As I mentioned above the difference between each stair total is 6. This shows that 6n is somewhere in the formula. I will now see what relation 6n has to the stair totals.
The value for 6n and the stair total are different. The total is larger which means the formula must be 6n+?. I will see what the difference between the total and 6n is to get the value of ?:
You have to add 44 to 6n to get the total of the stair. So now you can see the formula for the three-step stair is 6n+44 like I stated above.
This formula should work for obvious reasons. If you look at position 1 the biggest number is 21 and the smallest 1, meaning there is a difference of 20 between them. This will be the same anywhere on the grid. In the stair below there is a difference of 20 between 94 and 74. This relationship between the numbers remains the same anywhere on the grid and that is why my formula works anywhere.
I will now test to see if my formula, 6n+44, does work.
The total for this stair is:
74+75+76+84+85+94=488
For this stair shape n=74. I will now substitute numbers into my formula 6n+44.
=(6x74)+44
=444+48
=488
As you can see the answer gained from my equation is the right stair total for the stair given. I will now test for another stair to prove my formula does work
54+55+56+64+65+74=368
n=54
=(54x6)=44
=324+44
=368
I have now found and proven a general formula for a three-step stair on a 10x10 grid. I will now extend my investigation further.
Part 2:
I will now extend my investigation by posing this question.
Will my formula, 6n+44, work on different sized stair shapes on the 10x10 grid? If not can a formula be deduced to find out the total for any size stair on a 10x10 grid.
Since I already have results for three-step stairs I will begin my investigation with a four-step stair. A four-step stair on the grid will look like this
Position 1
The total for this stair is:
1+2+3+4+11+12+13+21+22+31=120
Position 2
The total for this stair is:
2+3+4+5+12+13+14+22+23+32=130
Position 3
The total for this stair is:
3+4+5+6+13+14+15+23+24+33=140
Position 4
The total for this stair is:
4+5+6+7+14+15+16+24+25+34=150
I will now tabulate these results and those for other 4 step stairs.
I will see whether 6n+44 will work to find the stair total for 4 steps. I will use position 1 as an example.
n=1
=6n+44
=(6x1)+44
=6+44
=50
As you can see the formula does not apply to four step stairs. Now I will try to find a formula that does apply to four steps.
Since this time there is a difference of 10 between the totals I now times n by 10 in my equation. I will see what 10n looks like compared to the totals.
The same way as with the three steps 10n does not equal the total it is larger. The difference is 110 so the formula for four step stairs is 10n+110. I will now test my new formula.
61+62+63+64+71+72+73+81+82+91=720
n=61
=10n+110
=(10x61)+110
=720
You can now see that my formula for the four step stairs is correct. I will now try five step stairs.
Position 1
The total for this stair is:
1+2+3+4+5+11+12+13+14+21+22+23+31+32+41=235
Position 2
The total for this stair is:
2+3+4+5+6+12+13+14+15+22+23+24+32+33+42=250
Position 3
The total for this stair is:
3+4+5+6+7+13+14+15+16+23+24+25+33+34+43=265
Position 4
The total for this stair:
4+5+6+7+8+14+15+16+17+24+25+26+34+35+44=280
Position 5
The total for this stair:
5+6+7+8+9+15+16+17+18+25+26+27+35+36+45=295
Position 6
The total for this stair:
6+7+8+9+10+16+17+18+19+26+27+28+36+37+46=310
For five step stairs I believe that I will have to work out a new formula. Since the totals increase by 15 each time in my equation I will have to 15n.
When comparing 15n to the totals I find:
I have to add a number to 15n to get the stair total. From my table you can see that this number is 220.
The formula to find the total of any five-step stair o a 10x10 grid is:
15n+220
I will test my formula.
n=55
55+56+57+58+59+65+66+67+68+75+76+77+85+86+95=1045
=15n+220
=(15x55)+220
=825+220
=1045
So my formula for five-step stairs does work as shown above.
I will now try to find a general formula to find any size step on a 10x10 grid. To try and find a pattern I will keep n the same, n=1, and change the stair size.
For now in all stairs n=1
One step stair
The total is
1+0=1
Two step stair
The total is:
1+2+11=14
Three step stair
The total for this stair is:
1+2+3+11+12+21=50
Four step stair
The total for this stair is:
1+2+3+4+11+12+13+21+22+31=120
Five step stair
The total for this stair is:
1+2+3+4+5+11+12+13+14+21+22+23+31+32+41=235
Six step stair
The total for this stair is:
1+2+3+4+5+6+11+12+13+14+15+21+22+23+24+31+32+33+41+42+51=406
Seven step stair
The total for this stair is:
1+2+3+4+5+6+7+11+12+13+14+15+16+21+22+23+24+25+31+32+33+34+41+42+43+51+52+61=644
I will now try to find a pattern with my results.
From looking at this table I am able to recognize a new idea, that in the final formula the number of 11 and height³.
I will make a new table to try and find out more.
(h=height of the square)
As you can see no pattern is obvious from this table.
When I was looking at the formula, 6n+44 I noticed that when you multiply it by 2 the n value is a multiple of the height of the stair. I decided to check if this was the same for the other formulas.
The relationship is the same and I put in h² to see if there was a pattern. From the table you can see that the third row equals h²+h. This could not be used in a formula because it twice as large so to use it in my general formula it would have to be:
n(h²+h)
2
This is the first part of my general formula but it is not complete and I need a second half.
From looking at the differences in stair total and the totals from 11h³-11h I believe I have found the second part of my equation. When looking at the total of 11h³-11h when h=2 I saw it equaled 66. The formula for a two-step square is 3n+11. 66/11=6. I wondered if dividing 11h³-11h by six on other heights would work, To check I made a table:
So every time when 11h³-11h is divided by 6 it gives the right number in the second part of each formula. So I now have a complete general formula to find out any sized stair anywhere on a 10x10 number grid. The formula is:
n+(h²+h) + (11h³-11h)
2 6
However I will need to test this formula:
T: Total of the stair
n: Stair number (the number in the bottom left had corner of the stair)
h: The height of the stair
n=72 Three Step Stair:
T=72+73+74+82+83+92=476
T=n((h²+h)/2)+(11h³-11h)/6
T=72((3²+3)/2)+(11x3³-11x3)/6
T=72((12)/2)+(264)/6
T=72(6)+44
T=432+44
T=476
For this one the answer is correct now I will try another sized stair
n=55 Five Step Stair:
T=55+56+57+58+59+65+66+67+68+75+76+77+85+86+95
=1045
T=n((h²+h)/2)+(11h³-11h)/6
T=55((5²+5)/2)+(11x5³-11x5)/6
T=55((30)/2)+(1320)/6
T=55(15)+220
T=825+220
T=1045
Now since I have tested it twice I am sure that my formula is correct and will work for any size stair anywhere on in the 10x10 grid. Now looking back I see what else I could have done to improve my investigation. Instead of changing stair height I could have instead changed the width of the grid or the height. I could also have changed the width and length simultaneously i.e. 5x5, 6x6, 7x7.
