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Filling the Gaps

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
It may be helpful to look at the problem What Numbers Can We Make?

In which columns do the square numbers appear?
In which columns do the squares of even numbers appear?  Can you explain why?
And the squares of odd numbers?  Can you explain why?
How can we describe the numbers in a particular column?
You might want to start by looking at the completely empty columns.