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Mr Smith and Mr Jones are two maths teachers. By asking questions, the answers to which may be right or wrong, Mr Jones is able to find the number of the house Mr Smith lives in... Or not!

# Generating Triples

##### Age 14 to 16 Challenge Level:

Charlie has been investigating square numbers. He decided to organise his work in a table: Charlie noticed some special relationships between certain square numbers:

$$3^2+4^2=5^2$$ $$5^2+12^2=13^2$$

Sets of integers like ${3,4,5}$ and ${5,12,13}$ are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle.

He wondered whether he could find any more...

Can you extend Charlie's table to find any more sets of Pythagorean Triples where the hypotenuse is 1 unit longer than one of the other sides?
Do you notice any patterns?
Can you make any predictions?

Can you find a formula that generates Pythagorean Triples like Charlie's?
Can you prove that your formula works?

Alison has been working on Pythagorean Triples where the hypotenuse is 2 units longer than one of the other sides.
So far, she has found these:
$$4^2 + 3^2 = 5^2$$ $$6^2+8^2=10^2$$ $$8^2+15^2=17^2$$

Some of these are just scaled-up versions of Charlie's triples, but some of them are new and can't be divided by a common factor (these are called primitive triples).

Can you find more Pythagorean Triples like Alison's?

Can you find a formula for generating Pythagorean Triples like Alison's?
Can you prove that your formula works?

Here are some follow-up questions you might like to consider:

• Can you find Triples where the hypotenuse is 3 units longer than one of the other sides? Or 4 units longer? Or...?
• Can you say anything about when such triples will be primitive triples?

For a challenging extension investigation, why not take a look at Few and Far Between?