Number Grids Coursework (MA1)
Experiment 1
Part 1
Grid Size = 10*10
Box Size = 2*2
Experiment
)
2 * 23 = 276
3 * 22 = 286
286 - 276 = 10
2)
9 * 20 = 180
0 * 19 =190
90 - 180 = 10
3)
89 * 100 = 8900
90 * 99 = 8910
8910 - 8900 = 10
4)
* 12 =12
2 * 11 = 22
22 - 12= 10
5)
81 * 92 = 7452
91 * 82 = 7462
7462 - 7452 = 10
Table Of Results
Equation
Sum A Answer
Sum B Answer
Difference
276
286
0
2
80
90
0
3
8900
8910
0
4
2
22
0
5
7452
7462
0
Conclusion
Whatever the answers are of the two sums the difference between the difference between them is 10.
X
X+1
X+10
X+11
This box is how the numbers lay out in the box of a 2*2 box
The algebra to explain this is below;
(x+10) (x+1) - (x) (x+11) = d (difference)
Therefore x squared +10x + 10 + x - x squared + 11x = d
Therefore 10 + x - x = d
Therefore d=10
Prediction
43
44
53
54
I predict when multiplied in the same way the difference between the two sums will be 10, I will now test this to check that I am correct;
Prediction)
43* 54 = 2322
44 * 53 = 2332
2332 - 2232 = 10 I Was Correct
Experiment 1
Part 2
Grid Size = 10*10
Box Size = 3*3
Experiment
)
* 23 = 23
3 * 21 = 63
63 - 23 = 40
2)
8 * 30 = 240
0* 28 = 280
280 - 240 = 40
3)
71 * 93 = 6603
73 * 91 = 6643
6643 - 6603 = 40
4)
78 * 100 = 7800
80 * 98 = 7840
7840 - 7800 = 40
5)
45 * 67 = 3015
47 * 65 = 3055
3055 - 3015 = 45
Table Of Results
Equation
Sum A Answer
Sum B Answer
Difference
23
63
40
2
240
280
40
3
6603
6643
40
4
7800
7840
40
5
3015
3055
40
Conclusion
Whatever the answers are of the two sums the difference between the difference between them is 40.
X
X+2
X+20
X+22
This box is how the numbers lay out in the box of a 3*3 box
The algebra to explain this is below;
(x+20) (x+2) - (x) (x+22) = d (difference)
Therefore x squared + 2x + 40 + 20x - x squared + 22x = d
Therefore x squared + 22x + 40 - x squared + 22x = d
Therefore 40 + 22x - 22x = d
Therefore d=40
Prediction
(NB: These Are ONLY The Numbers In The Corners)
4
6
24
26
I predict when multiplied in the same way the difference between the two sums will be 40, I will now test this to check that I am correct;
Prediction)
4* 26 = 104
6 * 25 = 144
44 -104 = 40 I Was Correct
Experiment 1
Part 3
Grid Size = 10*10
Box Size = 4*4
Experiment
)
* 34 = 34
4 * 34 = 124
24 - 34 = 90
2)
7 * 40 = 280
0 * 37 = 370
370 - 280 = 90
3)
61 * 94 = 5734
91 * 64 = 5824
5824 - 5734 = 90
4)
67 * 100 = 6700
97 * 70 = 6790
6790 - 6700 = 90
5)
34 * ...
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Part 3
Grid Size = 10*10
Box Size = 4*4
Experiment
)
* 34 = 34
4 * 34 = 124
24 - 34 = 90
2)
7 * 40 = 280
0 * 37 = 370
370 - 280 = 90
3)
61 * 94 = 5734
91 * 64 = 5824
5824 - 5734 = 90
4)
67 * 100 = 6700
97 * 70 = 6790
6790 - 6700 = 90
5)
34 * 67 = 2278
64 * 37 = 2368
2368 - 2278 = 90
Table Of Results
Equation
Sum A Answer
Sum B Answer
Difference
34
24
90
2
280
370
90
3
5734
5824
90
4
6700
6790
90
5
2278
2368
90
Conclusion
Whatever the answers are of the two sums the difference between the difference between them is 90.
