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Rationals Between

What fractions can you find between the square roots of 56 and 58?

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Root to Poly

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

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Consecutive Squares

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

Guesswork

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Although this problem lends itself to discussion (and use of) binary arithmetic, it is not essential.

Is it possible to create different sets of cards that you can use to identify numbers smaller than or larger than $63$?