Absolutely right, TPHacker! . . . *blush*
Everybody, please ignore this egregious incorrect solution . . .
The cubes are: n³
The triangular numbers are: n(n + 1)/2
Then we have: n³ = n(n+1)/2 ---> 2n³ - n² - n = 0
Factor: n(n - 1)(2n + 1) = 0
Since n is a positive integer, the only root is: n = 1