Rectangle Counts

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Mensa Daily Puzzlers

For over 15 years Mensa Page-A-Day calendars have provided several puzzles a year for my programming pleasure.  Coding "solvers" is most fun, but many programs also allow user solving, convenient for "fill in the blanks" type.  Below are Amazon  links to the two most recent years.

Mensa 365 Puzzlers  Calendar 2017

Mensa 365 Puzzlers Calendar 2018

(Hint: If you can wait, current year calendars are usually on sale in January.)


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Here's a puzzle sent to me by a viewer who says it is from a set of questions written by computer pioneer and inventor Sir Clive Sinclair. They were published many years ago as "Mensa Steps" in a magazine, perhaps "Design Technology" as the viewer recalls ,

"If you draw a nine by nine square, thus giving yourself eighty-one small squares in total;  how many rectangles can you count in total? "

One of my standard problem solving strategies is to "simplify"; solve a smaller version of the program and hope that it provides a clue to solving the larger one. The program  generates the number of rectangles for grids from from 1x1 to 9x9. It does it in a nested loop by summing all rectangle counts from 1x1 to NxN for each grid size.  We count rectangles of size H x V where  H and V represent horizontal and vertical dimensions and range from  1 to N. For each H there are  H+1-A rectangles across and for each V there are N+1-V  rectangles down.  The total numer of rectangles is the sum of all products (N+1-H)*(N+1-V) for all H and V combinations from 1 to N.

I did not discover the generating function based solely on the results, but if you search on the sequences of results (1,9,36,100,225,441 ...) you'll find lots of fascinating information including the formula in the Online Encyclopedia of Integer Sequences page at Also notice that the numbers in the sequence are perfect squares of the sequence 1,3,6,10,15.... These are called "triangular numbers" with the property that the Nth member is the number of dots in an equilateral triangle with N dots per side! (Think of  4th entry for example, being  represented by 10 bowling pins arranged in the traditional triangle with 4 pins per side.)  See for more interesting information.

I was planning to post this program on the Beginner's page since there are less than 20 lines of code, but decided on the Math Topics section after recognizing  the interesting math.

 Running/Exploring the Program 

bullet Download source
bullet Download  executable


Created March 6,,2012

Modified May 11, 2018



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