower bound but without an upper bound like =SUM(B2:B), which works fine in Google Sheets. To exclude the first few rows from summation, you can use one of the following workarounds.
- Sum the entire column and then subtract the cells you don't want to include in the total (cells B1 to B3 in this example):
- Remembering the worksheet size limits, you can specify the upper bound of your Excel SUM formula based on the maximum number of rows in your Excel version.
For example, to sum column B without the header (i.e. excluding cell B1), you can use the following formulas:
- In Excel 2007, Excel 2010, Excel 2013, and Excel 2016:
- In Excel 2003 and lower:
How to sum rows in Excel
Similarly to totaling a column, you can sum a row in Excel by using the SUM function, or have AutoSum to insert the formula for you.
For example, to add values in cells B2 to D2, use the following formula:
How to sum multiple rows in Excel
To add values in each row individually, just drag down your Sum formula. The key point is to use relative (without $) or mixed cell references (where the $ sign fixes only the columns). For example:
To total the values in a range containing several rows, simply specify the desired range in the Sum formula. For example:
=SUM(B2:D6) - sums values in rows 2 to 6.
=SUM(B2:D3, B5:D6) - sums values in rows 2, 3, 5 and 6.
How to sum a whole row
To sum the entire row with an indefinite number of columns, supply a whole-row reference to your Excel Sum formula, e.g.:
Please remember that you shouldn't enter that 'Sum of a row' formula in any cell of the same row to avoid creating a circular reference because this would result in a wrong calculation, if any:
To sum rows excluding a certain column(s), total the entire row and then subtract irrelevant columns. For example, to sum row 2 except the first 2 columns, use the following formula:
Use Excel Total Row to sum data in a table
If your data is organized in an Excel table, you can benefit from the special Total Row feature that can quickly sum the data in your table and display totals in the last row.
A big advantage of using Excel tables is that they auto-expand to include new rows, so any new data you input in a table will be included in your formulas automatically. If can learn about other benefits of Excel tables in this article: 10 most useful features of Excel tables.
To convert an ordinary range of cells into a table, select it and press Ctrl + T shortcut (or click Table on the Insert tab).
How to add a total row in Excel tables
Once your data is arranged in a table, you can insert a total row in this way:
- Click anywhere in the table to display the Table Tools with the Design tab.
- On the Design tab, in the Table Style Options group, select the Total Row box:
Another way to add a total row in Excel is to right click any cell within the table, and then click Table > Totals Row.
How to total data in your table
When the total row appears at the end of the table, Excel does its best to determine how you would like to calculate data in the table.
In my sample table, the values in column D (rightmost column) are added automatically and the sum is displayed in the Total Row:
To total values in other columns, simply select a corresponding cell in the total row, click the drop-down list arrow, and select Sum:
If you want to perform some other calculation, select the corresponding function from the drop-down list such as Average, Count, Max, Min, etc.
If the total row automatically displays a total for a column that doesn't need one, open the dropdown list for that column and select None.
Note. When using the Excel Total Row feature to sum a column, Excel totals values only in visible rows by inserting the SUBTOTAL function with the first argument set to 109. You will find the detailed explanation of this function in the next section.
If you want to sum data both in visible and invisible rows, do not add the total row, and use a normal SUM function instead:
How to sum only filtered (visible) cells in Excel
Sometimes, for more effective date analysis, you may need to filter or hide some data in your worksheet. A usual Sum formula won't work in this case because the Excel SUM function adds all values in the specified range including the hidden (filtered out) rows.
If you want to sum only visible cells in a filtered list, the fastest way is to organize your data in an Excel table, and then turn on the Excel Total Row feature. As demonstrated in the previous example, selecting Sum in a table's total row inserts the SUBTOTAL function that ignores hidden cells.
Another way to sum filtered cells in Excel is to apply an AutoFilter to your data manually by clicking the Filter button on the Data tab. And then, write a Subtotal formula yourself.
The SUBTOTAL function has the following syntax:
SUBTOTAL(function_num, ref1, [ref2],…)
- Function_num- a number from 1 to 11 or from 101 to 111 that specifies which function to use for the subtotal.
You can find the full list of functions on support.office.com. For now, we are interested only in the SUM function, which is defined by numbers 9 and 109. Both numbers exclude filtered-out rows. The difference is that 9 includes cells hidden manually (i.e. right-click > Hide), while 109 excludes them.
So, if you are looking to sum only visible cells, regardless of how exactly irrelevant rows were hidden, then use 109 in the first argument of your Subtotal formula.
- Ref1, Ref2, … - cells or ranges that you want to subtotal. The first Ref argument is required, others (up to 254) are optional.
In this example, let's sum visible cells in range B2:B14 by using the following formula:
And now, let's filter only 'Banana' rows and make sure that our Subtotal formula sums only visible cells:
Tip. You can have Excel's AutoSum feature to insert the Subtotal formula for you automatically. Just organize your data in table (Ctrl + T) or filter the data the way you want by clicking the Filter button. After that, select the cell immediately below the column you want to total, and click the AutoSum button on the ribbon. A SUBTOTAL formula will be inserted, summing only the visible cells in the column.
How to do a running total (cumulative sum) in Excel
To calculate a running total in Excel, you write a usual SUM formula with a clever use of absolute and relative cells references.
For example, to display the cumulative sum of numbers in column B, enter the following formula in C2, and then copy it down to other cells:
The relative reference B2 will change automatically based on the relative position of the row in which the formula is copied:
You can find the detailed explanation of this basic Cumulative Sum formula and tips on how to improve it in this tutorial: How to calculate running total in Excel.
How to sum across sheets
If you have several worksheets with the same layout and the same data type, you can add the values in the same cell or in the same range of cells in different sheets with a single SUM formula.
A so-called 3-D reference is what does the trick:
The first formula adds values in cell B6, while the second formula sums the range B2:B5 in all worksheets located between the two boundary sheets that you specify (Jan and Apr in this example):
You can find more information about a 3-d reference and the detailed steps to create such formulas in this tutorial: How to create a 3-D reference to calculate multiple sheets.
Excel conditional sum
If your task requires adding only those cells that meet a certain condition or a few conditions, you can use the SUMIF or SUMIFS function, respectively.
