Triangle numbers

problem
Triangles within pentagons

Age

14 to 16

Challenge level

Show that all pentagonal numbers are one third of a triangular number.
problem
Triangles within squares

Age

14 to 16

Challenge level

Can you find a rule which relates triangular numbers to square numbers?
problem
Triangles within triangles

Age

14 to 16

Challenge level

Can you find a rule which connects consecutive triangular numbers?
problem
Picturing triangular numbers

Age

11 to 14

Challenge level

What do you notice about the sum of two identical triangular numbers?
problem
A square deal

Age

7 to 11

Challenge level

Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.
problem
Graphing number patterns

Age

7 to 11

Challenge level

Does a graph of the triangular numbers cross a graph of the six times table? If so, where? Will a graph of the square numbers cross the times table too?
problem
Iff

Age

14 to 18

Challenge level

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
problem
Triangular triples

Age

14 to 16

Challenge level

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.
problem
Hot pursuit

Age

11 to 14

Challenge level

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...
problem
Sam again

Age

11 to 14

Challenge level

Here is a collection of puzzles about Sam's shop sent in by club members. Perhaps you can make up more puzzles, find formulas or find general methods.