### Use the following rule

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 the top right numbers in the square. Calculate the following difference between these numbers.INVESTIGATE!The first thing I’m going to do is work out the rule for a 10 x 10 grid. To do this I’m going to work out what the difference is between each row using 2 x 2, 3 x 3, 4 x 4, and 5 x 5 grids inside the main 10 x 10 one.10 x 10 grid1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495969798991002 x 2 3 x 323 x 14 = 322 33 x 15 = 49513 x 24 = 312 difference = 10 13 x 35 = 455 difference = 4048 x 39 = 1872 82 x 64 = 524838 x 49 = 1862 difference = 10 62 x 84 = 5208 difference = 4096 x 87 = 8352 88 x 70 = 616086 x 97 = 8342 difference = 10 68 x 90 = 6120 difference = 404 x 4 5 x 541 x 14 = 574 46 x 10 = 46011 x 44 = 484 difference = 90 6 x 50 = 300 difference = 16067 x 40 = 2680 62 x 26 = 161237 x 70 = 2590 difference = 90 22 x 66 = 1452 difference = 16092 x 65 = 5980 95 x 59 = 560562 x 95 = 5890 difference = 90 55 x 99 = 5445 difference = 160I can also use algebra to work out the differences.2 x 2:Xx + 1x + 10x + 11(x + 10)(x + 1) => xï¿½ + 11x + 10( x)(x + 11) => xï¿½ + 11x difference = 103 x 3Xx + 1x + 2x + 10x + 11x + 12x + 20x +21x + 22(x + 20)(x + 2) => xï¿½ + 22x + 40(x)(x + 22) => xï¿½ + 22x difference = 404 x 4Xx + 1x + 2x + 3x + 10x + 11x + 12x + 13x + 20x + 21x + 22x + 23x + 30x + 31x + 32x + 33(x + 30)(x + 3) => xï¿½ + 33x + 90(x)(x + 22) => xï¿½ + 33x difference = 905 x 5Xx + 1x + 2x + 3x + 4x + 10x + 11x + 12x + 13x + 14x + 20x + 21x + 22x + 23x + 24x + 30x + 31x + 32x + 33x + 34x + 40x + 41x + 42x + 43x + 44(x + 40)(x + 4) => xï¿½ + 44x + 160(x)(x +44) => xï¿½ + 44x difference = 160I will now attempt to find the nth term for this.Grid size: 2 ? 2 3 x 3 4 x 4 5 x 510 40 90 160 / / /1st difference: 30 50 70 / /2nd difference: 20 20The first differences are not the same so we have to take a second difference.

The co-efficient of n is half the second difference. So the co-efficient of n is therefore 10 and as we had to take a second difference n will therefore be squared.10nï¿½I now have to take 10nï¿½ away from the difference.E.g. if n is 2 then 10nï¿½ is 40 so if I take 40 away from the difference of the 2 x 2 grid I get the 10nï¿½ part.2 x 2 : difference = 10 n = 210 x 2ï¿½ = 40.

10 – 40 = -303 x 3: difference = 40 n = 310 x 3ï¿½ = 90 40 – 90 = – 504 x 4: difference = 90 n = 410 x 4ï¿½ = 160 90 – 160 = – 705 x 5: difference = 160 n = 510 x 5ï¿½ = 250 160 – 250 = – 90-30 -50 -70 -90 / / /-20 -20 -20The co-efficient of n this time is -20. As I didn’t have to take a second difference n will not be squared.If for example n = 2 to get -20 back up to -30 I will have to add 10 (2 x -20 = -40 + 10 = -30). So I should get 10nï¿½-20n+10. To see if this works I shall have to use a couple of examples:If n = 3 3 x -20 = – 60 + 10 = -50If n = 4 4 x -20 = – 80 + 10 = -70These both abide by the rule so it must work.-20n + 10I will add this to the end of the 10nï¿½to get:10nï¿½ – 20n + 10This factorizes to 10(n-1)ï¿½ therefore the nth term for a 10 x 10 grid is 10(n-1)ï¿½I’m now going to do the same but using a 9 x 9 grid.1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980812 x 2 3 x 320 x 12 = 240 33 x 17 = 56111 x 21 = 231 difference = 9 15 x 35 = 525 difference = 3652 x 44 = 2288 49 x 33 = 161743 x 53 = 2279 difference = 9 31 x 54 = 1581 difference = 3666 x 58 = 3828 79 x 63 = 497757 x 67 = 3819 difference = 9 81 x 61 = 4941 difference = 364 x 4 5 x 574 x 50 = 3700 46 x 14 = 64447 x 77 = 3619 difference = 81 10 x 50 = 500 difference = 14455 x 31 = 1705 50 x 18 = 90028 x 58 = 1624 difference = 81 14 x 54 = 756 difference = 14433 x 9 = 297 76 x 44 = 33446 x 36 = 216 difference = 81 40 x 80 = 3200 difference = 144I can also work these differences out using algebra.2 x 2xx + 1x + 9x + 10(x + 9)(x + 1) => xï¿½ + 10x + 9(x)(x + 10) => xï¿½ + 10x difference = 93 x 3xx + 1x + 2x + 9x + 10x + 11x + 18x + 19x + 20(x + 18)(x + 2) => xï¿½ + 20x + 36(x)(x + 20) => xï¿½ + 20x difference = 364 x 4xx + 1x + 2x + 3x + 9x + 10x + 11x + 12x + 18x + 19x + 20x + 21x + 27x + 28x + 29x + 30(x + 27)(x + 3) => xï¿½+ 30x + 81(x)(x + 30) => xï¿½ + 30x difference = 1445 x 5xx + 1x + 2x + 3x + 4x + 9x + 10x + 11x + 12x + 13x + 18x + 19x + 20x + 21x + 22x + 27x + 28x + 29x + 30x + 31x + 36x + 37x + 38x + 39x + 40(x + 36)(x + 4) => xï¿½ + 40x + 144(x)(x + 40) => xï¿½ + 40x difference = 144I will now attempt to find the nth term for this.

