Number Stairs.

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Number Stairs

On the 10 by 10 number grid above I have drawn a stair shape. This is known as a three-step stair.

The total of the numbers inside the stair is:

                                55+45+46+35+36+37=254

So the stair total for this shape is 254.

The stair number for this shape is stair 35 as the number in the bottom left hand corner is 35.

        Now we can investigate the relationship between the position of the stair shape on the grid and the stair total. The diagram below is stair 1 as the number in the bottom left hand corner is 1.

As we see, the stair total for this 3-step stair1 on this 10 by 10-number grid is 50 as 1 + 2 + 3 + 11 + 12 + 21 = 50.

Below is the stair shape one square to the right of stair1. This stair is called stair2 as the number in the bottom left hand corner is 1. Stair2 is a translation of stair1 one square to the right.

The stair total for this stair shape is 2 + 3+ 4 + 12 + 13 + 22 = 56.

Now we are going to find the step total for stair3 (a translation of stair2 one step to the right).

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

Now we can find the stair total for stair4 (which is a translation of stair 3 one square to the right).

The stair total for this stair shape is 4 + 5 + 6 + 14 + 15 + 24 = 68.

Now we can compare the stair totals of stairs 1, 2, 3, and 4 in order to find a pattern. If we look at their stair totals it is easy to see that as the stair number moves one square to the right the stair total increases by 6.

Stair Number:  1                         2                        3                        4

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Stair total:      50                         56                        62                        68                

Difference:                     6                           6                               6

Now I know this pattern I can predict that the stair total for stair 5 will be 74 as 68+6=74.

Here is the stair to prove my prediction.

The stair total for this stair shape is 5 + 6 + 7 + 15 + 16 + 25 = 74

We can therefore say that every ...

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