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Filling the Gaps
Age 14 to 16
Challenge Level 
Why do this problem?
This problem provides an interesting context in which students can apply algebraic techniques and ideas about modular arithmetic. It also gives them a taste of an area of number theory that they might study if they go beyond the school curriculum.
Possible approaches
This problem follows on nicely from What Numbers Can We Make?
"Mathematicians have been interested in which numbers can be written as a sum of square numbers. Here are the numbers that we can make as a sum of two squares."
Show grid.
"There don't seem to be any obvious patterns here. But the numbers are only in ten columns because we're used to grids like this. Perhaps we should try a different number of columns instead, like nine."
Show grid.
"Can we see any patterns this time?"
[Students might notice the two empty vertical columns. They might also notice a diagonal pattern (top right to bottom left). Suggest that this diagonal pattern could become vertical if we make each row one shorter.]
Show grid.
Hand out this copy of the grid and give students some time, working in pairs, to look for patterns, make predictions, and explain those predictions. You might want to encourage students to start by looking at the three completely empty columns.
Possible prompts if students are having difficulties providing convincing/rigorous explanations:
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?
Bring the class together to pool ideas, and then offer this grid with sums of three squares for further investigation. Some students might also like to consider what will happen when we add four squares.
Possible support
Ensure that students have worked on What Numbers Can We Make?
Possible extension
Suggest that students look for patterns that they can explain in the nine-column grid.
Students could also experiment with grids with different numbers of columns.
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Novemberish
a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.