How many squares on a chess board

This is my first piece of mathematics coursework. In this project I will be finding out how many squares on a 8×8 chess board. In this project I will be including labelled drawings, clear enough to allow a non mathematician to follow, clearly written work, initial drawings of each of my steps and a finished formula justified. Finally an extension on my board making it rectangular.

Plan* Introduction* Spider diagram* I will be collecting data from different sized grids ranging from 2×2 to 8×8.* Explanation of how to add up the squares on the grid* Further calculations* Create a table of values.* Formula and formula justified* Draw a graph* Extension* Conclusion* EvaluationTree diagramExplanation of how to add up the squares on the grid1.

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Count how many singular squares there are=162. Count how many double squares there are.=93. Continue this until you complete the whole shape in this case 4×4.

4. Once you have completed this you should have 208 squares on the 8×8 grid.Further calculations1x1=12×2=53×3=144×4=305×5=….

..1x11x1=1 =12x22x2=4(2-1)x(2-1)=1×1=1 =4+1=53x33x3=9(3-1) (3-1)=2×2=4(2-1) (2-1)=1×1=1 =9+4+1=144x44x4=16(4-1)x(4-1)=3×3=9(3-1)x(3-1)=2×2=4 =16+9+4+1=30(2-1)x(2-1)=1×1=1Check 4×42 2 2 2= (4) + (4-1) + (3-1) + (2-1)= 16+9+4+1=30Use of Algebra2 2 2 2(nxn) = (n) + (n-1) + (n-2) + (n-3)Estimating 5×52 2 2 2 2= (4) + (5-1) + (5-2) + (5-3) + (5-4)= 25+16+9+4+1= 55Table of valuesx y D1 D2 D31 1+42 5 +5+9 +23 14 +7+16 +24 30 +9+25 +25 55 +11+36 +26 91 +13+53 +27 140 +15+628 204Finding the formulaThe cubic formula of type3 2Y= Ax + Bx + Cx + D(x+1)3 2Y= Ax1 + Bx1 + Cx1 + D=1A+B+C+D=4 1(x+2)3 2Y= Ax2 + Bx2 + Cx2 + D=58A+4B+2C+D=5 2(x+3)3 2Y= Ax3 + Bx3 + Cx3 + D=1427A+9B+3C+D =14 3(x+4)3 2Y= Ax4 + Bx4 + Cx4 + D=3064A+16B+4C=D=30 4The four equations to be used are:1. A+B+C+D=12. 8A+4B+2C+D=53. 27A+9B+3C+D=144. 64A+16B+4C=D=304-3= 37A+7B+C=16 53-2= 19A+5B+C=9 62-1= 7A+3B+C=4 75-6= 18A+2B=7 86-7= 12A+2B=5 98-9= 6A=2=0.33’=13Sub A= 1 in 931 A+2B=53(12×1)+2B=534+2B=52B=5-42B=1B= 12Sub A and D in 77A+3B+C=4(7) (1) + (3) (1) + C=43 27+3+C=43 2C= 4-7-31 3 2C= 4×6 – 7×2 – 3×31 3 2C= 24 – 14 – 96 6 6C= 24-14-96C= 16Sub A,B,C in 1A+B+C+D=11 + 1 + 1 + D=13 2 6D= 1 – 1 – 1 – 13 2 61×6 – 1×2 – 1×3 – 11 3 2 66 – 2 – 3 – 16 6 6 66-2-3-16=0Justifying the formulaX=43 2Ax + Bx + Cx + D3 2 3 2(1) (4) + (1) (4) + (1) (4) + 0 OR 4 + 4 + 43 2 6 3 2 664 + 16 + 4 = 23 2 6 364×2 + 16×3 + 2×23 2 3128 + 48 + 4 + 06 6 6128 + 48 + 4 + 06= 180 = 306Justifying the formula of a higher number3 220 + 20 + 203 2 6=2870 SquaresNumber of squares in a rectangle3x13x1=3(3-1)x(1-1)=0 =33x23x2=6(3-1)x(2-1)= 2×1=2(2-1)x(1-1)=0 =83x33x3=9(3-1)x(3-1)= 2×2=4(2-1)x(1-1)= 1×1=1 =143x43x4=12(3-1)x(4-1)= 3×2=6(2-1)x(3-1)= 1×2=2(1-1)(2-1) =21The sequence number in this arrangement:Stage 1 Stage 2 Stage 3Rectangle: 3×2 3×3 3x4Sequence: 8 14 20+6 +6Stage 1 =8Stage 1= 1×6=6+2 =8Formula is (x6+2)N th rule= nx6+2=6n+2The reason in this that you don’t get square numbers as answers is because the shapes are not square.

Extension 2The cubic formula of type3 2Y= Ax + Bx + Cx + D1. A+B+C+D=22. 8A+4B+2C+D=83.

27A+9B+4C+D=204. 64A+16B+4C+D=404-3= 37A+7B+C=20 53-2= 19A+5B+C=12 62-1= 7A+3B+C=6 75-6= 18A+2B=8 86-7= 12A+2B=6 98-9= 6A=22/6= 0.33’=13Sub A= 1 in 8318A+2B=818Ax 1 +2B=836+2B=82B= 8-62B=2B= 1Sub A ; B in 77A+3B+C=67x 1 + 1×3+C=632 1 A+ 3B+C=636-5= 23C=23Sub A, B ; c in 1A+B+C+D=21 + 1 +2 + D =23 3D=0ConclusionThe results of my investigation has lead me to believe the following conclusion.

As the size of the grid 2×2 increases to 8×8, so does the number of squares. Using my algebraic equation the formula was obtained. When tested against a known number it seemed to be working satisfactory.

EvaluationAt the beginning of my project the work seemed to be very simple but as the project progressed I found it got harder and harder. If I hade more time to do my project I would have continued on my extension and maybe, tried to work out how many squares in a triangle, and a formula for working that out. In this project I have gained much more knowledge about numbers, shapes and creating formulas.