Math Challenge

A complete solution

1. 01. The problem

   Find a number that can be expressed as two different sums of two positive cubes.

2. 02. Build the cubes

   Try the pairs (1, 12) and (9, 10). A cube with side n has volume n³.

3. 03. Calculate each pair

   1³ = 1 and 12³ = 1728, so their sum is 1729.
   9³ = 729 and 10³ = 1000, so their sum is also 1729.

4. 04. Verify the equality

   $1^3+{12}^3=9^3+{10}^3=1729$

   Both pairs contain different positive integers, and both give the same sum.