For example, the following SUMIF formula adds only those amounts in column B that have "Completed" status in column C:
To calculate a conditional sum with multiple criteria, use the SUMIFS function. In the above example, to get the total of "Completed" orders with the amount over $200, use the following SUMIFS formula:
=SUMIFS(B:B,C:C,"completed",B:B, ">200" )
You can find the detailed explanation of the SUMIF and SUMIFS syntax and plenty more formula examples in these tutorials:
- SUMIF function in Excel: examples for numbers, dates, text, blanks and not blanks
- SUMIF in Excel - formula examples to conditionally sum cells
- How to use Excel SUMIFS and SUMIF with multiple criteria
Note. The Conditional Sum functions are available in Excel versions beginning with Excel 2003 (more precisely, SUMIF was introduced in Excel 2003, while SUMIFS only in Excel 2007). If someone still uses an earlier Excel version, you'd need to make an array SUM formula as demonstrated in Using Excel SUM in array formulas to conditionally sum cells.
Excel SUM not working - reasons and solutions
Are you trying to add a few values or total a column in your Excel sheet, but a simple SUM formula doesn't compute? Well, if the Excel SUM function is not working, it's most likely because of the following reasons.
1. #Name error appears instead of the expected result
It's the easiest error to fix. In 99 out of 100 cases, the #Name error indicates that the SUM function is misspelled.
2. Some numbers are not added
Another common reason for a Sum formula (or Excel AutoSum) not working are numbers formatted as text values. At first sight, they look like normal numbers, but Microsoft Excel perceives them as text strings and leaves them out of calculations.
One of the visual indicators of text-numbers are the default left alignment and green triangles in top-left corner of the cells, like in the right-hand sheet in the below screenshot:
To fix this, select all problematic cells, click the warning sign, and then click Convert to Number.
If against all expectations that does not work, try other solutions described in: How to fix numbers formatted as text.
3. Excel SUM function returns 0
Apart from numbers formatted as text, a circular reference is a common source of problem in Sum formulas, especially when you are trying to total a column in Excel. So, if your numbers are formatted as numbers, but your Excel Sum formula still returns zero, trace and fix the circular references in your sheet (Formula tab > Error Checking > Circular Reference). For the detailed instructions, please see How to find a circular reference in Excel.
4. Excel Sum formula returns a higher number than expected
If against all expectations your Sum formula returns a bigger number than it should, remember that the SUM function in Excel adds both visible and invisible (hidden) cells. In this case, use the Subtotal function instead, as demonstrated in How to sum only visible cells in Excel.
5. Excel SUM formula not updating
When a SUM formula in Excel continues to show the old total even after you've updated the values in the dependent cells, most likely Calculation Mode is set to Manual. To fix this, go to the Formulas tab, click the dropdown arrow next to Calculate Options, and click Automatic.
Well, these are the most common reasons for SUM not working in Excel. If none of the above is your case, check out other possible reasons and solutions: Excel formulas not working, not updating, not calculating.
This is how you use a SUM function in Excel. If you want to have a closer look at the formula examples discussed in this tutorial, you are welcome to download a sample Excel SUM workbook. I thank you for reading and hope to see you on our blog next week.
You may also be interested in
- How to use SUMIF in Excel - formula examples to conditionally sum cells
- How to use Excel SUMIFS and SUMIF with multiple criteria - formula examples
- How to add and subtract time in Excel
- How to add and subtract dates, days, weeks, months and years
- How to get mean, median and mode in Excel
- Excel MEDIAN formula examples
How do you sum in Excel excluding hidden rows? ›
For example you want to sum only visible cells only, please select the cell you will place the summing result at, type the formula =SUMVISIBLE(C3:C12) (C3:C13 is the range where you will sum only visible cells) into it, and press the Enter key. And then the summing result is calculated with ignoring all hidden cells.How do you add a conditional sum in Excel? ›
If you want, you can apply the criteria to one range and sum the corresponding values in a different range. For example, the formula =SUMIF(B2:B5, "John", C2:C5) sums only the values in the range C2:C5, where the corresponding cells in the range B2:B5 equal "John."How do I count only visible cells in Excel? ›
If you're using Excel 2007 or later, you can also use the COUNTA function to count filtered rows. This function counts all non-blank cells in a range. So if you have a column of data that has some blank cells, COUNTA will only count the non-blank cells.What Excel functions is used to get the total value of numbers in rows or columns? ›
If you need to sum a column or row of numbers, let Excel do the math for you. Select a cell next to the numbers you want to sum, click AutoSum on the Home tab, press Enter, and you're done. When you click AutoSum, Excel automatically enters a formula (that uses the SUM function) to sum the numbers.How do I SUM only visible cells not hidden? ›
Just organize your data in table (Ctrl + T) or filter the data the way you want by clicking the Filter button. After that, select the cell immediately below the column you want to total, and click the AutoSum button on the ribbon. A SUBTOTAL formula will be inserted, summing only the visible cells in the column.How do you sum using conditions? ›
To sum with multiple criteria, either use the SUMIFS function (adds up cells that meet all the conditions) or build a SUMIF formula with multiple OR criteria (sums cells that meet any of the conditions).How do you sum by two criteria in Excel? ›
In other words, you want the formula to sum numbers in that column if they meet the conditions. That cell range is the first argument in this formula—the first piece of data that the function requires as input.
Count ignore hidden cells and rows with excel functions
Select a blank cell you will place the counting result into, type the formula =SUBTOTAL(102,C2:C22) (C2:C22 is the range where you want to count ignoring manually hidden cells and rows) into it, and press the Enter key.
Tips: To sum the visible cells based on multiple criteria, you just need to add the criteria like this: =SUMPRODUCT(SUBTOTAL(3,OFFSET(reference,ROW(reference)-MIN(ROW(reference)),,1)),(criteria_range1=criteria1)*(criteria_range2=criteria2)*(criteria_range3=criteria3)*(sum_range)).How do you count filtered cells? ›
To count filtered rows in Excel, you can use the SUBTOTAL function that allows you to select its count feature in the first argument and the range from which you want to count filtered rows. After using this formula, when you apply a filter on that range and filter cells, it will only count the visible rows.
