56x74 = 4144 difference = 40
X = N x (N x 22) = (N x N) + (N x 22) = N² + 22n
Y = (N + 2) x (N + 20) = (N x N) + (N x 20) + (N x 2) + (2 x 20) = N² + 22N + 40
4 x 4
15x48 = 720
18x45 = 810 difference = 90
61x95 = 5795
64x91 = 5824 difference = 90
57x90 = 5130
60x87 = 5228 difference = 90
X = N x (N + 33) = (N x N) + (N x 33) = N² + 33N
Y = (N+3) x (N+30) = (N x N) + (N x 30) + (N x 3) + (3 x 30) = N² + 33N + 90
5 x 5
1x45 = 45
5x41 = 205 difference = 160
6x50 = 300
10x46 = 460 difference = 160
51x95 = 4845
55x91 = 5005 difference = 160
X = N x (N + 44) = (N x N) + (N x 44) = N² + 44N
Y = (N+4) x (N+40) = (N x N) + (N x 40) + (N x 4) + (4 x 40) = N² + 44n + 160
-- I will now attempt to work out the difference of a grid (6 x 6) without doing the multiplication before hand to test my equation--
X = N x (N + 55) = (N x N) + (N x 55) = N² + 55N
Y = (N + 5) x (N + 50) = (N x N) + (N x 50) + (N x 5) + (5 x 50) = N² + 55n + 250
From the equation I predict that the difference in the 6x6 grid will be 250
6x6
33 x 88 = 2904
38 x 83 = 3154 difference = 250
14 x 69 = 966
19 x 64 = 1216 difference = 250
Below I have worked out the Nth term for all the grid sizes (2x2 – 5x5)
11 x 9
2 x 2
24x36 = 864
25x35 = 875 difference = 11
60x72 = 4320
61x71 = 4331 difference = 11
21x33 = 693
22x32 = 704 difference = 11
X = N x (N + 12) = (N x N) + (N x 12) = N² + 12N
Y = (N+1) x (N + 11) = (N x N) + (N x 11) + (N x1) + (1 x 11) = N² + 12N + 11
3 x 3
27x51 = 1377
29x49 = 1421 difference = 44
67x91 = 6097
69x89 = 6141 difference = 44
X = N x (N + 24) = (N x N) + (N x 24) = N² + 24N
Y = (N + 2) x (N + 22) = (N x N) + (N x 22) + (N x 2) + (2 x 22) = N² + 24N + 44
4 x 4
14x50 = 700
17x47 = 799 difference = 99
51x87 = 4437
54x84 = 4536 difference = 99
X = N x (N + 36) = (N x N) + (N x 36) = N² + 36N
Y = (N + 3) + (N + 33) = (N x N) + (N x 33) + (N x 3) + (3 x 33) = N² + 36N + 99
5 x 5
17x65 = 1105
21x61 = 1281 difference = 176
47x95 = 4465
51x91 = 4641 difference = 176
X = N x ( N + 48) = (N x N) + (N x 48) = N ² + 48N
Y = (N + 4) x (N + 44) = (N x N) + (N x 44) + (N x 4) + (4 x 44) = N² + 48N + 176
-- I will now attempt to work out the difference of a grid (8 x 8) without doing the multiplication before hand to test my equation--
X = N x (N + 71) = (N x N) + (N x 71) = N² + 71N
Y = (N + 5) x (N + 55) = (N x N) + (N x 55) + (N x 5) + (6 x 55) = N² + 71N +275
From the equation I predict that the difference in the 8x8 grid will be 275
25x85 = 2125
30x80 = 2400 difference = 275
21x71 = 1491
16x76 = 1216 difference = 275
Below I have worked out the Nth term for all the grid sizes (2x2 – 5x5)
8 x 9
2 x 2
19x28 = 532
20x27 = 540 difference = 8
52x61 = 3172
53x60 = 3180 difference = 8
X = N x (N + 9) = (N x N) + (N x 9) = N² + 9N
Y = (N + 1) x (N + 8) = (N x N) + (N x 8) + (N x 1) + (8 x 1) = N² + 9N + 8
3 x 3
42x60 = 2520
44x58 = 2552 difference = 32
21x39 = 819
23x37 = 851 difference = 32
X = N x (N + 18) = (N x N) + (N x 18) = N² + 18N
