8 · Test the parity rule

One color, three rooms. Toggle which rooms hold the color, and watch which walls a string of that color may cross.

Try this. Search for a pattern that opens all three walls. There are only eight patterns, so you will be done soon, and none of them works: that is the whole proof that no triangle can touch three rooms.

A wall is usable for a color when exactly one of its two rooms holds that color. Around a triangle, held would have to alternate with missed three times, and three is odd.