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.

Binary Squares

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

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.