There are **21** NRICH Mathematical resources connected to **Triangle numbers**, you may find related items under Properties of Numbers.

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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

Can you describe this route to infinity? Where will the arrows take you next?

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

Take ten sticks in heaps any way you like. Make a new heap using one from each of the heaps. By repeating that process could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Can you find any two-digit numbers that satisfy all of these statements?

Watch the video to see how Charlie works out the sum. Can you adapt his method?

Show that all pentagonal numbers are one third of a triangular number.

Can you find a rule which relates triangular numbers to square numbers?

Can you find a rule which connects consecutive triangular numbers?

What would you do if your teacher asked you add all the numbers from 1 to 100? Find out how Carl Gauss responded when he was asked to do just that.

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

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...

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.

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?

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

I have forgotten the number of the combination of the lock on my briefcase. I did have a method for remembering it...