What is the formula for sum in Excel? ›
=SUM(A2:A10) Adds the values in cells A2:10. =SUM(A2:A10, C2:C10) Adds the values in cells A2:10, as well as cells C2:C10.What is the formula of subtotal in Excel? ›
|=SUBTOTAL(9,A2:A5)||The sum of the subtotal of the cells A2:A5, using 9 as the first argument.||303|
|=SUBTOTAL(1,A2:A5)||The average of the subtotal of the cells A2:A5, using 1 as the first argument.||75.75|
- Select a cell in a table.
- Select Design > Total Row.
- The Total row is added to the bottom of the table. ...
- From the total row drop-down, you can select a function, like Average, Count, Count Numbers, Max, Min, Sum, StdDev, Var, and more.
The same goes for hidden cells; since the SUM function takes a consecutive range (unless manually inputted with separate cells), hidden cells will also be included by the SUM function in counting the total.How do you sum data based on multiple criteria? ›
You use SUMIFS in Excel to find a conditional sum of values based on multiple criteria. The SUMIFS function was introduced in Excel 2007 and is available in all subsequent versions of Excel 2010, 2013, 2016, 2019, 2021, and Excel 365.How do I sum multiple columns in Excel with criteria? ›
SUMIFS + SUMIFS to sum multiple columns
To sum cells that match multiple criteria, you normally use the SUMIFS function.
Sum if cell contains text
If you are looking for an Excel formula to find cells containing specific text and sum the corresponding values in another column, use the SUMIF function. Where A2:A10 are the text values to check and B2:B10 are the numbers to sum. To sum with multiple criteria, use the SUMIFS function.
Things to remember about the SUM Function
The easiest way to apply the function is to simply select a cell next to the numbers that we wish to add and click on AutoSum on the Home tab. We can then press the Enter key and the SUM formula is automatically inserted.
If you clicked IF, the Function arguments dialog box displays the arguments for the IF function. To nest another function, you can enter it into the argument box. For example, you could enter SUM(G2:G5) in the Value_if_true box of the IF function. Enter any additional arguments that are needed to complete your formula.Can you use two conditions in a Sumif? ›
If you need to create a sum based on more than one criteria, you can use SUMIFS. It accomplishes the same task as SUMIF but allows you to set anywhere from one to 255 conditions. I have created a workbook with an example of using SUMIFS.
Can I use Sumifs for rows and columns? ›
If you only need to sum matching columns (not rows) you can use a formula like this. To sum matching rows only, you can use the SUMIFS function.How do I exclude hidden in subtotal in Excel? ›
As you remember, Subtotal formulas with function_num 101 to 111 ignore all hidden cells - filtered out and hidden manually. So, when you use Excel's Hide feature to remove irrelevant data from view, use function number 101-111 to exclude values in hidden rows from subtotals.How do I sum only visible cells in sheets? ›
If we attempt to use the SUM() function to sum the points column of the filtered rows, it will actually return the sum of all of the original values: What is this? Instead, we can use the SUBTOTAL() function: This function takes the sum of only the visible rows.What is the aggregate function in Excel? ›
The AGGREGATE function is designed for columns of data, or vertical ranges. It is not designed for rows of data, or horizontal ranges. For example, when you subtotal a horizontal range using option 1, such as AGGREGATE(1, 1, ref1), hiding a column does not affect the aggregate sum value.How do you exclude cells from sum? ›
If cells you need to exclude from the summing locating in discontinuous cells, for example, you need to exclude values in cell A3 and A6 while summing range A2:A7, please apply this formula: =SUM(A2:A7)-A3-A6.What is the Sumifs function in Excel? ›
The SUMIFS function, one of the math and trig functions, adds all of its arguments that meet multiple criteria. For example, you would use SUMIFS to sum the number of retailers in the country who (1) reside in a single zip code and (2) whose profits exceed a specific dollar value. Play.


2-D vs. 3-D Fixes
(an article by Ed Weston*)
Most receivers calculate a horizontal position after they acquire three satellites. To do this they must make presumptions on their altitude. Usually the last calculated altitude is used. If the altitude assumed by the receiver is in error, the accuracy of the horizontal position is impacted. For example, if the receiver assumed mean sea level and it was at 10,000 feet, a horizontal error of more than a mile could result. On the other hand, if the assumed value is very accurate the horizontal error may be less than if four satellites are tracked.
I?m going to try to present a simplified description of the GPS position calculation procedure to help the reader understand why the above is true. If a receiver is tracking four satellites, it is able to make simultaneous pseudo-range measurements from each. Each pseudo-range can be thought of as being equal to the range to the satellite plus the offset in the receiver?s clock, scaled to distance units. Since the receiver can calculate the satellite?s position from the Navigation Message, there are four unknowns in the pseudo-range equations: the three components of? the receiver?s position, and the offset in the receiver?s clock. To determine its location it then must solve the four equations for the four unknowns.? The receiver uses suitable algorithms to perform this arithmetic [Note for the more mathematically inclined. You?ve noticed that the equations are non-linear which makes the task non-trivial. If a rough estimate of position is available a Taylor series expansion about the estimated position is generally used. The partial derivatives of the pseudo-range wrt to receiver position are equal to the satellite-to-receiver unit line of sight vector.]
If the receiver ?knows? one of the three position components, then measurements from only three satellites are required to solve for the other two components and the receiver clock offset (3 equations and 3 unknowns).swiss-clock.me If, for example, the altitude is known perfectly, this is equivalent to tracking a satellite directly above (or below) the user, and having a perfect pseudo-range measurement to it. Since a real fourth satellite will never provide a perfect pseudo-range, the three-satellite solution will be better than the four-satellite solution (assuming similar geometry quality, i.e., PDOP). If on the other hand the assumed altitude is in error, then the two sides of the equations will be equal only if a compensating error is added to the horizontal position components and clock offset.
How does a receiver calculate its position if more than four satellites are tracked? Well, without getting into the mathematics it will be a little difficult to explain, but I?ll try. This description will be conceptual, that is should not be taken literally. As was described above, with four satellites (and good geometry) the position (and necessarily the clock offset) can be calculated. If a measurement from a fifth satellite is available, the receiver can calculate what the measurement should be (expected pseudo-range is range plus clock offset, based on the four satellite solution). The difference between the actual pseudo-range and the expected pseudo-range indicates whether the fifth satellite ?agrees? with the four-satellite position solution. That is, if the difference is zero the five-satellite solution is the same as the four-satellite solution. If it is non-zero, then the value is used to adjust the four satellite position (and clock offset) to produce the five satellite position. How is the adjustment done? Well mathematically, linear algebra techniques are used to produce a least square solution. Conceptually think about it like this. You guess a position (and clock) solution, then use it to calculate the five expected pseudo-range values as described above. You then square each of the pseudo-range differences (expected minus actual pseudo-range) and sum them up. This is a measure of the quality of the five-satellite solution. You try many different position solutions, calculating the quality measure for each. The one with the smallest quality measure is your least squares five satellite position. A similar technique is used to calculate a least squares solution for more than five satellites.
