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# Guesswork

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Age 14 to 16

Challenge Level

Ask a friend to choose a number between $1$ and $63$ and make a mental note of it without telling you.

Show each of the six cards in turn to your friend and ask them whether it contains your number.

Alternatively, there is an animation at the end of this problem in which the computer attempts to guess your number.

Using this information and the cards it is possible to say what the number is.

For example your friend might choose the number $21$.

This appears on three cards:

The one starting with a $4$

The one starting with a $1$

And the one starting with a $16$.

That is interesting! $4+1+16 =21$

Is this always the case?

Why?

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?