Because there is an instant pattern in the first difference – the totals go up
in 6, I need find out what 6n would come to:
Stair Number (n) 1 2 3 4 5 6 7 8
6n 6 12 18 24 30 36 42 48
6n doesn’t give me the correct answer so I add 44 to 6n and get the Stair Total:
Stair Number (n) 1 2 3 4 5 6 7 8
Stair Total (Tn) 50 56 62 68 74 80 86 92
6n 6 12 18 24 30 36 42 48
+ 44 44 44 44 44 44 44 44 44
6n + 44 50 56 62 68 74 80 86 92
Testing the Formula
To ensure the formula works in every case, I tested it in eight situations where
‘n’ was different each time.
Using the formula 6n + 44
If n = 1 then
· 6 x 1 + 44 = 6 + 44 = 50
If n = 12 then
· 6 x 12 + 44 = 72 + 44 = 116
If n = 23 then
· 6 x 23 + 44 = 138 + 44 = 182
If n = 34 then
· 6 x 45 + 44 = 270 + 44 = 314
If n = 56 then
· 6 x 56 + 44 = 336 + 44 = 380
If n = 67 then
· 6 x 67 + 44 = 402 + 44 = 446
If n = 78 then
· 6 x 78 + 44 = 468 + 44 = 512
Stair Number (n) 1 12 23 34 45 56 67 78
Stair Total (Tn) 50 116 182 248 314 380 446 512
6n + 44 50 116 182 248 314 380 446 512
All the results using the formula are correct, so I can come to the conclusion
that the formula for the total of a stair:
· On a 10 by 10 grid
· Which travels downwards from left to right
· With a height of 3 squares
Is:
6n + 44
Part Two (Extension)
Investigate further the relationship between the stair total and other step
stairs on the number grids.
I am going to investigate patterns and relationships in:
· The position of the stairs
· The height of the stairs
· The width of the grid
I will then put everything together and produce a universal formula.
Throughout this section, the symbols for the variable inputs will be as follows:
Tn = Total of stair
n = Stair number (the number in the bottom left had corner of the stair)
h = Number of squares high the stair is
w = Width of grid (number of squares)
Relationships between different stair heights on a 10 by 10 grid
To find a pattern, I kept ‘n’ constant (n = 1) and I changed the height, ‘h’.
Height (h) 1 2 3 4 5 6
Total (T1) 1 14 50 120 235 406
Now I need to find the differences between these numbers:
Height (h) 1 2 3 4 5 6
Total (T1) 1 14 50 120 235 406
Difference 1 13 36 70 115 171
Difference 2 23 34 45 56
Difference 3 11 11 11
I repeated this with n = 33
Height (h) 1 2 3 4 5 6
Total (T33) 33 110 242 440 715 1078
Difference 1 77 132 198 275 363
Difference 2 55 66 77 88
Difference 3 11 11 11
The third difference for both of them is 11, which tells me the formula may have
something to do with 11 and h3. I made a table to see if there were any
patterns:
(n = 1)
Height (h) 1 2 3 4 5 6
+11h3 11 88 297 704 1375 2376
-11h -11 -22 -33 -44 -55 -66
Total 0 66 264 660 1320 2310
h3 1 8 27 64 125 216
Total (T1) 1 14 50 120 235 406
There were no obvious patterns so to help myself, I made this grid:
This pattern will apply anywhere on a 10 by 10 grid
If h = 1 then Tn =
1n + 0
If h = 2 then Tn =
3n + 11
If h = 3 then Tn =
6n + 44
If h = 4 then Tn =
10n + 107
If h = 5 then Tn =
15n + 211
If h = 6 then Tn =
21n + 366
There is a pattern – the ‘h’ triangle number multiplies ‘n’ (so if h = 3, the
3rd triangle would multiply n). The formula for triangle numbers is:
h2 + h
2
So the first part of the formula will be:
n (h2 + h)
2
If I include this in a table, I may get some better results:
Height (h) 1 2 3 4 5 6
Stair Number (n) 1 1 1 1 1 1
Triangle Number of h 1 3 6 10 15 21
Triangle Number of h x n 1 3 6 10 15 21
11h3 – 11h 0 66 264 660 1320 2310
(11h3 11h)ק6 0 11 44 110 220 385
(Triangle Number of h x n) + (11h3 11h)ק6 1 14 50 120 235 406
Total (T1) 1 14 50 120 235 406
I included 11h3 11h from my last table. I divided 11h3 11h by 6. This is
because 6 was the number that got 11h3 11h closest to the required total. I
noticed that if you add the Triangle Number of h x n and (11h3 11h)ק6
together you get the Total. So my formula for any height Number stair on a 10 by
10 grid is:
n (h2 + h) + (11h3 11h)
2 6
Testing the formula
Tn = Total of stair
n = Stair number (the number in the bottom left had corner of the stair)
h = Number of squares high the stair is
w = Width of grid (number of squares)
Test One – Square 25, Height 3
Total = 25 + 26 + 27 + 35 + 36 + 45 = 194
T25 = 25 x ((32 + 3) ק 2) + ((11 x 33 11 x 3) ק 6)
T25 = 25 x (11 ק 3) + (297 33) ק 6
T25 = 25 x 6 + 44
T25 = 194
Test Two Square 68, Height 3
Total = 68 + 69 + 70 + 78 + 79 + 88 = 194
... thats all ive don