n+n+1+n+2+ n+G+n+G+1+n+2G=T
Simplifying the above equation using means of collecting up will give me:
6n+4G+4=T
For this grid, it would be: 6n+4*10+4=T
Next I will use a 9 by 9 grid, using the same method as before to obtain an appropriate formula.
Using the same method as above, I can develop a formula.
Again, you can see that the ‘T’ column is increasing by 6 each time the ‘n’ number is increased by 1.
The ‘T’ column increases by 6 each time, so we multiply the ‘n’ number by the increase:
- 6n, and then add the remaining amount to end up with the ‘T’ number.
To check this formula I will calculate another sum to which I know the solution to:
Using another method, I will ensure that my formula is correct and if possible simplify the expression given.
n+n+1+n+2+ n+G+n+G+1+n+2G=T
Simplifying the above equation using means of collecting up will give me:
6n+4G+4=T
For this grid, it would be: 6n+4*9+4=T
After completing the 9 by 9 grid I will extend my investigation further by going on to an 8 by 8 grid. Again using the same methods as before to develop a formula significant to the 8 by 8 grid.
Using the same method as the previous, I will develop a formula.
Again, you can see that the ‘T’ column is increasing by 6 each time the ‘n’ number is increased by 1.
The ‘T’ column increases by 6 each time, so we multiply the ‘n’ number by the increase:
- 6n, and then add the remaining amount to end up with the ‘T’ number.
To check this formula I will calculate another sum to which I know the solution to:
Using another method, I will ensure that my formula is correct and if possible simplify the expression given.
n+n+1+n+2+ n+G+n+G+1+n+2G=T
Simplifying the above equation using means of collecting up will give me:
6n+4G+4=T
For this grid, it would be: 6n+4*8+4=T
I have developed formulas for each grid (8 by 8, 9 by 9 and 10 by 10), I have come to the simple conclusion that the general formula 6n+4G+4=T applies to all grids as long as the stair number remains the same at 3 (3-step stair).
Developing further on the generic subject matter, I will find solutions on moving the ‘n’ number either along the x axis or along the y axis.
I will use the G grid because from my previous working I can righteously say that this grid can be used to substitute any grid as long as the ‘G’ number remains relevant to the grid size.
If I move the ‘n’ number along the ‘x’ axis for x places:
It would be correct to say that:
6n+4G+4+6x=T
The 6 being there because there are six squares in the 3-step stair.
I will test this using the 10 by 10 grid:
T=7+8+9+17+18+27=86
T = 1+2+3+11+12+21=50
6*1+4*10+4+6*6=86
The ‘n’ number had moved 6 places along the ‘x’ axis to the right.
If it had moved to the left the number would be negative (-6).
The expression is correct.
If I move the ‘n’ number along the ‘y’ axis for y places:
It would be correct to say that:
6n+4G+4+6yG=T
The 6 being there because there are six squares in the 3-step stair.
I will test this using the 10 by 10 grid:
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T=1+2+3+11+12+21=56
T=51+52+53+61+62+71=350
6*1+4*10+4+6*5*10=350
The ‘n’ number had moved 5 places upward, along the y axis.
If the ‘n’ number had moved downwards then the number would be negative (-y).
I am now going to advance the investigation by using different size stair shapes on the ‘n’ grid.
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Instead of using the number grids I have decided to use the ‘n’ grid. This will save a lot of time and unnecessary calculations as this grid will give me a generic formula that will work with any size grid as long as the grid number is kept the same in the formula.
Above I will develop a 4-step stair shape formula that will work with any grid:
N+n+1+n+2+ n+3+ n+G+ n+G+1+ n+G+2+ n+2G+ n+2G+1+ n+3G=T
Using means of collecting up I can simplify this formula to: 10n+10G+10 = T
I will check that this formula is correct using the 10 by 10 grid:
From the table I will see whether the formula I derived from the ‘n-grid’ is correct.
1) 10*1+10*10+10=120
2) 10*2+10*10+10=130
From the formulae I have just used I am assuming the rest of the T numbers in the table are correct due to the results I have obtained from the sums above.
I will now do the same but using a 5-step stair:
Above I will develop a 5-step stair shape formula that will work with any grid:
N+n+1+n+2+ n+3+n+4+ n+G+ n+G+1+ n+G+2+ n+G+3+n+2G+ n+2G+1+n+2G+2+ n+3G+n+3G+1+n+4G=T
Using means of collecting up I can simplify this formula to: 15n+20G+20 = T
I will check that this formula is correct using the 10 by 10 grid:
Here I am using the formula to see whether it works…
- 15*1+20*10+20=235
- 15*2+20*10+20=250
- 15*3+20*10+20=265
From the sums I have calculated above you can see that the formula works and is accurate as it follows my table completely.
I assume and believe that the formula will work with all problems that use a 5 step stair.
Here I will generate another formula using a 6-step stair shape.
Again, using the ‘n’ grid as a generic base.
Above I will develop a 6-step stair shape formula that will work with any grid:
N+n+1+n+2+ n+3+n+4++n+5+ n+G+ n+G+1+ n+G+2+ n+G+3+n+G+4+n+2G+ n+2G+1+n+2G+2+n+2G+3+ n+3G+n+3G+1+n+3G+2+n+4G+n+4G+1+n+5G=T
Using means of collecting up I can simplify this formula to: 21n+35G+35 = T
I will check that this formula is correct using the 10 by 10 grid:
Here I am using the formula to see whether it works…
- 21*1+35*10+35=406
- 21*2+35*10+35=427
- 21*3+35*10+35=448
From the sums I have calculated above you can see that the formula works and is accurate as it follows my table completely.
I assume and believe that the formula will work with all problems that use a 6-step stair.
