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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?

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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.

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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?

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

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

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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...

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

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

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

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

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

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

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Can you find a rule which connects consecutive triangular numbers?

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

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

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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?

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

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

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

Challenge Level

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