Triangle numbers

problem

Sam Again

Age

11 to 14

Challenge level

1 out of 3

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.
problem

Reciprocal Triangles

Age

16 to 18

Challenge level

2 out of 3

Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.
problem

Triangular Triples

Age

14 to 16

Challenge level

2 out of 3

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

3 out of 3

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

Alphabet Blocks

Age

5 to 11

Challenge level

1 out of 3

These alphabet bricks are painted in a special way. A is on one brick, B on two bricks, and so on. How many bricks will be painted by the time they have got to other letters of the alphabet?
problem

Swimming Pool Tiles

Age

7 to 11

Challenge level

1 out of 3

This activity creates an opportunity to explore all kinds of number-related patterns.
problem

Cat Food

Age

7 to 11

Challenge level

1 out of 3

Sam sets up displays of cat food in his shop in triangular stacks. If Felix buys some, then how can Sam arrange the remaining cans in triangular stacks?
problem

Graphing Number Patterns

Age

7 to 11

Challenge level

1 out of 3

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

A Square Deal

Age

7 to 11

Challenge level

2 out of 3

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

Human Food

Age

7 to 14

Challenge level

1 out of 3

Sam displays cans in 3 triangular stacks. With the same number he could make one large triangular stack or stack them all in a square based pyramid. How many cans are there how were they arranged?