Number Grid Investigation

I was first given A 10×10 grid, counting from 1-100. Inside the grid was A 2×2 box surrounding the numbers, 12, 13, 22 and 23;123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100I was asked to;* Find the product of the top left number and bottom right number in the box.* Do the same with the top right and bottom left numbers n the box.* Calculate the difference between these numbers.I did this,12 x 23 = 27613 x 22 = 286 286 – 276 = 10The difference between them was 10I decided to try it with A 3×3 box surrounding the numbers;12131422232412 x 34 = 40814 x 32 = 448 448 – 408 = 40The difference between them was 40I also tried this with A 4×4 box, and A 5×5 box:(4×4) 12 x 45 = 54015 x 42 = 630 630 – 540 = 90(5×5) 12 x 56 = 67216 x 52 = 832 832 – 672 = 160I decided to try and find A pattern between these numbers, I thought that since the boxes grows in an even number, so should the totals.Put into order they looked like this:Side of box length 2 3 4 5Difference in numbers 10 40 90 160Difference 30 50 7020 20There was A definite pattern between the numbers, this gave me the idea that I might be able to find A formula for the difference.I thought that if I were to start working out a formula, I should name some of the variables:Side of box = MTop of box = NSide of grid = XBottom of grid = YDifference = LI now had to start by trying to work out the sequence of the numbers:M = 2 3 4 5L = 10 40 90 16030 50 7020 20To work out A sequence, we first see how many times we have to look for an equal difference, in this case we have to go look twice, this means that M will be squared;M2We next look at the difference, in this case 20, we halve it, this will be multiplied by M;M2 x 10This should be the formula, I will test it;42 x 10 = 160This is clearly wrong, as this is the answer for the next part, but I can see what’s wrong, so I can fix it. The problem is that I am going to have to take 1 away from M before it is multiplied by 10 to get the correct formula;(M – 1)2 x 10, or factorised, 10(M – 1)2This will now work, (4 – 1)2 x 10 = 90I have found the formula for any square box, but the box must be square, as in the formula I only have one variable for size of box, and if the two sides are different, the formula will take the side put in to be both sides.This is A problem, I have decided to work on rectangular boxes.Rectangular BoxesWhen working out the rectangular boxes, I must keep one side the same, otherwise I will never find A pattern in totals, so I will work out the following boxes, keeping one side at 2 squares;123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100I will not write here all of my working, as it would take up too much space, so I will just write the results;N x M = L2x2 = 103×2 = 204×2 = 305×2 = 406×2 = 50There is A certain pattern here, as each time we add 1 to N, the difference goes up by 10.Instead of taking M to be the variable in this formula, I will use N as it is the one changing;N –> 2 3 4 5 6L –> 10 20 30 40 5010 10 10 10As I only had to look once for A common difference, N will not have to be squared.The formula turned out like this:10 (N – 1)I tested this for the 7th part of the ‘sequence’,10 (7 – 1) = 60, which, as you can see from the numbers above, is the next part of the sequence, I have found A formula for rectangles where M = 2.