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'Up and Down Staircases' printed from http://nrich.maths.org/

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Many of you worked on this problem and sent us solutions.

Chloe from Watton Junior School explained how she solved the first part of the question:

I worked this out practically, by building a model using lego mega blocks. By doing this I was able to understand that however many steps up/down you require you must therefore have that amount of blocks in the middle and then you have one less block in the columns on either side until the last step, which of course will be one block.

Kaan from Erenkoy Isik School in Turkey also built staircases himself. He sent us this diagram:

images of staircases with 1 step, 2 steps, 3 steps etc

So, Chloe and Kaan agree that for an up-and-down staircase with 5 steps up and 5 steps down, you need 25 blocks.

Kaan goes on to say:

I concluded that, if I multiply the number of steps with itself, I can find the number of blocks. For example;

If I want to build

6 step up and down staircase, I will need 6 x 6 = 36 blocks

7 step up and down staircase, I will need 7 x 7 = 49 blocks

8 step up and down staircase, I will need 8 x 8 = 64 blocks and ...

Rachel from Histon and Impington Infant School adds a little more about why this works:

I noticed that if I take one half of the staircase and put it on top of the other, I have a square shape. This shape is 5 bricks across and 5 bricks down.

However many steps you have up and down is the same as the number of bricks across and down in the square. So if the number of steps is N, then the number of bricks is N x N.

This is a fantastic explanation Rachel, thank you. If you haven't seen it already, there is an animation of this in the Hint.