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# A box is drawn around four numbers. Find the product of the top left number and the bottom right number in this box. Do the same with the top right and bottom left num

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Introduction

Assignment A box is drawn around four numbers. Find the product of the top left number and the bottom right number in this box. Do the same with the top right and bottom left numbers. Calculate the difference between these products. Investigate further 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 The product difference from the two by two square is ten Part of the investigation is to multiply the top left hand corner number with the bottom right hand corner number. ...read more.

Middle

To investigate this theory I am going to increase the size of the square within the grid. Firstly I am going to conduct the same test on a 3x3 grid 67 68 69 77 78 79 87 88 89 67 x 89 = 5963 69 x 87 = 6003 6003 - 5963 = 40 From this and my previous finding I can state that all 3 x 3 squares with a ten by ten grid have a product difference of 40. The next stage of my project was to expand the squares yet again. This time to 4x4. 34 35 36 37 44 45 46 47 54 55 56 57 64 65 66 67 7 8 9 10 17 18 19 20 27 28 29 30 37 38 39 40 37 x 10 = 370 64 x 37 = 2368 40 x 7 = 280 67 x 34 = 2278 = 90 = 90 From this I have concluded that a 4 x 4 square will always have a product outcome of 90. ...read more.

Conclusion

28 19 x 28 = 532 20 x 27 = 540 = product difference = 8 To test for accuracy I will conduct this again. 46 47 54 55 46 x 55 = 2530 47 x 54 = 2538 = product difference = 8 3x3 square 17 18 19 27 28 29 37 38 39 17 x 39 = 663 19 x 37 = 703 product difference of 40 n n+2 n+16 n+18 (n+2(n+16) - n(n+18) N2+ 18n+40 - n�-18n = 40 Difference on a 2x2= 8 =1x8 = n�x8 Difference on a 3x3= 40 =4x8 = n�x8 = N( -1) 10 x 10 square = N (-1) 8x8 square = N (-1) Conclusion From my investigation I have found out that the same formula can be used to find diagonal squares within a larger grid. This formula can stay the same even with varying grid and square sizes. I have used algebraic equations to visually represent my findings and calculations so that the formulas and methods are clearly shown and worked out as I went through. ?? ?? ?? ?? Richard Aldridge ...read more.

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