Rotom’s Pokédex Rectangle

At a glance

• Part 1: The four hidden corner numbers always total 138.

Key idea

On a 1–100 chart, moving right adds some number of ones and moving down adds some number of tens. Start with a top-left corner called 𝑎. If the horizontal move adds 𝑟 and the vertical move adds 𝑑, the four corners, shown left then right in each row, are:

1. Prove the forced total

Add the top-left and bottom-right corners:

𝑎 +(𝑎 +𝑟 +𝑑) =2𝑎 +𝑟 +𝑑.

Add the top-right and bottom-left corners:

(𝑎 +𝑟) +(𝑎 +𝑑) =2𝑎 +𝑟 +𝑑.

The two opposite-corner sums are equal for every rectangle that follows rows and columns. Eevee and Squirtle are opposite corners and their numbers add to 69, so the other opposite pair must also add to 69.

69 +69 =138

Thus the total of all four hidden numbers is always 138.

Why every rectangle works

Changing the rectangle changes 𝑎, 𝑟, or 𝑑, but both diagonal sums still contain exactly two copies of 𝑎, one horizontal move, and one vertical move. This shared structure—not the particular picture—forces the answer.

Check

For example, 24 and 45 can be opposite corners because they occupy different rows and columns and add to 69. The other two corners are 25 and 44, which also add to 69. All four total 24 +25 +44 +45 =138.