Grid size: 2 ? 2 3 x 3 4 x 4 5 x 59 36 81 144 / / /1st difference: 27 45 63 / /2nd difference: 18 18Again the first differences are not the same so we have to take a second difference. The co-efficient of n is half the second difference. So the co-efficient of n is therefore 9 and as we had to take a second difference n will therefore be squared.9nï¿½I now have to take 9nï¿½ away from the difference.

2 x 2 : difference = 9 n = 29 x 2ï¿½ = 36 9 – 36 = -273 x 3: difference = 36 n = 39 x 3ï¿½ = 81 36 – 81 = – 454 x 4: difference = 81 n = 49 x 4ï¿½ = 144 81 – 144 = – 635 x 5: difference = 144 n = 59 x 5ï¿½ = 225 144 – 225 = – 81-27 -45 -63 -81 / / /-18 -18 -18The co-efficient of n this time is -18. Like before because I haven’t taken a second difference n will not be squared.If for example n = 2 then to get -18 back to -27 you would have to add 9 (2 x -18 = -36 + 9 = -27)To see if this rule works I will have to sub in a couple of examples.If n = 3 3 x -18 = -54 + 9 = -45If n = 4 4 x -18 = -72 + 9 = -63These show that the rule works.-18n + 9I will add this to the end of the 9nï¿½to get:9nï¿½ – 18n + 9This factorizes to 9(n-1)ï¿½ therefore the nth term for a 9 x 9 grid is9(n-1)ï¿½8 x 8 grid123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263642 x 2 3 x 320 x 13 = 260 22 x 8 = 17612 x 21 = 252 difference = 8 6 x 24 = 144 difference = 3238 x 31 = 1178 33 x 19 = 62730 x 39 = 1170 difference = 8 17 x 35 = 595 difference = 3250 x 43 = 2150 62 x 48 = 297642 x 51 = 2142 difference = 8 46 x 64 = 2944 difference = 324 x 4 5 x 529 x 8 = 232 34 x 6 = 2045 x 32 = 160 difference = 72 2 x 38 = 76 difference = 12843 x 22 = 946 60 x 32 = 192019 x 46 = 874 difference = 72 28 x 64 = 1792 difference = 12849 x 28 = 1372 41 x 13 = 53325 x 52 = 1300 difference = 72 9 x 45 = 405 difference = 128I can also work out these differences using algebra.2 x 2xx + 1x + 8x + 9(x + 8)(x + 1) =; xï¿½ + 9x + 8(x)(x + 9) =; xï¿½ + 9x difference = 83 x 3xx + 1x + 2x + 8x + 9x + 10x + 16x + 17x + 18(x + 16)(x + 2) =; xï¿½ + 18x + 32(x)(x + 18) =; xï¿½ + 18x difference = 324 x 4xx + 1x + 2x + 3x + 8x + 9x + 10x + 11x + 16x + 17x + 18x + 19x + 24x + 25x + 26x + 27(x + 24)(x + 3) =; xï¿½ + 27x +72(x)(x + 27) =; xï¿½ + 27x difference = 725 x 5xx + 1x + 2x + 3x + 4x + 8x + 9x + 10x + 11x + 12x + 16x + 17x + 18x + 19x + 20x + 24x + 25x + 26x + 27x + 28x + 32x + 33x + 34x + 35x + 36(x + 32)(x + 4) =; xï¿½ + 36x + 128(x)(x + 36) =; xï¿½ + 36x difference = 128I will now attempt to find the nth term for this.

