4 68 6
5 74 6
6 80 6
7 86 6
Each time the stair total increases by 6. This takes the previous total to the next total.
To get the formula I multiplied each stair total by 6. I noticed that after each multiplication the only difference between the number and the ST was 44.
Therefore the formula is Dn+(A-D)
6n+(50-6)
6n+44
The stair number is multiplied by 6 and 44 is added to make the stair total.
The formula allows the stair total to be obtained from only the star number.
Part 2
As said in my plan I intend to try different step sizes on the same 10 x 10 grid. I will try a 2-step stair.
From my previous investigation I figured that the 6 at the start of the equation is related to the fact that there a 6 squares in a 3-step stair. I can now assume that the first number of a 2-step stair will b 3 as there are 3 squares in a 2-step stair.
2-step stair
SN = 1
ST = 14
SN = 2
ST = 17
SN = 3
ST =20
SN = 4
ST = 23
SN ST Difference
1 14 3
2 17 3
3 20 3
4 23 3
5 26 3
6 29 3
7 32 3
8 35 3
9 38 3
10 41 3
The difference from 1 ST to the next is + 3. Therefore to start off the rule there must be a 3n. From that you must add 11 to reach the ST.
Formula = 3n+11
5-Step stair
As before I intent to change the size of the stair, leaving the grid size the same (10 x 10). I will change the size of the stair to a 5-step stair. I predict the first number will be 15, as there are 15 square in a 5-step stair.
SN = 1
ST = 235
SN = 2
ST = 250
SN = 3
ST = 265
SN = 4
ST = 280
SN ST Difference
1 235 15
2 250 15
3 265 15
4 280 15
5 295 15
6 310 15
7 325 15
8 340 15
9 355 15
10 370 15
To make the start of the equation there will be a 15n. From that (15n) the remaining number to make the ST must be covered.
Formula = 15n + 220
General Rules
Now that I have rules for 3 different stair sizes I will look for the general rules for all stair sizes. This will be a formula to get all the ST’s from only the SN’s.
The total of the difference is 20 + 10 + 11+ 1 +2 = 44
The difference when added together makes the number that must be added at the end of the formula (i.e.6n + 44).
I will see if this applies to all stair sizes.
The total of the differences is 11. This is also the end of the rule(3n + 11).
Total difference = 220
Formula = 15n +220
The rule for finding the differences works.
I will try to find the rules for larger stair numbers.
As there are 21 squares I will say the start of the rules is 21n. The total of the differences is 385. The formula can is 21n + 385.
