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Number Grid Investigation

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Introduction

Number Grid Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Use the following rule: Find the product of the top left number and the bottom right number in the square. Do the same thing with the bottom left and top right numbers in the square. Calculate the difference between these numbers. Investigate. Method First I found the product of the top left and bottom right numbers: 23*34=782 Then I found the product of ...read more.

Middle

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 64*86=5504 66*84=5544 5544-5504=40 I tried this again with this square. 48 49 50 58 59 60 68 69 70 48*70=3360 50*68=3400 3400-3360=40 I then worked out what the difference was by using the method I used before: X X+2 X+20 X+22 ((X+2)(X+20))-(X(X+22))=Difference (X�+20X+2X+40)-(X�+X22)=Difference (X�+22X+40)-(X�+X22)=Difference Difference=40 This shows that the difference of a 3 by 3 square is always 40 on a grid with ten squares in each row. ...read more.

Conclusion

Difference=10(2*3) Difference=60 10*29=290 13*26=338 339-290=49 X X+(W-1) X+10(H-1) X+10(H-1) +(W-1) If the grid is 8 squares long you would have to go 8 squares along from a square to get to the one beneath it. This means the number would be greater by 8. This shows that if the grid was 8 squares along the top, the formula for the bottom left square in the rectangle would be X+8(H-1). I used this bit of information to try working out the formula the same way as before only replacing the 10's with L's to represent the amount of squares the grid length is. X X+(W-1) X+L(H-1) X+L(H-1) +(W-1) Difference=(X+(W-1))*(X+L(H-1)) - (X(X+L(H-1)+(W-1)) Difference=X�+X(L(H-1))+X(W-1)+(W-1)L(H-1) - X�+X(L(H-1))+X(W-1) Difference=L(H-1)(W-1) This is how I tested the formula: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 Difference=L(H-1)(W-1) Difference=6(2*4) Difference=48 13*29=377 17*25=425 425-377=48 Difference=48 The final formula I got was Difference=L(H-1)(W-1) ...read more.

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