Stage: 5 Challenge Level: 
Find all the 3 digit numbers such that by adding the first digit, the square of the second digit and the cube of the third digit you get the original number, for example $1 + 3^2 + 5^3 = 135$. Can you find ways of reducing the search so that you do not need to test 900 cases?
[This problem appears in Ken Follett's spy thriller of the same name].
Manipulating algebraic expressions/formulae. Diophantine equations. Factorisation (algebraic). Number bases. Mathematical reasoning & proof. Proof by exhaustion. Inequality/inequalities. Divisibility. Place value. Modulus arithmetic.
Published December 2000.
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