Geometry on Shai Asher

The Cleaning Robot Puzzle: Five Squares at Every Corner

Fri, 25 Sep 2026 09:30:00 +0300

Put five squares round every corner of the cleaning robot’s floor instead of four, and within ten steps of the dock there are 16,737 fields instead of 221.

This is Part 7 of the cleaning robot puzzle, and the last of four in which the floor stops being flat. In Part 6, three squares met at the corners of a box, each corner was 90° short, and three left turns brought the robot home. This part goes the other way. Five squares meet at every corner, each corner has 90° too many, and the floor, which can no longer lie flat, turns out to have room for far more fields than any flat floor.

The Cleaning Robot Puzzle: Cleaning a Box

Fri, 25 Sep 2026 09:00:00 +0300

Send the cleaning robot up the walls of a swimming pool and it finds that driving forward and turning left three times can bring it back to where it started.

This is Part 6 of the cleaning robot puzzle. Part 4 glued the room into a torus and Part 5 gave it a twist, but both floors were still flat: at every corner, four squares met, exactly as on a tiled kitchen floor. This part folds the floor round the corners of a box, where only three squares meet. That one missing square changes what “turning left” means, rules out a compass on the whole surface, and breaks the checkerboard again.