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Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

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Binary Squares

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

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Learn about Number Bases

We are used to writing numbers in base ten, using 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Eg. 75 means 7 tens and five units. This article explains how numbers can be written in any number base.

Basic Rhythms

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

All the results can be checked by calculation. For an explanation and proof it is most efficient to use a general method that works for all bases. Moreover algebra reveals the structure that is obscured when using particular numbers to check special cases. Writing $b$ for the base then each expression can be written in terms of $b$ and, by collecting like terms in the expression, or equating the powers of $b$, it can be shown that each expression holds. All that is required is some care over the algebraic manipulation.