Task B: Diagonal Difference
Name: Terry Curtis Centre Name: Gaynes School
Form: 11s2 Centre Number: 12847
Candidate Number: 7045 Teacher Name: Mr Nash
Index Page
Page 1: Title Page
Page 2: Index Page
Page 3: Statement
Page 4: Initial Investigation, 3 x 3 grids inside a 8 x 8 grid
Page 5: Initial Investigation, 4 x 4 grids inside a 8 x 8 grid
Page 6: Initial Investigation, 5 x 5 grids inside a 8 x 8 grid
Page 7: Initial Investigation, 2 x 2 grids inside a 8 x 8 grid
Page 8: Justifying My Results, form for a 3 x 3 and 4 x 4 grid inside a 8 x 8 gird
Page 9: Justifying My Results, form for a 2 x 2 and 5 x 5 gird inside a 8 x 8 grid
Page 10: Justifying My Results, prediction for a 6 x 6 grid inside a 8 x 8 grid
Page 11: Justifying My Results, justifying my prediction
Page 12: Justifying My Results, nth term found for the results of a 8 x 8 grid
Page 13: Justifying My Results, formula found for any grid inside a 8 x 8 gird
Page 14: Justifying My Results, Formula tested on previous problems
Page 15: Further Investigation, 6 x 6 grid
Page 16: Further Investigation, 6 x 6 grid
Page 17: Justifying My Results, formula found for any gird inside 6 x 6 grid
Page 18: Justifying My Results, formula tested for any grid inside a 6 x 6 grid
Page 19: Further Investigation, 7 x 7 grid
Page 20: Further Investigation, 7 x 7 grid
Page 21: Justifying My Results, formula found for any gird inside 7 x 7 grid
Page 22: Justifying My Results, formula tested for any grid inside a 7 x 7 grid
Page 23: Further Investigation, formula found for any grid inside any grid
Page 24: Justifying My Results, formula tested on previous grids of all sizes
Page 25 : Justifying My Results, formula tested on grids of all sizes not yet done
Page 26: Justifying My Results, formula tested on grids of all sizes not yet done
Page 27: Further Investigation, formula found for any rectangle grid inside any square grid.
Page 28: Justifying My Results, formula tested on grids of all sizes not yet done
Page 29: Justifying My Results, formula tested on grids of all sizes not yet done
Page 30: Conclusion
Statement
I have been told to consider the following table of numbers:
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
I was then told that Sarah had picked a 3 x 3 grid from the above table and wrote it below:
0
1
2
8
9
20
26
27
28
From this 3 x 3 grid inside the 8 x 8 grid, Sarah has noticed that when you multiply the opposite corners the difference between the products is 32.
For example:
10 x 28 = 280
12 x 26 = 312
The Diagonal Difference is:
312 - 280 = 32
I have now been asked to investigate the diagonal difference for other 3 x 3 grids and to investigate further.
Initial Investigation
8 x 8 grid:
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
Four 3 x 3 grids:
2
3
9
0
1
7
8
9
6
7
8
4
5
6
22
23
24
1 x 19 = 19 6 x 24 = 144
3 x 17 = 51 8 x 22 = 176
51 - 19 = 32 176 - 144 = 32
41
42
43
49
50
51
57
58
59
46
47
48
54
55
56
62
63
64
41 x 59 = 2419 46 x 64 = 2944
43 x 57 = 2451 48 x 62 = 2976
2419 - 2451 = 32 2976 - 2944 = 32
Sarah's theory is justified, when any 3 x 3 grid consist with in a 8 x 8 grid , the difference between the products of the two opposite corners multiplied is 32.
Initial Investigation
I will now investigate weather or not the number is the same when I multiply the opposite corners and find the difference of the products within a 4 x 4 grid, which consist inside a 8 x 8 grid.
