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Triangle Numbers

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

In the multiplication square below, the first eleven triangle numbers $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$, $66$ have been identified.

Can you see a pattern?
Can you predict the positions of the next triangle numbers?
Check to see if you're right.

Multiplication square

Can you explain why triangle numbers take up these positions in a multiplication square?

Thank you to Paul Stephenson from The Magic Mathworks Travelling Circus who provided the idea for this problem.