Rotom’s Pokédex Rectangle

Problem 10: Rotom’s Pokédex Rectangle

The total is 138.

On a hundred chart, moving one square right adds 1, and moving one row down adds 10. Start at the top-left corner of any rectangle. If moving to the right adds some ones and moving to the bottom adds some tens, its corners have this structure:

```text top-left top-left + right move

top-left + down move top-left + right move + down move ```

Now compare the diagonals:

• The top-left and bottom-right corners together contain two copies of the top-left number, one right move, and one down move.
• The top-right and bottom-left corners together contain those exact same pieces.

So the sums of the two opposite-corner pairs are always equal. This is the diagonal-sum invariant.

Eevee and Squirtle are opposite, and their numbers add to 69. Therefore Pikachu and Charmander, the other opposite pair, must also add to 69:

69 + 69 = 138

The individual hidden numbers can change when Rotom chooses a different rectangle, but the four-number total cannot change.