* Ed Weston is a GPS analyst and has worked on GPS for DOD since 1975.


Decay at 16+
All radioactive decay occurs in such a way as the
number of atoms 'decaying' in unit time (dN/dt) is directly proportional
to the number of undecayed radioactive atoms present in the sample at
that time (N).
The mathematics of statistics applies (a very large
number of 'events' needs to be considered). A constant of proportionality
the decay constant) replaces the proportionality symbol, thereby making
the equation simpler:
With time the number of undecayed atoms decreases.
The activity decreases therefore the change in activity is negative.
Adding the negative sign allows us to have a positive value for the
activity. Note the negative sign!
The type of equation is called a differential equation.
Calculus can be used to integrate this giving and expression that incorporates
both the initial number of atoms present and the final number present
after a time interval 't'
When solution of this equation is:
Thus we have a mathematical expression that shows
that radioactive decay is exponential.
is half of N0
time t is the time taken for half of the sample to undergo spontaneous
disintegration or decay. This is termed the half life and is given the
Let us follow the mathematics of this through
for a half life value:
Now let us insert the number of atoms present in
terms of the original number:
How this is achieved should be understood. If necessary
look in the mathematics
section at the background on use of logs.
It is important that l can be found directly from
the half life and vice versa is remembered.
A similar calculation can be done to find the time
taken for a certain percentage of a sample to decay.
Try the following question:
The half-life of cobalt 61 is 100 minutes.
How long does it take for the activity of the sample to fall to
80% of its initial value?
Try this question:
or to find the percentage left after a certain time
The half-life of radioactive zinc is
50 hours. Find the percentage loss of activity which occurs for
a sample each hour.
spreadsheets to investigate the half life idea in a little


I love creating “out of the box” ways for teaching tough concepts. Division is one of those concepts that may be challenging for children to learn, especially children who are more “right brain” learners.
This division resource includes directions for using a cute Mrs. Owl who has had a great fall harvest and wants to share her beans with her friends. Students will enjoy helping her divide up the beans among her friends using eye catching Math Mats for all of the division facts from 1 – 9.
This pack includes a practice sheet to go along with each mat.
It also includes a super simple way of teaching how to set up a division problem using the “house.” In this resource, the house actually has a door and the friends stand outside and knock (the number you are dividing by goes outside the house). Mrs. Owl is inside the house. She has all the beans (the total number you are dividing goes inside the house). She sets the beans up on a high shelf while she divides them out (the answer goes on top).
My time is limited these days, so I mostly just try to create things for my own children. I created this resource for my daughter and she is really enjoying sorting those beans! Hope your students/children love it too!


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Thanks to a tip from Bob Atkey, I’ve recently been looking at coeffects as a means to prove program functions monotone. This post contains an introduction to coeffects, along with an explanation of how they can be used to reason about monotonicity. My goal is to cover the key ideas of coeffects without getting into a full treatment of their semantics. The particular coeffect system that I will examine is taken from Coeffects: A calculus of context-dependent computation by Tomas Petricek, Dominic Orchard, and Alan Mycroft.
The content of this post depends on some basic category theory. In particular, it involves functors, natural transformations, and cartesian closed categories. We write objects using capital letters etc. We write arrows as lower-case letters etc. Categories are written using bold capital letters etc. Natural transformations are written as lowercase greek letters: , etc. The component of a natural transformation at an object is written , where the natural transformation has been parenthesized and subscripted by the object. We write functors using capital letters etc. We write the composition of two functors and as and the application of a functor to an object as .
In a structural type-and-coeffect system, a function has a type of the form , where and are ’s domain and codomain types, and is a coeffect scalar. The coeffect scalar denotes some ``contextual capability’’ which augments the domain . We will formalize the notion of a contextual capability in the next section, and in the meantime it will stay intuitive.
Petricek et al. provide an example targeted at dataflow languages such as Lustre. In this example, time is modeled as a sequence of discrete steps, and variables denote value streams which contain one element per time step. A variable occurrence denotes its stream’s value at the current time step, and values at previous timesteps can be accessed by wrapping occurrences of the variable in a syntactic form called pre. As depicted above, a coeffect scalar in this system is a natural number denoting a lower bound on the number of cached values at consecutive previous timesteps for a particular stream. Note that certain contextual capabilities are stronger than others. For example, in the dataflow system, the coeffect scalar allows us to read a stream’s value at 1 timestep prior, while the coeffect scalar is stronger than because it allows us to read a stream’s value both 1 and 2 timesteps prior.
It may be helpful to open a new browser window fixed to Figure 2, so that its rules may be quickly recalled when required by later sections.
We can assume that a function constant of type ``utilizes’’ at most the contextual capabilities indicated by its scalar . But an upper bound on the contextual requirements of a lambda abstraction must be proven, using the type-and-coeffect system of Figure 2. A type-and-coeffect judgment has the form . Here , , and are the familiar typing context, expression, and type. is a vector of coeffect scalars, associating one scalar to each entry in ; we refer to such a vector simply as a coeffect. We write the concatenation of coeffects and as .
The ABS rule says that under context and coeffect , an abstraction has type whenever we can type-check its body under the the extended context and extended coeffect . The key to ensuring that our abstraction does not ``overuse’’ the capability denoted by is to restrict, within the body , both the positions in which the variable may occur and the positions in which it may be discarded; the weaker the capability a scalar denotes, the more restrictive it is syntactically. The particular set of restrictions a scalar makes is dependent on a parameter to the structural type-and-coeffect system called the coeffect scalar structure.
