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
Now I will start the investigation
There are a large number of possible starting points in 10x10 square but I will start with the smallest square possible and make my way up so that the investigation does not get too complicated.
2x2 square
n n+1 n(n+11) - [(n+1) (n+10)]
1 2
n2 +11n - n2 +10n + n+10
11 12
n+10 n+11 n2 +11n – n2 +11n +10
11 x 2 = 22 10
'font-size:14.0pt; '>12 x 1 = 12
'font-size:14.0pt; '>22 – 12 = 10
In both ways the difference is 10
3x3 square
n n+2 n(n+22) – [(n+2) (n+20)]
1 2 3
11 12 13 n2+22n - n2+20n +2n +40
21 22 23
n+20 n+22 n2 +22n - n2 +22n +40
21 x 3 = 63 40
23 x 1 = 23
63 – 23 = 40
In both ways the difference is 40
4x4 square
n n+2 n(n+33) - [(n+3) (n+30)]
1 2 3 4
11 12 13 14 n2+33n - n2+30n +3n +90
21 22 23 24
31 32 33 34 n2+33n - n2+33n + 90
n+30 n+33 10
31 x 4 = 124 In both ways the difference is 90
34 x 1 = 31
124 – 34 = 90
5x5 square
n n+4 n(n+44) - [(n+4) (n+40)]
1 2 3 4 5
11 12 13 14 15 n2+44n - n2+40n + 4n +160
21 22 23 24 25
31 32 33 34 35 n2+44n - n2+44n +160
41 42 43 44 45
n+40 n+44 160
41 x 5 = 205
45 x 1 = 45
205 – 45 = 160
In both ways the difference is 160
6x6 square
n n+5 n(n+55) - [(n+5) (n+50)
1 2 3 4 5 6
11 12 13 14 15 16 n2+55n - n2+50n + 5n + 250
21 22 23 24 25 26
31 32 33 34 35 36 n2+55n - n2+55n + 250
41 42 43 44 45 46
51 52 53 54 55 56 250
n+50 n+55
51 x 6 = 306
56 x 1 = 56
306 – 56 = 250
In both ways the difference is 250
