This problem follows on from Number Pyramids.

In the number pyramid below, the number in the bottom left hand corner determines all the other numbers.
Try entering some different numbers.
What patterns do you notice?

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Here are some questions to consider:

• Which numbers is it possible to get at the top, if you start with a whole number in the bottom left hand corner?
• Can you explain why some numbers are impossible to get at the top, if you start with a whole number in the bottom left hand corner?
• Given the number at the top, can you find a quick way of working out the number in the bottom left hand corner?

Test out your observations and insights.
How would your insights change if you used negative numbers? Decimals?

Can you justify any generalisations that you have reached?
Perhaps you could use algebra to explain your thinking.

Can you adapt your insights so that they apply to pyramids with different sequences on the bottom layer?
What if the numbers on the bottom layer go up in 2s? Or 3s? Or start at 17 and go up in 7s? Or...
This spreadsheet might be useful for exploring such pyramids with four or five layers.
You could adapt it to work on even larger pyramids.

For a challenging extension, why not explore Function Pyramids, in which the structure of the pyramid is based on a more complicated function than addition.

This problem features in Maths Trails - Generalising, one of the books in the Maths Trails series written by members of the NRICH Team and published by Cambridge University Press. For more details, please see our publications page .