A coeffect scalar structure consists of
- An underlying set of scalars
- Binary composition and contraction operators and on
- A coeffect scalar , the most restrctive scalar that permits a variable to appear in an hole context, i.e. the most restrictive scalar such that is derivable
A coeffect scalar , the most restrictive scalar which constrains a variable in a way that permits it to be discarded, i.e. whenever is derivable so is
- A preorder on , where means that is more restrictive than
We require and to be monoids. Further, for all we require these distributivity equalities
Finally, both and should be monotone separately in each argument, that is implies
The coeffect scalar structure for the dataflow coeffect system is , where is the set of natural numbers, is the addition operator on natural numbers, is the operator which produces the greater of two natural number arguments, and is the standard comparison operator for natural numbers. It’s instructive to consider the system of Figure 2 instantiated with this structure, in which the VAR axiom says that a variable occurrence does not require the capability to read that variable’s stream at prior time steps. Assuming is the sole base type, the construct, which allows us to access a stream at the previous time step, is then implemented as a built-in function constant of type .
To get a feel for the APP rule, consider this derivation.
The conclusion concatenates disjoint typing contexts and coeffects from its function and argument premises. While the scalar in the left premise () matches its portion of the conclusion (the in ), the scalar in the right premise () differs from its portion of the conclusion (the in ). The right premise of Figure 3 places the variable in an empty context, where it required to provide its value at the current timestep, and prior timesteps (thus the scalar ). In the left premise, is a function requiring access to its argument at 1 prior timestep (thus the scalar ). Hence, the conclusion requires access to at prior timesteps. The essential power of type-and-coeffect systems is to reason about such composition of contextual capabilities, and this is handled with operator, which in the dataflow coeffect scalar structure is the operator on natural numbers. In the APP rule, and may in general contain multiple entries. When a vector coeffect rather than a scalar is used for the second argument of , we apply componentwise: for all coeffects and all scalars we have
Since the APP rule concatenates disjoint typing contexts and coeffects, one might wonder how distinct occurrences of a single variable within the same term are typed. It’s accomplished by contracting two variables in context and combining their associated contextual capabilities. This requires an application of the CTXT typing rule, which allows us to prove a typing judgment with context-and-coeffect using a typing derivation with context-and-coeffect , obtained from a structural manipulation of . The right premise allows us to apply a structural manipulation rule of the form ; in particular, to contract two variables (and their associated capabilities) we must use the CONTR rule.
Formalizing contextual capabilities
So far we have been keeping the notion of a contextual capability intuitive. We will now briefly summarize the formalization of contextual capabilities. Recall that a standard typed lambda calculus can be modeled as a cartesian closed category , where for every type , the denotation of is an object of . In particular, the interpretation of a function type is the exponential object . Furthermore, the interpretation of a typing judgment is an arrow .
To model type-and-coeffect judgments categorically, we interpret the scalar coeffect structure as a category using the standard interpretation of preorders as categories. However, we use the transpose of for the preorder. That is, the objects of are the elements of , and for each with , has one arrow with domain and codomain . We consider a strictly monoidal category with monoidal product and unit .
We interpret a structural type-and-coeffect system (instantiated with a scalar coeffect structure ) with respect to a cartesian closed category and an indexed comonad functor
Writing for the application to the scalar , D is associated with a counit natural transformation
and a family of comultiplication natural transformations
making the following diagrams commute for all objects of
In true categorical fashion, a contextual capability is defined not by its structure, but how it behaves. In particular, the contextual capability denoted by is the endofunctor . operates on an object of by ``placing it into a context with capability ’’. It operates on an arrow of , which for our purposes represents a transformation, by converting it into a transformation that preserves the contextual capability .
is the transformation of pulling an object out of a context, while for scalars , is the transformation of decomposing a single context into a context-in-a-context. Whenever such that , the functorality of gives a natural transformation which helps justify the context manipulation rule called SUB.
The bottom left corner of the top diagram therefore states that a transformation which
- Decomposes a context of capability into an empty context containing a context of capability .
- Then takes the inner context of capability out of the empty context.
is equivalent to the identity transformation, which does nothing. The bottom diagram shows that context decomposition is in a sense associative.
With the indexed comonad , a function type is interpreted as the exponential . A single-variable type-and-coeffect judgment is interpreted as an arrow of type . Of course, we also need a way to interpret type-and-coeffect judgments with multiple variables in context, as well as interpretations of structural rules such as contraction and weakening. For this we need structural indexed comonads, a topic outside the scope of this post. For a full account of categorical semantics for coeffects, see Petricek.
At this point, you might be wondering about the goal of all this indexed comonad business. Furthermore, if the coeffect in a type-and-coeffect judgment just denotes some applications of endofunctors, why bother making those applications explicity in the judgment system? In other words, why have a judgment of the form rather than ? The answer is that the domain has been decomposed into a portion that is handled manually by the programmer and a portion that is handled automatically by the indexed comonad. To apply a function of type , the programmer must place an expression of type in the argument position of the application. Importantly, the programmer does not supply an argument of type ; the contextual capability is automatically piped in to the function application.
To convey the intuition behind this automatic piping of contextual capabilities, we present the notion of an indexed coKleisli category.
If is an indexed comonad, the indexed coKleisli category determined by , written is defined such that
The objects of are the same as the objects of .
For all , and objects and of , an arrow is considered as an arrow of of .
Given arrows and of with underlying arrows and in , we define the composition as the arrow generated from the arrow defined as .
For each object of , the identity arrow in has the underlying arrow in .
In an indexed coKleisli category, we can compose arrows without regard to the scalars attached to their underlying arrows. In this way, programming in a type-and-coeffect system is kind of like composing arrows in a coKleisli category.
To provide some further intuition, in an algorithmic type-and-coeffect system, in the judgment is an output position, much like ; in constrast, is an input position. The contextual capabilities required to execute expression must be synthesized from the structure of rather than inherited from ’s context.
We can interpret dataflow coeffects by letting , the category of sets and functions, and using the indexed comonad where for all , sets and , and functions , we have
We place the type into a context of capability by pairing it up with its n-ary self-product, where the nth component of the product represents the value of the stream at the nth previous consecutive timestep. To transform a function into an equivalent function which preserves the contextual capability , we must apply the function not only at the current timestep, but also at the cached consecutive previous timesteps.
We pull a value of type out of a context with 0 cached previous consecutive timesteps simply by getting its value at the current timestep.
