maths stairs

Luke Griffiths

Number Stairs part 2

I am going to be investigating the relationship between stair totals using 3 by 3 size step stairs on different size grids. I am looking for an equation that will link the stair total to the size of the grid. I am going to do 3 different grid sizes and then predict the 4th. If I am successful I will use the formula. If I am unsuccessful I shall try again. I will use 1 as the corner squares, testing on 2, n as the term for the corner square, and g as the grid size. How I got the formula is explained in part 1. (Diagrams are above)

I am unable to do a ‘1 by 1’, ‘2 by 2’ as there is not enough room to get 1 result.

n+n+1+n+2+n+3+n+4+n+7=6n+16

I have worked out that if the corner square is 1 in a 3 by 3 grid the total will be.

1+2+3+4+5+8=22

I worked this out by adding all of the numbers inside the stair and finding the total.

I came up with the formula of 16+6n. As there is still the same number of ‘n’ in the diagram, the only change is that of the numbers. This has to be tested on the previous as there is no other 3 step stairs possible on a 3 by 3 grid.

16+(6*1)=22

This is the same ...

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n+n+1+n+2+n+3+n+4+n+7=6n+16

I have worked out that if the corner square is 1 in a 3 by 3 grid the total will be.

1+2+3+4+5+8=22

I worked this out by adding all of the numbers inside the stair and finding the total.

16+(6*1)=22

This is the same and therefore this is the is right as there is nothing to contradict it.

I have worked out that if the corner square is 1 in a 4 by 4 grid the total will be.

1+2+3+5+6+9=26

I worked this out by adding all of the numbers inside the stair and finding the total.

I came up with the formula of 20+6n. As there is still the same number of ‘n’ in the diagram, the only change is that of the numbers. To test this I will do both on the

I predict that if the corner square is 2 in a 4 by 4 grid the total will be.

(6*2)+20=32

To prove I used the other method as well.

2+3+4+6+7+10=32

Therefore the 4 by 4 grid formula for a 3 step stair is 6n+20.

I have now moved on to a 5 by 5 grid. If the corner square is 1 then the total will be.

1+2+3+6+7+11=30

I have worked out the formula of 24+6n. I did this by taking 6(6*1) from 30, the first result. I then came up with 24, because of this I found that 6n+24=30 for this particular result.

To prove this I predict that if the corner square is 2 in a 5 by 5 grid then the total will be.

24+(6*2)=36

2+3+4+7+8+12=36.

Therefore there is a formula, and now that I have 3 different formulas I believe I can see a pattern, this is that there is an increase of 4 every increase in grid size. I think therefore there is an increase of 4g which I take to be grid size. I also think that they fit into the equation as shown below, because there are 4g, there would be an increase of 4 per grid increase.

However to prove this I predict that for the next grid increase, 6 by 6, the formula will be 6n+28.

Which would therefore mean that the formula would be (6*1)+28=34

1+2+3+7+8+13=34

Therefore this constant increase is apparent and therefore I can use this to draw up a table of results, which is shown below.

Now I need a general formula for a 3 step stair on any grid size. I will use g to represent the grid size.

The diagram is following. I have seen that the grid size is also the amount the n increases by from line to line. Therefore n+g will be the correct number proved by:

n=1, g=3

n+g=4

4 is the number directly above n so therefore is n+g therefore the formula is correct.