Triangle numbers

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

Problem

Triangle Numbers printable worksheet

 

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.

 

Image
A 12 by 12 multiplication grid, with numbers 1,3,6,10,15,21,28,36,45,55,66 highlighted.

 

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.