Y = (N + 2) x (N + 16) = (N x N) + (N x 16) + (N x 2) + (16 x 2) = N² + 18N + 32
4 x 4
19x46=874
22x43=946 Difference = 72
34x61=2074
37x58=2146 Difference = 72
X = N x (N+27) = (N x N) + (N x 27) = N² + 27N
Y = (N + 3) x (N + 24) = (N x N) + (N x 24) + (N x 3) + (3 x 24) = N² + 27N + 72
-- I will now attempt to work out the difference of a grid (5 x 5) without doing the multiplication before hand to test my equation--
X = N x (N + 36) = (N x N) + (N x 36) = N² + 36N
Y = (N + 4) x (N + 32) = (N x N) + (N x 32) + (N x 4) + (4 x 32) = N² + 36N + 128
From the equation I predict that the difference in the 5x5 grid will be 128
18 x 54= 972
22 x 50= 1100 Difference = 128
26 x 62=1612
30 x 58=1740 Difference = 128
Below I have worked out the Nth term for all the grid sizes (2x2 – 5x5)
The rectangles you can have in a 10 x 10 grid are:
2 x 3 3 x 6 4 x 10 7 x 8
2 x 4 3 x 7 5 x 6 7 x 9
2 x 5 3 x 8 5 x 7 7 x 10
2 x 6 3 x 9 5 x 8 8 x 9
2 x 7 3 x 10 5 x 9 8 x 10
2 x 8 4 x 5 5 x 10 9 x 10
2 x 9 4 x 6 6 x 7
2 x 10 4 x 7 6 x 8
3 x 4 4 x 8 6 x 9
3 x 5 4 x 9 6 x 10
--I am going to investigate further by working out the differences of rectangles with 3 in them (only the ones which are blue)--
10 x 10
2 x 3
12x24 = 288
14x22 = 308 difference = 20
45x57 = 2565
47x55 = 2585 difference = 20
X = N x (N + 12) = (N x N) + (N x 12) = N² + 12N
Y = (N + 2) x (N + 10) = (N x N) + (N x 10) + (N x 2) + (2 x 10) = N² + 12N + 20
3 x 4
13x36 = 468
16x33 = 528 difference = 60
53x76 = 4028
56x73 = 4088 difference = 60
X = N x (N + 23) = (N x N) + (N x 23) = N² + 23N
Y = (N + 3) + (N + 20) = (N x N) + (N x 20) + (N x 3) + (3 x 20) = N² + 23N + 60
3 x 5
23x47 = 1081
27x43 = 1161 difference = 80
14x38 = 532
18x34 = 612 difference = 80
X = N x (N + 24) = (N x N) + (N x 24) = N² + 24N
Y = (N + 4) + (N + 20) = (N x N) + (N x 20) + (N x 4) + (4 x 20) = N² + 34N + 80
3 x 6
44x69 = 3036
49x64 = 3136 difference = 100
22x47 = 1034
27x42 = 1134 difference = 100
X = N x (N + 25) = (N x N) + (N x 25) = N² + 25N
Y = (N + 5) + (N + 20) = (N x N) + (N x 20) + (N x 5) + (4 x 20) = N² + 25N + 100
3 x 7
13x39 = 507
19x33 = 627 difference = 120
43x69 = 2967
49x63 = 3087 difference = 120
X= N x (N + 26) = (N x N) + (N x 26) = N² + 26N
Y = (N + 6) + (N + 20) = (N x N) + (N x 20) + (N x 6) + (6 x 20) = N² + 26N + 120
-- I will now attempt to work out the difference of a grid (3 x 8) without doing the multiplication before hand to test my equation--
X = N x (N + 27) = (N x N) + (N x 27) = N² + 27N
Y = (N + 7) + (N + 20) = (N x N) + (N x 20) + (N x 7) + (7 x 20) = N² + 27N + 140
From the equation I predict that the difference in the 3x8 grid will be 140
3 x 8
12x39 = 468
19x32 = 608 difference = 140
42x69 = 2898
49x62 = 3038 difference = 140
Below I have worked out the Nth term for all the grid sizes (3x4 – 3x7)
I can now use these to work out a general rule for rectangle grids on a
10 x 10 Grid.
I have worked out a rule for the rectangular grid which seems to work
10 x (L – 1) (W - 1)
Zuhaib Ahmad
