10X+110
We can check this by doing another step square
If use the formula for all the numbers in red we get 10 x 5 + 110 = 160 and if we add up all the numbers: -
5+6+7+8+15+16+17+25+26+35=160
So as we can see the formula works.
Now I will do a 5x5 step square on a 10x10 grid
5x5
I will now draw a 10x10 grid and mark on in blue the 5x5 step square.
The formula for all these numbers are as shown below.
Here as shown all the X’s add up to 15 and all the numbers add up to 220
So the formula for a 5x5 step square is 15X+220
We can prove this by looking at the same grid but using different numbers this time colored in red.
Using the formula we can get the total of all the numbers in the step square.
15 x 6 + 220=310
And if we add 6+7+8+9+10+16+17+18+19+26+27+28+36+37+46=310
So the formula works as the answers match.
Now I have 3 step squares results I can start to work out the formula.
Part 3
The formula’s I have got are as shown: -
6X+44
10X+110
15X+220
We can also estimate by the pattern that it is going up in the 6 step formula is 21X + 385 so we can use this to help us workout the formula
I will now start by finding out the formula for the +N’s on the end then by doing the X’s
6X+44
10X+110
15X+220
21X + 385
as you can see by the third difference that they are all 11 so we divide the numbers by 11 to get
2
3
4
5
And as you can see by the third difference that it is 1 this means that it is a N (cubed)
Now if we look at N (cubed)
1 8 27 64 125 216
As we look at the third difference we notice this is six. But the answer we got for the third difference was 1. This means that our number is a sixth smaller than n (cubed) So we put it as.
N 6
But as we put a number into the formula we notice that the number still isn’t right
3 6=4.6
- 6=10.67
- 6=20.83
As we look at the results we can tell that there is a point something after the number we want. We notice that this number is a sixth of N so what we do is
N 6
So if we take this away from the first part of the formula we get the number we want so the formula for the last bit of the formula is
N --- N b 6 6
But to finish off the formula we need to x 11 as we divided the original + N’s by 11 so the formula for the last bit is
(N --- N) x 11 b 6 6
Now we will try and work out the formula for the amount of X’s in the Step Square
We do this by trying to get the difference of the amount of X’s that we know already from the previous numbers.
6 10 15 21
As we can see the number is one and is the second difference this means its N (squared) So we now look at the N (squared) pattern to see how it is in relationship to it
1 4 9 16
As we notice from the second difference here that it is 2 but our number is 1 this means that ours is half of N (squared)
So the formula for this part is
½ N
To get the next bit we need to substitute N into ½ N and take that away from X ( the amount of X’s in a Step Square) to see how much we are off from X
N 3 4 5
1/2N 4 ½ 8 12 1/2
X 6 10 15
X- ½ N ½ 2 2 1/2
As we can see the first difference of X- ½ N is ½ this means that it Is ½ N so the altogether formula for the X’s is
½ n + ½ n
We can now put together the whole formula together
(½ n + ½ n) + (N --- N) x 11 b 6 6 6
To see if this works we now check it .
Marked off is a 3x3 step square the total of this square is 1+2+3+11+12+21=50
Using the formula to check this it’s
(½ 3 + ½ 3) + (3 --- 3) x 11 b 6 6 6
= 4.5 + 1.5 + 11(4.5 – 0.5)
= 50
So as we can see the formula works
So the formula for any size step square in a 10x10 grid is
(½ n + ½ n) + (N --- N) x 11 b 6 6 6
Which has been proved and tested to show that it works so we know that this is right and will work for future use.
