The Duke's Castle
A genuine puzzle: prove that no path through the castle grid visits every open square exactly once (a parity/checkerboard argument).
Teacher note
Why it's impossible: color the grid like a checkerboard. The entrance and exit squares are the same color (call it black). Every step moves to the other color, so a path that starts and ends on black uses exactly one more black square than white. But the blocked squares remove three white squares and only one black, leaving 23 black and 21 white: two more black than white. No path can visit them all.
Use this cartoon
Free for classrooms, worksheets, slides and other non-commercial use under CC BY-NC 4.0, with credit to Ben Orlin.
Cartoon by Ben Orlin, Math with Bad Drawings. https://cartoons.mathwithbaddrawings.com/2025-06-12-the-duke-s-castle/ (CC BY-NC 4.0)Transcript
"Your job is to mop the duke's castle, stone by stone, never missing or revisiting a square." [A grid with three black (impassable) squares; entrance bottom-left, exit top-right] "You're quite sure the task is impossible. Can you convince the duke?" [Three failed attempts, each missing a square]