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# Sums, Squares and Substantiation

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What Does it All Add up To?

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Impossible Sums

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Difference of Odd Squares

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Direct Logic

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Adding Odd Numbers (part 2)

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KS5 Proof Shorts

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Creating convincing arguments or "proofs", to show that statements are always true, is a key mathematical skill. The problems in this feature offer you the chance to explore number patterns and create proofs to show that these are always true.

Many of the problems in this feature include proof sorting activities which challenge you to rearrange statements in order to recreate clear, rigorous proofs. There are also a selection of "dodgy proofs" where your challenge is to find out where the logic breaks down.

Plus magazine has a selection of interesting articles about proofs here.

Age 11 to 18

Challenge Level

If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?

Age 14 to 18

Challenge Level

Which numbers cannot be written as the sum of two or more consecutive numbers?

Age 14 to 18

Challenge Level

$40$ can be written as $7^2 - 3^2.$ Which other numbers can be written as the difference of squares of odd numbers?

Age 16 to 18

Challenge Level

Can you work through these direct proofs, using our interactive proof sorters?

Age 16 to 18

Challenge Level

Can you use Proof by Induction to establish what will happen when you add more and more odd numbers?

Age 16 to 18

Challenge Level

Here are a few questions taken from the Test of Mathematics for University Admission (or TMUA).

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of these resources.*