X
X+3
X+30
X+33
This box is how the numbers lay out in the box of a 4*4 box
The algebra to explain this is below;
(x+30) (x+3) - (x) (x+33) = d (difference)
Therefore x squared + 3x + 90 + 30x - x squared + 33x = d
Therefore x squared + 33x + 90 - x squared + 33x = d
Therefore 90 + 33x - 33x = d
Therefore d=90
Prediction
(NB: These Are ONLY The Numbers In The Corners)
64
67
94
97
I predict when multiplied in the same way the difference between the two sums will be 90, I will now test this to check that I am correct;
Prediction)
64 * 97 = 6208
67 * 94 = 6298
6297 - 6208 = 90 I Was Correct
Experiment 1
Part 4
Grid Size = 10*10
Box Size = 5*5
Experiment
)
* 45 = 45
5 * 41 = 205
205 - 45 = 160
2)
56 * 100 = 5600
60 * 96 = 5760
5760 - 5600 = 160
3)
51 * 95 = 4845
55 * 91 = 5005
5005 - 4845 = 160
4)
6 * 50 = 300
0 * 46 = 460
460 - 300 = 160
5)
23 * 67 = 1541
27 * 63 = 1701
701 - 1541 = 160
Table Of Results
Equation
Sum A Answer
Sum B Answer
Difference
45
205
60
2
5600
5760
60
3
4845
5005
60
4
300
460
60
5
541
701
60
Conclusion
Whatever the answers are of the two sums the difference between the difference between them is 160.
X
X+4
X+40
X+44
This box is how the numbers lay out in the box of a 5*5 box
The algebra to explain this is below;
(x+40) (x+4) - (x) (x+44) = d (difference)
Therefore x squared + 4x + 160 + 40x - x squared + 44x = d
Therefore x squared + 44x + 160 - x squared + 44x = d
Therefore 160 + 44x - 44x = d
Therefore d = 160
Prediction
(NB: These Are ONLY The Numbers In The Corners)
43
47
83
87
I predict when multiplied in the same way the difference between the two sums will be 160, I will now test this to check that I am correct;
Prediction)
43 * 87 = 3741
47 * 83 = 3901
3901 - 3741 = 160 I Was Correct
Experiment 1
Part 5
Grid Size = 10*10
Box Size = 6*6
Experiment
)
* 56 = 56
6 * 51 = 306
306 - 56 = 250
2)
45 * 100 = 4500
50 * 95 = 4750
4750 - 4500 = 250
3)
5 * 60 = 300
0 * 55 = 550
550 - 300 = 250
4)
41 * 96 = 3936
46 * 91 = 4186
4186 - 3936 = 250
5)
31 * 86 = 2666
36 * 81 = 2916
2916 - 2666 = 250
Table Of Results
Equation
Sum A Answer
Sum B Answer
Difference
56
306
250
2
4500
4750
250
3
300
550
250
4
3936
4186
250
5
2666
2916
250
Conclusion
Whatever the answers are of the two sums the difference between the difference between them is 250.
X
X+5
X+50
X+55
This box is how the numbers lay out in the box of a 6*6 box
The algebra to explain this is below;
(x+50) (x+5) - (x) (x+55) = d (difference)
Therefore x squared + 5x + 250 + 50x - x squared + 55x = d
Therefore x squared + 55x + 250 - x squared + 55x = d
Therefore 250 + 55x - 55x = d
Therefore d = 250
Prediction
(NB: These Are ONLY The Numbers In The Corners)
43
48
83
88
I predict when multiplied in the same way the difference between the two sums will be 160, I will now test this to check that I am correct;
Prediction)
43 * 88 = 3784
48 * 83 = 3984
3984 - 3784 = 250 I Was Correct
OVERALL Conclusion
After doing the last five parts of this experiment I have found that if you times the difference by two and plus ten you get the next difference except on the first number where you must ad the difference, times it by two and add ten to the difference. An algebra equation, which explains this, is shown below;
0 (G - 1) = D
Therefore G = D
(NB G = Grid And D = Difference)
I will now vary the size of the grid and see how this changes the results
Experiment 2
Grid Size = 11*11
Box Size = 2*2
Experiment
)
* 13 = 13
2 * 12 = 24
24 - 13 = 11
2)
0 * 22 = 220
1 * 21 = 231
231 - 220 = 11
3)
00 * 112 = 11,200
01 * 111 = 11,211
1,211 - 11,200 = 11
4)
09 * 121 = 13,189
10 * 120 = 13,200
3,200 - 13,189 = 11
5)
60 * 72 = 4320
61 * 71 = 4331
4331 - 4320 = 11
Table Of Results
Equation
Sum A Answer
Sum B Answer
Difference
3
24
1
2
220
231
1
3
1,200
1,211
1
4
3,189
3,200
1
5
4320
4331
1
Conclusion
Whatever the answers are of the two sums the difference between the difference between them is 11.