Having cached previous consecutive timesteps is equivalent to having cached the length- suffixes at the previous consecutive timesteps.
Finally, we can discard cached values. For we have
Monotonicity as coeffect
Instantiating the coeffect calculus of Figure 2 with the following coeffect scalar structure results in a system which allows us to prove functions monotone.
- The scalar set is , where denotes the capability to be transformed in an equivalence-class-preserving manner, to be transformed monotonically, to be transformed antitonically, and to be transformed in an order-robust manner. (An order robust function maps two inputs values related by the symmetric transitive closure of its domain to two isomorphic elements of its codomain.)
- The composition and contraction operators and are defined in Figure 6.
- The scalar is .
- The scalar is .
- The preorder on scalars is depicted in Figure 7.
For the underlying cartesian closed category of our semantics, we choose , the category of preordered sets and monotone functions. For preordered sets and , and monotone function , our indexed comonad is defined as
- is the dual of , that is, the preordered set defined such that for all we have if and only if .
- is the order obtained by removing from all edges which do not have a symmetric counterpart in .
- is the symmetric transitive closure of .
For all preorders and all we have
Furthermore, if then we have
These definitions are a bit anticlimactic, but they make sense when one considers that and
The natural transformations of monotonicity coeffects have a much different flavor than those of the dataflow coeffects, in that they have no significant “runtime content”. While it’s tempting to think of a contextual capability as some extra information that is paired with each input value supplied to a function, monotonicity coeffects show that this really isn’t the case.
I will demonstrate this system using an example from page 2 of Consistency Analysis in Bloom: a CALM and Collected Approach. Let be the type of natural numbers, let be the type of finite sets of natural numbers, ordered by inclusion, and let be the two element preordered set containing the boolean values true and false, ordered such that . The example requires us to prove the expression monotone, where
- is a variable of type
- is a function of type , which produces the minimum element contained in the argument
- is the standard “less than” operator on naturals, of type
Renaming “<” to “lt”, rewritten to prefix notation is . The following abbreviations allow our proofs to fit on the page.
Using Petricek’s coeffect system, proof of the judgment follows.
Coeffects are a simple yet extremely general framework enabling type systems which can prove, among many other things, that a program function is monotone. I think that recently proposed programming frameworks such as Bloom, LVars, and Lasp can benefit from this power. In addition, I believe all of these frameworks would also benefit from the ability to prove that a program function is a semilattice homomorphism. For many semilattice datatypes, a monotonicity-aware type system such as the one presented here can be used for this purpose, as a monotone function from a certain (common) kind of semilattice’s join-irreducible elements can be converted into a homomorphism from the semilattice itself.
While this post focused primarily on semantics, there are pragmatic issues which need addressing.
First, the coeffect calculus presented here in non-algorithmic. However, it’s not hard to imagine an algorithmic variant, developed under the same strategy as algorithmic linear type systems: a type-and-coeffect judgment would include an additional vector of output coeffects, denoting that portion of the input capability which is not actually utilized by the term. It may be necessary to place some additional restrictions on the scalar coeffect structure, but I expect that the scalar coeffect structure for monotonicity coeffects would satisfy these restrictions.
Second, even with an algorithmic system, it isn’t currently clear to me that this approach is going to be mentally tractable for programmers, as there’s a fair amount of coeffect shuffling that happens to the left of the turnstile. In my opinion, a type system is most useful as a design tool, a mental framework that a programmer uses to compose a complex program from simple parts. I’m ultimately looking for such a mental framework, rather than a mere verification tool, for proving program functions monotone.
It’s exciting to imagine a statically typed variant of BloomL. Perhaps coeffects can serve as the basis for static BloomL’s type system.
If you found this post interesting, these links may interest you as well.
Datafun, a functional calculus inspired by Datalog, currently has the best monotonicity type system that I know of, as well as a working prototype. Interestingly, it features the ability to take the fixpoint of a monotone function under conditions which guarantee its existence.
Tomas Petricek created a really cool interactive web tutorial for coeffects.
The semantics of Petricek et al.’s coeffect calculus is based on category theory. I’ve found Awodey’s book one of the best sources on the subject. Presentations of categorical type system semantics can be daunting, but Abramsky and Tzevelekos’ lecture notes are a highly accessible exception to the rule.
Thanks to Michael Arntzenius and Chris Meiklejohn for providing feedback and catching errors in this post.


Using his "how many runs scored on each hit" system, he figured out that: A single is worth .46 of a run. A double is worth .79 of a run. A triple is worth 1.15 of a run A home run is worth 1.55 of a run. Pause once more to think about this. He only looked at 1,000 hits in 1916. He came up with a quirky system to figure out how many runs scored. Now, jump ahead 50 years. John Thorn and Pete Palmer wrote "The Hidden Game of Baseball," an all-time classic. In it, they introduced the Linear Weights system. For it, they used computer simulations and ALL the data available going back to 1901. And this is what they determined. A single is worth .46 of a run. A double is worth .80 of a run A triple is worth 1.02 of a run A home run is worth 1.40 of a run Numerous other people — Friend of Blog Tom Tango among them — have looked at Linear Weights, tweaked it, worked with it, and the numbers move a bit here and there. But the point here is that what F.C. Lane did through sheer will power, limited data and some rudimentary math skill was REMARKABLY accurate and at least a half century ahead of its time.


ITK Pre-Processing Algorithm Wrappers in Python/SobelEdgeDetectionFilter
Sobel Edge Detection Filter
This filter is implemented using the itkSobelEdgeDetectionImageFilter in ITK. This filter computes the gradient of the image using the Sobel operator and find the magnitude of the gradient vector. The magnitude is computed using the sum of the squares of the components of the gradient. Fig. 1 shows the result obtained by applying the sobel edge detection filter on the input image seg_orig.tif.
- 1. NumberOfThreads - this parameter gives the number of threads that the execution phase of the algorithm should be split into.


Muslims are famous for intricate and elegant geometrical designs decorating their historic buildings. Scholars contributed to the development of geometry, especially algebraic geometry and geometric algebra. They Inherited, developed and extended the work of the Greeks including the famous Euclid.
As Islam spread from nation to nation, Islamic artists combined their penchant for geometry with existing traditions, creating a new and distinctive Islamic art. This art expressed the logic and order inherent in the Islamic vision of the universe. 9th century Al-Mahani formulated the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. 10'th century Thabit ibn Qurra dealt with arithmetical operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.


