Squares of All the Digits

1. How many perfect squares use each of the digits $1$ to $9$ exactly once?

2. How many use all ten digits $0$ to $9$ exactly once?

3. Write $123456789$ as the difference of two squares.

Solution

Parts 1 and 2. There are $30$ squares using the digits $1$ to $9$ once each, the smallest being $11826^2 = 139854276$ and the largest $30384^2 = 923187456$. Allowing the zero as well, there are $87$ squares using all ten digits once each, the smallest being $32043^2 = 1026753849$. (One checks these by running through the squares in the relevant range; both counts are settled.)

Part 3. Any odd number is a difference of two consecutive squares, since $(k+1)^2 - k^2 = 2k + 1$. As $123456789$ is odd, set $2k + 1 = 123456789$, so $k = 61728394$ and $$123456789 = 61728395^2 - 61728394^2.$$