72 + 73 + 74 + 82 + 83 + 92 = 476
72 72+1 72+2 72+ 10 72+11 72+20
1 + 2 + 10 + 11 + 22 = 44
72x6+44=476
This is the formula
If I want to find 76 I don’t have to write all the number and add them.
I just need to do it 76x6+44=500 to find out I am right I have to work it out.
76+77+78+86+87+96=500 my predict is right
31+32+33+41+42+51=230
32+33+34+42+43+52=236
33+34+35343+44+53=242
I am predicting the next one that start with 34 the answer is going to be 248 to find out I am right I have to work it out.
34x6+44=248
N=34
34 + 35 + 36 + 44 + 45 + 54 =248
N n+1 n+2 n+10 n+11 n+20=6n+44 change 34 to N
8+9+10+18+19+28=92
18+19+20+28+29+38=152
28+29+39+38+39+48=212
38+39+40+48+49+85=272
Every time it goes up by 60. The next is 332 that start with 48. Now I have to fine I am right.
48+49+50+58+59+68=332
48x6+44=332
71+72+73+81+82+91=470
62+63+64+72+73+82=416
53+54+55+63+64+73=362
This time it is different it goes down by 52.
I am protecting the next one which start with 44 and the answer is 308.
44 + 45 + 46 + 54 + 55 + 64=308
N + n+1 n+2 + n+10+ n+11+ n+20=6n+44 N=44
44x6+44=308
Now we are going to see what happens when we move the stair shape up one square on the grid and if there is a pattern. As we already know that the stair total for stair1 is 50, we are now going to find the stair total for stair11 (a translation of stair1 one square up).
The stair total for this shape is 11 + 12 + 13 + 21 + 22 + 31 = 110
Now we are going to find the stair total for stair21 (a translation of square11 one square up).
The stair total for this stair shape is 21 + 22 + 23 + 31 + 32 + 41 = 170.
Now we are going to find the stair total for stair31 (a translation of square21 one square up).
The stair total for this stair shape is 31 + 32 + 33 + 41 + 42 + 51 = 230.
We can compare the stair totals for stairs 1, 11, 21, and 31 in order to find a pattern. If we look at their stair totals it is easy to see that as the stair shapes move one square up on the grid the stair total increases by 60.
This is shown in the diagram below.
Stair Number: 1 11 21 31
Stair total: 50 110 170 230
Difference: 60 60 60
I can now predict that the stair total for stair 41 is going to be 230 +60=290.
Here is the stair to prove my prediction is correct.
The stair total for this shape is 41 + 42 + 43 + 51 + 52 + 61 = 290.
We can therefore say that every time you move the shape one square down the total decreases by 60 and when you move one square up the grid then total increases by 60. Also when you move one square to the right of the grid the total increases by 6 and when you move one square to the left the total decreases by 6.
The reason the total increases by 60 when you move the shape one square up on the grid is because since there are 6 squares in a 3-step stair and each separate number increases by 10. 6 multiplied by 10 equals 60.
The reason the total increases by 6 when you move the shape one square to the right on the grid is because there are 6 squares in a 3-step stair and each separate number increases by 1. 1 multiplied by 6 equals 6.
We can now introduce algebra in order to find a common formula to find the stair total for and 3-step stair on a 10 by 10-number grid.
If this stair is stair x then the numbers in the stair will be
(x) + (x+2) + (x+3) + (x+10) + (x+11) + (x+20)
This can also be written as 6x + 44 = stair total
Using stair7 we are now going to use my formula to find out its stair total and see if my formula is correct. x is equal to 7 as 7 is the number in the bottom left hand corner of stair7 and x is the number in the bottom left hand corner of stair x.
6x + 44 = stair total
(6 x 7) + 44 = 86
So my formula says that the stair total of this stair is 86.
Now we can test this.
7 + 8 + 9 + 17 + 18 + 27 = 86
This is true. Therefore we can now say that the general formula for any 3-step stair on a 10 by 10-number grid is 6x + 44 = stair total.
We are now going to investigate further the relationship between stair totals and other step stairs on other number grids.
The numbers inside the stair and the formula for finding the stair total varies depending on the grid size:
For a 10 by 10 Grid:
But for a 9 by 9 Grid:
We can see that the formula for a 9 by 9 grid is 6x + 40 = stair total. This is 4 less than the formula for the 10 by 10 grid.
In order to prove this I will use stair20 on a 9by 9 grid.
6x + 40 = stair total
(6 x 20) + 40 = 160
20 + 21 + 22 + 29 + 30 + 38 = 160
I can therefore conclude that the general equation for a 3-step stair on a 9 by 9 number grid is 6x + 40 = stair total where x is the number in the bottom left hand corner of the stair shape.
We can now say that in order to find the equation for an 11 by 11 number grid all we have to do is add 4 to the equation of a 10 by 10 grid. If we do this then the equation will be 6x + 48 = stair total.
I am now going to check this equation with stair 12 on an 11 by 11 number grid.
6x + 48 = stair total
(6 x 12) + 48 = 120
12 + 13 + 14 + 23 + 24 + 34 = 120
This is true so I conclude that the stair total for any 3-step stair on an 11 by 11 number grid is 6x + 48= stair total.
We can now say that every time you increase the size of the grid by one (going from a 9by9 number grid to a 10by10 number grid) you have to add four more to the equation. Therefore if you decrease the size of the grid by one you must also decrease the equation by 4.
With this knowledge I can now find a general equation for any 3-step stair on any size grid.
Grid Size Formula
7 by 7 6x + 32
8 by 8 6x + 36
9 by 9 6x + 40
10 by 10 6x + 44
11 by 11 6x + 48
Masood Akbari 11.6HA page