ON PROPERTIES OF PRIME IDEAL GRAPHS OF COMMUTATIVE RINGS
The prime ideal graph of in a finite commutative ring with unity, denoted by , is a graph with elements of as its vertices and two elements in are adjacent if their product is in . In this paper, we explore some interesting properties of . We determined some properties of such as radius, diameter, degree of vertex, girth, clique number, chromatic number, independence number, and domination number. In addition to these properties, we study dimensions of prime ideal graphs, including metric dimension, local metric dimension, and partition dimension; furthermore, we examined topological indices such as atom bond connectivity index, Balaban index, Szeged index, and edge-Szeged index.
R. Diestel, Graph Theory (5th Edition). Springer, 2017.
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F. Mansoori, A. Erfanian, and B. Tolue, “Non-coprime graph of a finite group,” in AIP Conference Proceedings, American Institute of Physics Inc., Jun. 2016. doi: 10.1063/1.4954605.
E. Y. Asmarani, A. G. Syarifudin, I. G. A. W. Wardhana, and N. W. Switrayni, “The Power Graph of a Dihedral Group,” EIGEN MATHEMATICS JOURNAL, pp. 80–85, Jan. 2022, doi: 10.29303/emj.v4i2.117.
N. Nurhabibah, A. G. Syarifudin, I. G. A. W. Wardhana, and Q. Aini, “The Intersection Graph of a Dihedral Group,” EIGEN MATHEMATICS JOURNAL, pp. 68–73, Dec. 2021, doi: 10.29303/emj.v4i2.119.
A. Mukhtar, R. Murtaza, S. U. Rehman, S. Usman, and A. Q. Baig, “Computing the size of zero divisor graphs,” Journal of Information and Optimization Sciences, vol. 41, no. 4, pp. 855–864, May 2020, doi: 10.1080/02522667.2020.1745378.
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S. Humaira, P. Astuti, I. Muchtadi-Alamsyah, and A. Erfanian, “The matrix Jacobson graph of finite commutative rings,” Electronic Journal of Graph Theory and Applications, vol. 10, no. 1, pp. 181–197, 2022, doi: 10.5614/ejgta.2022.10.1.12.
K. Nozari and S. Payrovi, “The annihilator graph for modules over commutative rings,” Journal of Algebraic Systems, vol. 9, no. 1, Jun. 2021, doi: 10.22044/jas.2020.9194.1448.
H. M. Salih and A. A. Jund, “Prime ideal graphs of commutative rings,” Indonesian Journal of Combinatorics, vol. 6, no. 1, p. 42, Mar. 2022, doi: 10.19184/ijc.2022.6.1.2.
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F. Okamoto, B. Phinezy, and P. Zhang, “The local metric dimension of a graph,” Mathematica Bohemica, vol. 135, no. 3. pp. 239–255, 2010.
C. Wei, M. F. Nadeem, H. M. Afzal Siddiqui, M. Azeem, J. B. Liu, and A. Khalil, “On Partition Dimension of Some Cycle-Related Graphs,” Math Probl Eng, vol. 2021, 2021, doi: 10.1155/2021/4046909.
K. C. Das, “Atom-bond connectivity index of graphs,” Discrete Appl Math (1979), vol. 158, no. 11, pp. 1181–1188, Jun. 2010, doi: 10.1016/j.dam.2010.03.006.
M. Knor, R. Skrekovski, and A. Tepeh, “Mathematical Aspects of Balaban Index,” Communications in Mathematical and in Computer Chemistry, vol. 79, pp. 685–716, 2018.
M. Faghani and A. R. Ashrafi, “Revised and edge revised Szeged indices of graphs,” Ars Mathematica Contemporanea, vol. 7, no. 1, pp. 153–160, 2014, doi: 10.26493/1855-3974.269.44e.
Copyright (c) 2023 Rian Kurnia, Ahmad Muchlas Abrar, Abdul Gazir Syarifudin, Verrel Rievaldo Wijaya, Nur Ain Supu, Erma Suwastika
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Latest Comic: An old, out of touch man tries to reconnect with his gamer youth
By and large, furries are bi and large.
431 prime number, Sophie Germain prime, sum of seven consecutive primes (47 + 53 + 59 + 61 + 67 + 71 + 73), Chen prime, Eisenstein prime with no imaginary part
You loose sir, good day.
Quote from: Snoiper Bull on June 06, 2009, 01:24:26 PMYou loose sir, good day.Do I even wanna know what you did to him?


1. From city A, Jamil drives a car at an average speed of 60 km/hour to the city B which is 300 km far. One hour later, Aldo drives a car following the departure of Teguh to city B. determine the average speed of Aldo’s car so he can arrive in city B at the same time as Teguh!
2. A map is made such that a distance of 8 cm represents an actual distance of 120 km. Find :
a. the scale used
b. the actual distance if the distance on the map is 18 cm, and
c. the distance on the map if the actual distance is 675 km.
3. The distance between two cities on a map scaled at 1 : 18,000,000 is 20 cm Find the distance between the two cities on a map scaled at 1: 5,000,000 !
4. A drawing of a land is scaled at 1 : 500. If the measure of the land is 22.5 cm × 12 cm, find :
a. the actual measure of the land, and
b. the ratio of its area on the
5. A square land measures 72 m x 72 m. The lands is drawn to a measure of 12 cm x 12 cm. Determine :
a. the scale used, and
b. the ratio of its area on the drawing to its actual area
6. Simplify the following ratios!
a. 18 : 72
b. 2 ½ : 1¼
c. 25 cm : 1 m
d. 200 gr : 7 kg
e. 35 minutes : 1 ½ hours
7. Find the ratio of the following pricings!
a. Rp250 per unit to Rp3,600 per dozen
b. Rp50,000 per meter to Rp800 per cm
8. An elder sister is 8 year older than her brother. If the ratio of their ages is 5 : 3, find their ages!
9. The number of students in a class is 40 people, including 18 boys. Find the ratio of :
a. the number of female students to the number of male students
b. the number of female students to the number of all students
10. The sun of
11. A factory employee receives a wage of Rp17,500 per day. Later, his wage is increased by 6/5 times the previous wage. Determine the current wage of the employee!
