# Maths grid extension

This is a 10 x 10 size grid with a 3-stair shape in blue. This is called the stair total. The stair total for this stair shape is 25 + 26 + 27 + 35 + 36 + 45 = 194. To investigate the relationship between the stair total and the position of the stair shape, I will use the far-left bottom square as my stair number:

This is always the smallest number in the stair shape. It is 25 for this stair shape.

I will then translate this 3-stair shape to different positions around this 10 x 10 grid:

The stair-total for this stair shape is 26 + 27 + 28 + 36 + 37 + 46 = 200

The stair-total for this stair shape is 67 + 68 + 69 + 77 + 78 + 87 = 446

The stair-total for this stair shape is 68 + 69 + 70 + 78 + 79 + 88 = 452

The stair-total for this stair shape is 3 + 4 + 5 + 13 + 14 + 23 = 62

The stair-total for this stair shape is 4 + 5 + 6 + 14 + 15 + 23 = 68

I will then summarize these results in a table:

In order to find a formula that I can use to find the stair total when I am given the stair number, I am going to put the stair number as the position and the stair total as the term for the sequence:

I have noticed that there is an increase of 6 between two consecutive terms in this arithmetic sequence. Therefore the position-to-term rule must be 6n + or – something.

As can be seen, the term is always 6 times the position, plus 44.

Thus, for the bottom stair of any 3-stair shape on a 10 x 10 grid, the formula must be Un = 6n + 44, where ‘n’ is the stair number, and ‘Un is the term which is the stair total. I have realised that the ‘6n’ in the formula must come from the number of squares in the stair shape. This will be proven when I write the numbers inside the 3-stair shape on the 10 x 10 grid in terms of ‘n’.

I have represented ‘n’ as the stair number, and the other numbers in relation to the stair number ‘n’.

This means that no matter where I translate this stair shape around this grid, the values in terms of ‘n’ will always be the same. To find the stair-total, I have added the values inside the 3-stair shape:

To check if this formula works for other stair numbers, I will try another number, say, 55 to be ‘n’.

Stair-total         =         6n + 44

(found using        =         6(55) + 44

the formula)        =         330 + 44

=         374

Stair-total         =         55 + 56 + 57 + 65 +

(found by                66 + 75

=         374

This shows that my formula must work for all stair-numbers in 3-stair shapes on the 10 x 10 grid.

However, I have observed that you could just sub any number as ‘n’ into the formula and you could still get a stair total even if that stair shape cannot actually be drawn on the grid.

For instance you could put, say, 10 as ‘n’ into the formula like this:

However it is impossible to draw a stair shape with the bottom left-hand number as 10, simply because it would not fit.

PART 2

To investigate further the relationship between the stair totals and other step stairs on other number grids, I will first find a formula for calculating the stair total if you know the stair number of a 2-stair shape. I will start with 2-stair on a 5 x 5 grid because these are the smallest values for both variables. Then I can gradually increase the variables in a systematic order. The stair number will still be the square at the bottom left-hand corner, i.e. the smallest.

Stairs: 2

To find a formula for calculating the stair total if you know the stair number of a 2-stair shape on a  5 x 5 grid, I will draw out the 2-stair shape on a 5 x 5 grid in terms of ‘n’.

I have realised that the square above the stair-number is (n + 5) because of the grid size, which is 5. This explains why the square above the stair number in the 3-stair shape on the 10 x 10 grid is (n + 10) and the square above that is [n + (10 x 2)] or (n + 20).

Therefore I can redraw the 2-stair shape on a 5 x 5 grid in terms of ‘n’ and ‘g’.

The stair total for this 3-stair shape will now be

n + (n + 1) + ...