Grid size: 2 ? 2 3 x 3 4 x 4 5 x 58 32 72 128 / / /1st difference: 24 40 56 / /2nd difference: 16 16Again the first differences are not the same so we have to take a second difference. The co-efficient of n is half the second difference. So the co-efficient of n is therefore 8 and as we had to take a second difference n will therefore be squared.8nï¿½I now have to take 8nï¿½ away from the difference.2 x 2 : difference = 8 n = 28 x 2ï¿½ = 32 8 – 32 = -243 x 3: difference = 32 n = 38 x 3ï¿½ = 72 32 – 62 = -304 x 4: difference = 72 n = 48 x 4ï¿½ = 128 72 – 128 = -565 x 5: difference = 128 n = 58 x 5ï¿½ = 200 128 – 200 = -72-24 -40 -56 -72 / / /-16 -16 -16The co-efficient of n this time is -16. Because I don’t have to take a second difference n will not be squared.

If n = 2 then I will have to add 8 to get back to the original difference of -24 (2 x -16 = -32 + 8 = -24). To see if this rule works I have to sub in two other terms of n.If n = 3 3 x -16 = -48 + 8 = -40If n = 4 4 x -16 = -64 + 8 = -56This shows that this rule works.-16n + 8I will add this to the end of the 8nï¿½to get:8nï¿½ – 16n + 8This factorizes to 8(n-1)ï¿½ therefore the nth term for this is 8(n-1)ï¿½I can also work out the nth term of other grid sizes by using algebra to work out the differences of different size squares on the grid and then finding the nth term in the same way I have for the 10 x 10, 9 x 9 and the 8 x 8 grids.I am going to work out the nth term of a 7 x 7 grid.2 x 2xx + 1x + 7x + 8(x + 7)(x + 1) => xï¿½ + 8x + 7(x)(x + 8) => xï¿½ + 8x difference = 73 x 3xx + 1x + 2x + 7x + 8x + 9x + 14x + 15x + 16(x + 14)(x + 2) => xï¿½ + 16x + 28(x)(x + 16) => xï¿½ + 16x difference = 284 x 4xx + 1x + 2x + 3x + 7x + 8x + 9x + 10x + 14x + 15x + 16x + 17x + 21x + 22x + 23x + 24(x + 21)(x + 3) => xï¿½ + 24x +63(x)(x + 24) => xï¿½ + 24x difference = 635 x 5xx + 1x + 2x + 3x + 4x + 7x + 8x + 9x + 10x + 11x + 14x + 15x + 16x + 17x + 18x + 21x + 22x + 23x + 24x + 25x + 28x + 29x + 30x + 31x + 32(x + 28)(x + 4) => xï¿½ + 32x + 112(x)(x + 32) => xï¿½ + 32x difference = 112Grid size: 2 ? 2 3 x 3 4 x 4 5 x 57 28 63 112 / / /1st difference: 21 35 49 / /2nd difference: 14 14As I had to take a second difference the co-efficient of n will be halved but n will be squared.

This will give us 7nï¿½.Now I have to take away 7nï¿½ from the original difference.2 x 2: difference = 7 n = 27 x 2ï¿½ = 28 7 – 28 = -213 x 3: difference = 28 n = 37 x 3ï¿½ = 63 28 – 63 = -354 x 4: difference = 63 n = 47 x 4ï¿½ = 112 63 – 112 = – 495 x 5: difference = 112 n = 57 x 5ï¿½ = 175 112 – 175 = -63-21 -35 -49 -63 / / /-14 -14 -14The co-efficient of n now is -14 and as we didn’t have to take a second difference n will not be squared.If for example n = 2 to get this difference of -14 back up to the original difference I would of -21 I would have to + 7 ( 2 x – 14 = -28 + 7 = -21)To see if this works and it isn’t just a fluke I will have to use a couple of other examples.

If n = 3 3 x – 14 = -42 + 7 = -35If n = 4 4 x – 14 = -56 + 7 = -49I will now add -14n + 7 to the end of 7nï¿½ to get:7nï¿½- 14n +7This factorizes to 7(n-1)ï¿½, this is the nth term for a 7 x 7 grid.Gridnth term10 x 1010(n-1)ï¿½9 x 99(n-1)ï¿½8 x 88(n-1)ï¿½7 x 77(n-1)ï¿½As they seem to be falling into a pattern we can make a general nth term for all grid sizes. This is:g(n-1)ï¿½g being the grid size.

