- 20x5=100
- 21x5=105
- 22x5=110
- 23x5=115
The multiplied versions of the T-numbers go up by five each time as well as the T-totals, this seems to suggest that the formula will be derived from these two parts, because the difference between them on this grid will never be different. So if we subtract the T-totals from the multiplied T-numbers and then add the result to the formula, we should have the correct formula.
- 100-37=63
- 105-42=63
- 110-47=63
- 115-52=63.
The answers are all 63, another constant in the grid. I now predict that the formula for finding the T-totals from the T-numbers will be 5N-63. I will test this out with the highlighted shapes on the grid below.
I haven’t chosen T-shapes close to each other in case of the eventuality that the formula would only work on the first line. First I will work out if the formula works on the blue T-shape.
5N-63; 21x5=105 105-63=42 2++3+4+12+21=42.
The formula works for the blue T-shape, if the formula works for the other highlighted T-shapes in this grid I will have found the formula for the 9x9 grid.
Red T-shape; 5N-63 43x5=215 215-63=152 24+25+26+34+43=152
Green T-shape; 5N-63 47x5=235 235-63=172 28+29+30+38+47=172
Yellow T-shape; 5n-63 79x5=395 395-63=332 60+61+62+70+79= 332
All the formulae were correct; the formula for the 9x9 grid is 5N-63. Now I will find the formula of other grid sizes and use that information to find a formula, which fits all the grids. I will begin with a 10x10 grid, as it is only one step up from a 9x9 grid and the formula shouldn’t be that different.
I will apply the same method to this grid, I will find the difference between each T-total and apply that to the beginning of the formula, then times the T-numbers by this amount and subtract the T-totals from the multiplied t-numbers.
I will begin by finding out the t-totals of the first five t-shapes on the grid e.g. the ones that begin with 1 through to 5.
1st T-shape= 1+2+3+12+22=40
2nd T-shape= 2+3+4+13+23=45
3rd T-shape=3+4+5+14+24=50
4th T-shape=4+5+6+15+25=55
5th T-shape=5+6+7+16+26=60
Again the T-totals go up in 5, so therefore the beginning of the formula would be 5N like the previous formula. Now that we have found that the difference between each T-total is 5 we need to times the T-numbers by this amount. I will chose the same T-shapes, which I used previously to find the formula.
1st T-shape= 22x5= 110
2nd T-shape= 23x5= 115
3rd T-shape= 24x5= 120
4th T-shape= 25x5= 125
5th T-shape= 26x5= 130
Now we subtract the multiplied T-numbers from the T-totals to find out the end part of the formula.
1st T-shape=110-40=70
2nd T-shape=115-45=70
3rd T-shape=120-50=70
4th T-shape=125-55=70
5th T-shape=130-60=70.
So the formula for the 10x10 grid should be 5N-70. I will now test it on the T-shape below.
69x5= 345 345-70=275 48+49+50+59+69=275
The formula was correct. I will now find out the formula of a grid smaller than 9x9 to see what affect lessening the grid size has on the formula.
I will now find the formula for an 8x8 grid. First I will take the T-totals from the first three T-shapes
1st shape 1+2+3+10+18=34
2nd shape 2+3+4+11+19=39
3rd shape 3+4+5+12+20=44
The difference between the T-totals is five, the same as the 9x9 grid and the 10x10 grid. This suggests that differing grid sizes wont change the beginning part of the formula, just the end part.
Now I will find the t-numbers and multiply them by five
1st T-shape= 18x5= 90
2nd T-shape= 19x5= 95
3rd T-shape= 20x5=100
And then subtract the t-totals from the multiplied T-numbers to find the rest of the formula.
1st T-shape; 90-34=56
2nd T-shape; 95-39=56
3rd T-shape; 100-44=56
The end part of the formula is 56, making 5N-56, I will test it on the T-shape below.
28x5=140 140-56=84 11+12+13+20+28=84
The formula is correct. I will now attempt to seeif I can notice any patterns in the formulas and therefore predict what the formula for an 11x11 grid would be.
The formula’s go up by seven each time, therefore an 11x11 grid size would have the formula 5n-77, because the formula is seven higher than the 10x10 grid.
The formula should be 5n-77; I will test the formula on the blue highlighted T-shape
83x5=415 415-77=338 60+61+62+72+83=338
The formula is correct. Now using the information that each time grid sizes go up by 1 the formula goes up by seven I can work out the formula of any grid size. The 9x9 grid size was 5n-63 and 7x9 is 63, so therefore the grid size times seven is the end part of the formula, so the formula is 5n-ax7 where a is the grid size. I will try it on an 8x8 grid T-shape.
5n-ax7= 20x5-8x7=44 3+4+5+12+20=44.
Now that I have found out the formula for any grid size, I will rotate the T-shape and see what an affect this has on the formula. I have illustrated this in the grid below.
The graph may be a little confusing however the blue T-shape is the original, then it is rotated 90 degrees clockwise each time, so the red T-shape is the blue T-shape rotated 90 degrees and the green was the red rotated and so on. I predict that the blue and the green will have similar formulas because they are the same but rotated 180 degrees.
I will first find out the formula of the red T-shape because, the formula for the blue T-shape was already worked out on page 4. I will work out the answer by multiplying the T-number by five and subtracting the T-total from it.
41x5=205 34+41+42+43+52=212 212-205=7 5N-7
The formula for this T-shape is 5N-7. However to see if it works on other T-shapes, I will test it on the one below.
16x5=80 80+7=87 9+16+17+18+27=87
The formula works. Now I will experiment on the next, green T-shape again by multiplying the T-number by five and subtracting it from the T-total.
41x5=205 41+50+58+59+60=268 268-205=63
Again I will test the formula 5N-63 on the T-shape below.
57x5=285 74-75+76+66+57=348 348-285=63.
The formula is correct, and interestingly it is the exact opposite of the blue T-shapes, which was 5N+63. I therefore predict that the yellow T—shape will be the same as the red T-shapes, except if the red was an addition the yellow would be a subtraction, and visa versa. Since the red formula is 5N-7 the yellow should be 5N+7. I will now test this theory.
41x5=205 30+39+40+41+48=198 205-198=7
It is correct, therefore anything when rotated 180 degrees is the same but if it was an addition it would become a subtraction, and if it was a subtraction, it would become an addition. However I still have to test it on the T-shape below.
39x5=195 28+37+38+39+46=188 195-188=7
It is correct proving the theory works.