8 x 8 grid:
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
4 x 4 grids:
2
3
4
9
0
1
2
7
8
9
20
25
26
27
28
5
6
7
Name: Terry Curtis Centre Name: Gaynes School
Form: 11s2 Centre Number: 12847
Candidate Number: 7045 Teacher Name: Mr Nash
Index Page
Page 1: Title Page
Page 2: Index Page
Page 3: Statement
Page 4: Initial Investigation, 3 x 3 grids inside a 8 x 8 grid
Page 5: Initial Investigation, 4 x 4 grids inside a 8 x 8 grid
Page 6: Initial Investigation, 5 x 5 grids inside a 8 x 8 grid
Page 7: Initial Investigation, 2 x 2 grids inside a 8 x 8 grid
Page 8: Justifying My Results, form for a 3 x 3 and 4 x 4 grid inside a 8 x 8 gird
Page 9: Justifying My Results, form for a 2 x 2 and 5 x 5 gird inside a 8 x 8 grid
Page 10: Justifying My Results, prediction for a 6 x 6 grid inside a 8 x 8 grid
Page 11: Justifying My Results, justifying my prediction
Page 12: Justifying My Results, nth term found for the results of a 8 x 8 grid
Page 13: Justifying My Results, formula found for any grid inside a 8 x 8 gird
Page 14: Justifying My Results, Formula tested on previous problems
Page 15: Further Investigation, 6 x 6 grid
Page 16: Further Investigation, 6 x 6 grid
Page 17: Justifying My Results, formula found for any gird inside 6 x 6 grid
Page 18: Justifying My Results, formula tested for any grid inside a 6 x 6 grid
Page 19: Further Investigation, 7 x 7 grid
Page 20: Further Investigation, 7 x 7 grid
Page 21: Justifying My Results, formula found for any gird inside 7 x 7 grid
Page 22: Justifying My Results, formula tested for any grid inside a 7 x 7 grid
Page 23: Further Investigation, formula found for any grid inside any grid
Page 24: Justifying My Results, formula tested on previous grids of all sizes
Page 25 : Justifying My Results, formula tested on grids of all sizes not yet done
Page 26: Justifying My Results, formula tested on grids of all sizes not yet done
Page 27: Further Investigation, formula found for any rectangle grid inside any square grid.
Page 28: Justifying My Results, formula tested on grids of all sizes not yet done
Page 29: Justifying My Results, formula tested on grids of all sizes not yet done
Page 30: Conclusion
Statement
I have been told to consider the following table of numbers:
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
I was then told that Sarah had picked a 3 x 3 grid from the above table and wrote it below:
0
1
2
8
9
20
26
27
28
From this 3 x 3 grid inside the 8 x 8 grid, Sarah has noticed that when you multiply the opposite corners the difference between the products is 32.
For example:
10 x 28 = 280
12 x 26 = 312
The Diagonal Difference is:
312 - 280 = 32
I have now been asked to investigate the diagonal difference for other 3 x 3 grids and to investigate further.
Initial Investigation
8 x 8 grid:
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
Four 3 x 3 grids:
2
3
9
0
1
7
8
9
6
7
8
4
5
6
22
23
24
1 x 19 = 19 6 x 24 = 144
3 x 17 = 51 8 x 22 = 176
51 - 19 = 32 176 - 144 = 32
41
42
43
49
50
51
57
58
59
46
47
48
54
55
56
62
63
64
41 x 59 = 2419 46 x 64 = 2944
43 x 57 = 2451 48 x 62 = 2976
2419 - 2451 = 32 2976 - 2944 = 32
Sarah's theory is justified, when any 3 x 3 grid consist with in a 8 x 8 grid , the difference between the products of the two opposite corners multiplied is 32.
Initial Investigation
I will now investigate weather or not the number is the same when I multiply the opposite corners and find the difference of the products within a 4 x 4 grid, which consist inside a 8 x 8 grid.
8 x 8 grid:
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
4 x 4 grids:
2
3
4
9
0
1
2
7
8
9
20
25
26
27
28
5
6
7