For example on the 10 x 10 grid g = 10.The rectangles you can have in a 10 x 10 grid are:2 x 3 3 x 6 4 x 10 7 x 82 x 4 3 x 7 5 x 6 7 x 92 x 5 3 x 8 5 x 7 7 x 102 x 6 3 x 9 5 x 8 8 x 92 x 7 3 x 10 5 x 9 8 x 102 x 8 4 x 5 5 x 10 9 x 102 x 9 4 x 6 6 x 72 x 10 4 x 7 6 x 83 x 4 4 x 8 6 x 93 x 5 4 x 9 6 x 10I am going to investigate all the rectangles with five in it rather then changing both side lengths of the rectangle each time.2 x 5 3 x 5 4 x 5 6 x 5 7 x 5 8 x 5 9 x 5 10 x 51234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495969798991002 x 515 19 15 x 29 = 43525 29 25 x 19 = 475 difference = 4054 58 54 x 68 = 367264 68 64 x 58 = 3712 difference = 4085 89 95 x 89 = 845595 99 85 x 99 = 8415 difference = 403 x 52 6 22 x 6 = 13222 26 2 x 26 = 52 difference = 8035 39 55 x 39 = 214555 59 35 x 59 = 2065 difference = 8065 69 85 x 69 = 586585 89 65 x 89 = 5785 difference = 804 x 543 47 73 x 47 = 343173 77 43 x 77 = 3311 difference = 12056 60 86 x 60 = 516086 90 56 x 90 = 5040 difference = 12021 25 51 x 25 = 127551 55 21 x 55 = 1155 difference = 1206 x 534 38 84 x 38 = 319284 88 34 x 88 = 2992 difference = 20042 46 92 x 46 = 423292 96 42 x 96 = 4032 difference = 2006 10 56 x 10 = 56056 60 6 x 60 = 360 difference = 2007 x 531 35 91 x 35 = 318591 95 31 x 95 = 2945 difference = 24015 19 75 x 19 = 142575 79 15 x 79 = 1185 difference = 24024 28 84 x 20 = 235284 88 24 x 88 = 2112 difference = 2408 x 513 17 83 x 17 = 141183 87 13 x 87 = 1131 difference = 28016 20 86 x 20 = 172086 90 16 x 90 = 1440 difference = 2802 6 72 x 6 = 43272 76 2 x 76 = 152 difference = 2809 x 512 16 92 x 16 = 147292 96 12 x 96 = 1152 difference = 32014 18 94 x 18 = 169294 98 14 x 98 = 1372 difference = 3203 7 83 x 7 = 58183 87 3 x 87 = 261 difference = 32010 x 56 10 96 x 10 = 96096 100 6 x 100 = 600 difference = 3601 5 91 x 5 = 45591 95 1 x 95 = 95 difference = 3603 7 93 x 7 = 65193 97 3 x 97 = 291 difference = 360Rectangle: 2 x 5 3 x 5 4 x 5 6 x 5 7 x 5 8 x 5 9 x 5 10 x 540 80 120 200 240 280 320 360* 4 8 12 20 24 28 32 36/ / / / / / / /Multiples: 1 4 2 4 3 4 5 4 6 4 7 4 8 4 9 4*Here I can remove a factor of 10 from each difference.I can now use these to work out a general rule for rectangle grids on a10 x 10 grid.2 x 5 2 = width 5 = length2 – 1 = 1 5 – 1 = 4 1 and 4 are multiples of 4 which is the difference of a 2 x 5 rectangle with a factor of 10 removed.3 x 5 3 = width 5 = length3 – 1 = 2 5 – 1 = 4 2 and 4 are multiples of 8 which is the difference of a 3 x 5 rectangle with a factor of 10 removed.4 x 5 4 = width 5 = length4 – 1 = 3 5 – 1 = 4 3 and 4 are multiples of 12 which is the difference of a 4 x 5 rectangle with a factor of 10 removed.Using these I can assume a general rule for finding the difference:(w-1)(l-1) but because I took a factor of 10 out all this would have to be multiplied by 1010(w-1)(l-1)To show that this works I am going to use rectangles 6 x 5, 7 x 5 and8 x 5.6 x 5 10 x 5 x 4 = 2007 x 5 10 x 6 x 4 = 2408 x 5 10 x 7 x 4 = 280