n=smallest number in the boot.
v=total value of the boot.
To check my formula I will prove on another boot in the grid.
n = 9
4n = 4x9=36
36+26 = 62
9+14+19+20=62
To prove that 4n+26 is correct is to test it using n.
If I add these together n+n+5+n+10+n+11 it equals 4n+26.
This proves my formula for grid 5x5 is correct.
Grid 6x6
I found that in grid 6x6 that the formula for the added value of the boot is 4n+31. I discovered this by:
Working out the value of boots 1 to 4.
Boot 1= 1+7+13+14=35
Boot 2= 2+8+14+15=39
Boot 3= 3+9+15+16=43
Boot 4=4+10+16+17=47
Now I have this information I can begin to work on the sequence.
35 39 43 47
4 4 4
35-4=31 39-8=31 43-12=31 47-16=31
These all-equal 31!
This shows that the formula is 4n+31=v
n=smallest number in the boot.
v=total value of the boot.
To check my formula I will prove on another boot in the grid.
n = 15
4n = 4x15 = 60
60+31=91
15+21+27+28=91
To prove that 4n+31 is correct is to test it using n.
If I add these together n+n+6+n+12+n+13 it equals 4n+31.
This proves my formula for grid 6x6 is correct.
Grid 7x7
I found that in grid 7x7 that the formula for the added value of the boot is 4n+36. I discovered this by:
Working out the value of boots 1 to 4.
Boot 1= 1+8+15+16=40
Boot 2= 2+9+16+17=44
Boot 3= 3+10+17+18=48
Boot 3= 4+11+18+19=52
Now I have this information I can begin to work on the sequence.
40 44 48 52
4 4 4
40-4=36 44-8=36 48-12=36 52-16=36
These all-equal 36!
This shows that the formula is 4n+36=v
n=smallest number in the boot.
v=total value of the boot.
To check my formula I will prove on another boot in the grid.
n = 12
4n = 4x12 = 48
60+36=84
12+19+26+27=84
To prove that 4n+36 is correct is to test it using n.
If I add these together n+n+7+n+14+n+15 it equals 4n+36.
This proves my formula for grid 7x7 is correct.
Grid 8x8
I found that in grid 8x8 that the formula for the added value of the boot is 4n+41. I discovered this by:
Working out the value of boots 1 to 4.
Boot 1= 1+9+17+18=45
Boot 2= 2+10+18+19=49
Boot 3= 3+11+19+20=53
Boot 4= 4+12+20+21=57
Now I have this information I can begin to work on the sequence.
45 49 53 57
4 4 4
45-4=41 49-8=41 53-12=41 57-16=41
These all-equal 41!
This shows that the formula is 4n+41=v
n=smallest number in the boot.
v=total value of the boot.
To check my formula I will prove on another boot in the grid.
n = 27
4n = 4x27 = 108
108+41=149
27+35+43+44=149
To prove that 4n+41 is correct is to test it using n.
If I add these together n+n+8+n+16+n+17 it equals 4n+41.
This proves my formula for grid 8x8 is correct.
Linking the Formulas
Now that I have the formulas for the grids I will find a formula to link them all together.
5 x 5 grid formula: 4n + 26
6 x 6 grid formula: 4n + 31
7 x 7 grid formula: 4n + 36
8 x 8 grid formula: 4n + 41
The four before the n represents the number of squares in a boot that is why they are all four.
Now I have this information I can begin to work on the sequence for the size of the grid.
26 31 36 41
5 5 5
g = grid size
26-1=25 31-1=30 36-1=35 41-1=40
All need to add one to reach the sequence number!
This means the formula to work out the value of the boot and the grid size has to be 4n+5g+1.
I will test this formula on a 5x5 grid.
n=8 g=5
4x8=32 + 5x5=25 + 1
32+25+1=58
8+13+18+19=58
This proves my formula for the total value of the built and for the size of the grid is correct.