X
X+1
X+11
X+12
This box is how the numbers lay out in the box of a 2*2 box in a 11*11 grid, the algebra to explain this is below;
(x+1) (x+11) - (x) (x+12) = d (difference)
Therefore x squared + x + 11 + 11x - x squared + 12x = d
Therefore x squared + 12x + 11 - x squared + 12x = d
Therefore 11 + 12x - 12x = d
Therefore d = 11
Prediction
86
87
97
98
I predict when multiplied in the same way the difference between the two sums will be 11, I will now test this to check that I am correct;
Prediction)
86 * 98 = 8428
87 * 97 = 8439
8439 - 8428 = 11 I Was Correct
Experiment 3
Grid Size = 9*9
Box Size = 2*2
Experiment
)
* 11 = 11
2 * 10 = 20
20 - 11 = 9
2)
8 * 18 = 142
9 * 17 = 153
53 - 142 = 9
3)
71 * 81 = 5751
72 * 80 = 5760
5760 - 5751 = 9
4)
64 * 74 = 4736
65 * 73 = 4745
4745 - 4736 = 9
5)
30 * 40 = 1200
31 * 39 = 1209
209 - 1200 = 9
Table Of Results
Equation
Sum A Answer
Sum B Answer
Difference
1
20
9
2
42
53
9
3
5751
5760
9
4
4736
4745
9
5
200
209
9
Conclusion
Whatever the answers are of the two sums the difference between the difference between them is 9.
X
X+1
X+9
X+10
This box is how the numbers lay out in the box of a 2*2 box in a 9*9 box, the algebra to explain this is below;
(x+1) (x+9) - (x) (x+10) = d (difference)
Therefore x squared + x + 9 + 9x - x squared + 10x = d
Therefore x squared + 10x + 9 - x squared + 10x = d
Therefore 9 + 10x - 10x = d
Therefore d = 9
Prediction
37
38
46
47
I predict when multiplied in the same way the difference between the two sums will be 9, I will now test this to check that I am correct;
Prediction)
37 * 47 = 1739
38 * 46 = 1748
739 - 1748 = 9 I Was Correct
Experiment 4
Grid Size 8*8
Box Size 2*2
Experiment
)
* 10 = 10
2 * 9 = 18
8 - 10 = 8
2)
8 * 15 = 110
7 * 16 = 118
18 - 110 = 8
3)
49 * 58 = 2842
50 * 57 = 2850
2850 - 2842 = 8
4)
55 * 64 = 3520
56 * 63 = 3528
3528 - 3520 = 8
5)
28 * 37 = 1036
29 * 36 = 1044
044 - 1036 = 8
Table Of Results
Equation
Sum A Answer
Sum B Answer
Difference
0
8
8
2
10
18
8
3
2842
2850
8
4
3520
3528
8
5
036
044
8
Conclusion
Whatever the answers are of the two sums the difference between the difference between them is 8.
X
X+1
X+8
X+9
This box is how the numbers lay out in the box of a 2*2 box in a 9*9 box, the algebra to explain this is below;
(x+1) (x+8) - (x) (x+9) = d (difference)
Therefore x squared + x + 8 + 8x - x squared + 9x = d
Therefore x squared + 9x + 8 - x squared + 9x = d
Therefore 9 + 9x - 9x = d
Therefore d = 8
Prediction
1
2
9
20
I predict when multiplied in the same way the difference between the two sums will be 8, I will now test this to check that I am correct;
Prediction)
1 * 20 = 220
2 * 19 = 228
228 -220 = 8 I Was Correct
Experiment 5
Grid Size 7*7
Box Size 2*2
Experiment
)
* 9 = 9
2 * 8 = 16
6 - 9 = 7
2)
6 * 14 = 84
7 * 13 = 91
91 - 84 = 7
3)
36 * 44 = 1584
37 * 43 = 1591
591 - 1584 = 7
4)
41 * 49 = 2009
42 * 48 = 2016
2016 - 2009 = 7
5)
7 * 25 = 425
8 * 24 = 432
432 - 425 = 7
Table Of Results
Equation
Sum A Answer
Sum B Answer
Difference
9
6
7
2
84
91
7
3
584
591
7
4
2009
2016
7
5
425
432
7
Conclusion
Whatever the answers are of the two sums the difference between the difference between them is 7.
X
X+1
X+7
X+8
This box is how the numbers lay out in the box of a 2*2 box in a 9*9 box, the algebra to explain this is below;
(x+1) (x+7) - (x) (x+7) = d (difference)
Therefore x squared + x + 8 + 7x - x squared + 8x = d
Therefore x squared + 8x + 8 - x squared + 8x = d
Therefore 8 + 8x - 8x = d
Therefore d = 8
Prediction
7
8
24
25
I predict when multiplied in the same way the difference between the two sums will be 7, I will now test this to check that I am correct;
Prediction)
7 * 25 = 425
8 * 24 = 432
425 -432 = 7 I Was Correct
FINAL Conclusion
After investigation I have discovered that as the grid size in/decreases the difference of a 2*2 box will go up or down to match the size of the grid on one size (eg a 10*10 grid has a difference of ten when a 2*2 box inside the grid is multiplied)
The algebra below explains this;
G (( The Square Root Of B) - 1) Squared = D
SO
(The Square Root Of G) * ((The Square Root Of B) - 1 ) Squared = D
SO
0 * (2 - 1 Squared) = D
Therefore 10 * 1 Squared = D
Therefore 10 * 1 = D
Therefore D = 10
Lee Eastwood 10Q Coursework