12. Find the values of a and p in the following proportions!
a. a : 5 = 16 : 20
b. 6 : 5 = p : 30
13. The following expressions are direct proportions. Find the values of x and y!
a. 40 : (x + 3) = 5 : 2
b. 7 : 3 = 35 : (y + 5)
14. Determine the ratio a : b : c if a : b = 7 : 4 and b : c = !
15. In the following propositions, a : b is the inverse proportion of p : q. Find the value of x if :
a. a : b = and p : q = (x +10) : 24,
b. a : b : (x + 1) and p : q = (2x + 7) : 18!
16. The ratio of two numbers is 2 : 3. If each number is added by 4, then the new ratio is 3 : 4. Find the two numbers!
17. An employee can type 380 words in 20 minutes. How many words can the employee type in 2 ½ hours?
18. In a party, 36 kg of rice in needed to serve 150 guest. How many kilograms of rice are needed to serve 125 people?
19. The bus ticket price for each 15 km is Rp1,800. What is the bus ticket price for a distance of 50 km?
20. The weight of 40 cm3 of gold is 250 grams. What is the weight of 52 cm3 of gold?
21. The cost of staying in a hotel for 7 days is Rp875,000. What is the cost of staying in the hotel for 4 days?
22. Tanti bakes some breads filled with chocolate and some breads filled with cheese. To make 18 breads filled with chocolate requires 4 kg of wheat and to make 15 breads Filled with cheese requires 3 kg of wheat. How many kg of wheat is needed if Tanti wants to bake 216 breads filled with chocolate and 135 breads filled with cheese?
23. Mr. Hamid can buy 40 books for Rp1,200 each. If by the same amount of money the wants to get 50 books, then what is the price of each book which he should buy?
24. Twenty five workers can complete a job in 54 days. How many days can the job be completed if it is done by 18 workers?
25. A landlord estimates he can provide food for 135 person for 8 days. If the food supply is exhausted in 6 days, then how many additional residents are there ?
COMPETENCY TEST CHAPTER 5
(RATIO AND PROPORTIONS)
I. For each problem numbered 1 to 5 choose the right answer!
1. The Patroness towers in
a. 0.0452 m
b. 0.452 m
c. 4.52 m
d. 45.2 m
2. A map of
a. 10 km
b. 18 km
c. 100 km
d. 180 km
3. If 2 U.S. dollars is worth the same as Rp21,000, then Rp525,000 is worth the same as …
a. 50 dollars
b. 51 dollars
c. 52 dollars
d. 53 dollars
4. Let fuel consumption of a car is 2 liters for a 25 km distance. If the distance between
a. 4 ½ liters
b. 9 liters
c. 12 ½ liters
d. 18 liters
5. If the price of 1 dozen T-shirts is Rp120,000, then the price of 1 score of T-shirt is …
6. By using a car, the distance between two cities can be traveled in 10 hours at an average speed of 60 km/hr. the time needed to travel that distance if the average speed is 75 km/hr is…
a. 7 hours
b. 8 hours
c. 9 hours
d. 10 hours
7. A job is estimated to be completed by 12 people in 1 week. If the job must be completed in 4 days, then the number of additional workers required is …
a. 3 person
b. 6 person
c. 9 person
d. 12 person
8. A poultry man has food supply for 1,000 chickens for 2 week. If the adds 400 more chickens, the food supply will be exhausted in …
a. 9 days
b. 10 days
c. 11 days
d. 12 days
9. To make 75 cakes, it requires 0.75 kg of sugar. The amount of sugar required to make 100 cakes is …
a. 1 kg
b. 1,5 kg
c. 10 kg
d. 15 kg
10. A diskette can hold 1.4 MB of data, a data CD can hold up to 700 MB. The number of diskettes that is needed the match the capacity of a CD is …
a. 500 diskettes
b. 450 diskettes
c. 50 diskettes
d. 45 diskettes
11. In 2004 Mr. Dani earned Rp2,000,000. In 2005, his salary increased by 10%. The ratio of Mr. Dani’s salary in 2004 to that in 2005 is …
a. 2 : 2,1
b. 2 : 2,2
c. 1 : 2,1
d. 1 : 2,2
12. Users of the 5 km long toll road TMI – Cawang is charged Rp1,000. The ideal toll for Cipularang toll road which is 40 km long is …
13. The sum of Hasan’s and Husen’s saving is Rp3,500,000. If the ratio of Hasan ‘s to Husen’s saving is 3 : 2, then the amount of Hasan saving is …
14. The number of students in Nusantara Junior High School is 1,500 with the ratio of the number of male students to the number of female students is 3 : 2. The number of female students who have to be added so that the number of male students equals the number of female students is …
a. 200 people
b. 250 people
c. 300 people
d. 350 people
15. A drawing of a rectangular land measures 16 cm x 10 cm. If the drawing is made to a scale of 1 : 500, then the actual area of the land is ….
a. 400 m2
b. 800 m2
c. 4,000 m2
d. 8,000 m2
II. Do the following problems in detail!
1. A dormitory has food supply for 60 people for 25 days. If its residents increased by 5 persons, then in how many days will the food supply be exhausted?
2. Based on an observation on some circles whose radial are different in length, the area of each is found as follows.
a. Find the ratio of the area to the square of radius of each circle.
b. Is the value of the ratio for a circle equal to that for another circle? Do you familiar with that value?
c. Now make a conclusion of what you know about the number p (pi)!
3. Dewi made 2 types of bread, i.e.., the bread filed with banana and the bread filled with meat. To make 20 breads filled with banana she needs 5 ounces of wheat, and to make 15 breads filled with meat she needs 4 ounces of wheat. How many ounces of wheat Dewi needs to make :
a. 140 breads filled with banana?
b. 45 breads filled with meat?
4. In a ropewalking, a female participant walked at a speed of 6 km/hr so that she could reach the finish line in 5 hours. After 2 hours of walking, she paused for 30 minutes since she suffered from a muscular spasm. In order to reach the finish line according to her initial plan, at what speed did she had to walk?
5. A contractor estimates that he can have a building construction completed in 40 days by employing 15 workers. After 12 days of works, the works is paused for 1 week since he runs out of the building materials. How many workers must he add so that the building constructions can be completed in time?