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Sums, Squares and Substantiation

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Sums, Squares and Substantiation
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.

What does it all add up to?
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What does it all add up to?

Age
11 to 18
Challenge level
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If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?
Impossible sums
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Impossible sums

Age
14 to 18
Challenge level
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Which numbers cannot be written as the sum of two or more consecutive numbers?
Difference of odd squares
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Difference of odd squares

Age
14 to 18
Challenge level
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$40$ can be written as $7^2 - 3^2.$ Which other numbers can be written as the difference of squares of odd numbers?
KS5 Proof shorts
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KS5 Proof shorts

Age
16 to 18
Challenge level
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Here are a few questions taken from the Test of Mathematics for University Admission (or TMUA).
Direct logic
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Direct logic

Age
16 to 18
Challenge level
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Can you work through these direct proofs, using our interactive proof sorters?
Adding odd numbers (part 2)
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Adding odd numbers (part 2)

Age
16 to 18
Challenge level
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Can you use Proof by Induction to establish what will happen when you add more and more odd numbers?
Dodgy proofs
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Dodgy proofs

Age
16 to 18
Challenge level
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These proofs are wrong. Can you see why?