Rationals Between

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

Root to Poly

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

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?

Smith and Jones

Stage: 4 Challenge Level:

Mr Smith and Mr Jones are two maths teachers, who meet up one day. Mr Smith lives in a house with a number between 13 and 1300. He informs Mr Jones of this fact, and challenges Mr Jones to work out the number by asking closed questions.

Mr Jones asks if the number is bigger than 500. Mr Smith answers, but he lies.

Mr Jones asks if the number is a perfect square. Mr Smith answers, but he lies.

Mr Jones asks if the number is a perfect cube. Mr Smith answers and (feeling a little guilty) tells the truth for once.

Mr Jones says he knows that the number is one of two possibilities, and if Mr Smith just tells him whether the second digit is 1, then he'll know the answer. Mr Smith tells him and Mr Jones says what he thinks the number is. He is, of course, wrong.

What is the number of Mr Smith's house?