I am going to begin my investigation by trying to find the rule that allows you, if you are given a shape number, to work out the total for a shape on a 10x10 grid. I will begin by putting the shape numbers, numbers inside that shape and shape totals onto a grid. I already know number one from my example and I will use the method I used in my example to work out the subsequent numbers and totals. However, for times’ sake I will not draw out each shape.
10x10 grid:
From this grid I have noticed several things. Firstly, each time the shape moves along one square, for example from step shape one to step shape two the shape total increases by six. Also as the stair steps move up rows the shape total increases by 60. I have also noticed that there is a connection between N, the shape number, and all the other totals in the shape. These connections can be shown as the following:
From this it can be seen that if you have N then you can work out the shape total. The information is the grid can now be put into the formula:
6N+44= Shape total
I will test this formula on a shape chosen at random. I will use step shape 24. This shape, as part of the grid, would look as follows.
So if I put shape 24 into the formula it would be:
6x24+44=Total
144+44=Total
Total=188
So if my formula is correct then the total for the shape 24 is 188. I will add up the numbers in the shape and see if I am correct. 24 + 25 + 26 + 34 + 35 + 44 = 188. I was right. I now have the formula that will give me the total for any shape on a 10 x 10 grid.
I will now move onto a 9 x 9 sized grid. I will use the same method as I used on the 10 x 10 grid. Hopefully I should be able to use a similar method to find a formula. I will begin by putting the totals into a grid.
9 x 9 grid:
As before as you move the shape across the total increases by 6 but as you move it up the total increases by 42. This tends to suggest that the increase from moving the shape across differs according to the size of the step shape, which in this case I kept constant and the change when the shape is moved up depends on the size of the grid. I will now put the shape into the N form to work out a formula.
This method works. From this I can use the formula:
6N+40= Shape total
I will test this shape again on shape 24:
6 x 24 + 40= 184
If the same method to find the formula can be used for any grid size then shape 24 on a 9 x 9 grid should equal 84. 24+25+26+33+34+42 =184. I have found a rule for the 9 x 9 grid and I also know that my method will give me a rule on any grid size. I will now work out all the individual grid rules for grid 5 to 10 in an attempt to find put a rule for any grid size.
8 x 8 grid:
An 8 x 8 grid will use this pattern:
Giving this formula:
6n+36
I know this formula is correct so I wont test it.
7 x 7 grid:
A 7 x 7 grid will use this pattern:
Giving this formula:
6n+32
6 x 6 grid:
A 6 x 6 grid will use this pattern:
Giving this formula:
6n+28
5 x 5 grid:
A 5 x 5 grid will use this pattern:
Giving this formula:
6n+24
I am now going to try and find out a formula that, given the shape number, will allow you to work out a shapes total on any size grid.
From this I have noticed that the number is always 6N for a 6-box stair shape and that the ‘plus’ number is always the grid size times 4 plus 4. So if this were written into a rule it would be:
6N +4G+4
(G =the grid size)
I have also noticed that the 4 in the four G relates to the following:
In each case G repres-
ents 10.
