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How many squares are there on a chessboard? The aim of this investigation is to find out how many squares and rectangles of specific sizes can be found on a chessboard, and too see if there is a common sequence and algebraic formulae for each example. I have started with the simplest example that is to count the different combinations of squares on a range of boards from 2x2 board up to an 8x8 board. Results from the count 2x2 board 1x1=4 2x2=1 3x3 board 1x1=9 2x2=4 3x3=1 4x4 board 1x1=16 2x2=9 3x3=4 4x4=1 5x5 board 1x1=25 2x2=16 3x3=9 4x4=4 5x5=1 6x6 board 1x1=36 2x2=25 3x3=16 4x4=9 5x5=4 6x6=1 7x7 board 1x1=49 2x2=36 3x3=25 4x4=16 5x5=9 6x6=4 7x7=1 8x8 board 1x1=64 2x2=49 3x3=36 4x4=25 5x5=16 6x6=9 7x7=4 8x8=1 When the 8x8 boards results are analyzed a quadratic sequence can be identified i.e the second difference is a constant. term No of squares 1st diff 2nd diff 1 64 2 49 15 3 36 13 2 4 25 11 2 5 16 9 2 6 9 7 2 7 4 5 2 8 1 3 2 Using the general term for a quadratic sequence where A= 1/2 the constant 2nd difference, which is 2x 1/2 = 1 Yn=An + Bn + C So y1 = 1x1 + B x 1 + C = 64 So B +...

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