Algorithms on Shai Asher

The Cleaning Robot Puzzle: A Lower Bound from a Linear Program

Fri, 02 Oct 2026 08:00:00 +0300

Part 3’s fast route cleans a furnished 40×40 room in 1,327 commands, and all Part 3 could prove is that no route needs fewer than 1,037. Between the two lie 290 commands nobody could account for.

This is Part 8 of the cleaning robot puzzle, and the first of the parts that take the robot’s optimisation problems to the tools operations research really uses. Part 3 asked for the fewest commands that clean the whole floor, built a fast solver, an exact one that gives up on big rooms, and a lower bound from three simple facts. It ended on an open question: “On big furnished rooms the gap to the lower bound stays wide. The bound is weak there, and nobody knows how far from the best the fast routes really are.”

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.

The Cleaning Robot Puzzle: A Floor with a Twist

Fri, 25 Sep 2026 08:30:00 +0300

Give the cleaning robot’s floor a half twist, like a Möbius strip, and after one lap round the room its up is the map’s down, and the commands that cleaned every flat room walk it into the walls.

This is Part 5 of the cleaning robot puzzle. Part 4 glued the room’s edges into a torus, and the robot lost its ability to name fields by counting moves. This part glues two edges with a twist, and the robot loses something it never knew it had: a compass. On every floor so far, ^ meant the same direction everywhere. Here it doesn’t.

The Cleaning Robot Puzzle: A Room That Wraps Around

Fri, 25 Sep 2026 08:00:00 +0300

Glue the cleaning robot’s room into a doughnut, so that walking off the right edge brings it back on the left, and the blind robot from Part 3 never stops cleaning.

This is Part 4 of the cleaning robot puzzle, and the first of four parts in which the floor stops being flat. Part 1 cleaned a room with a depth-first walk, Part 2 added dirt, a bag and a battery, and Part 3 asked for the fewest commands, took the map away, added a dock and a second robot. Every one of those rooms was a flat grid with a wall round it.

The Cleaning Robot Puzzle: Four New Rules

Thu, 24 Sep 2026 13:00:00 +0300

Ask the cleaning robot for the fewest commands, take away its map, give it a dock that empties its bag, then add a second robot that must not collide with it.

Part 1 and Part 2 kept the robot’s world simple: it knew the map, it worked alone, and any complete route was good enough. This part takes those comforts away one at a time. Each question changes a single rule, and each change turns the puzzle into a different classic problem.

The Cleaning Robot Puzzle: Dirt, Capacity and a Battery

Thu, 24 Sep 2026 12:30:00 +0300

Every floor field gets some dirt, the robot gets a bag that fills up and then a battery that runs out, and the cleaning robot puzzle turns into subset sum, a connected knapsack and the orienteering problem.

In Part 1 the robot could clean everything, so any complete walk was good enough. This part changes the rules so that choices matter: the robot can’t take everything any more, so it has to decide what to take. It builds on Part 1’s depth-first walk and assumes you can read Python.

The Cleaning Robot Puzzle

Thu, 24 Sep 2026 12:00:00 +0300

One robot, rooms full of furniture, a 50,000-command budget — and one short depth-first search that cleans all of it.

This is Part 1, and it solves the original puzzle. Part 2 changes the rules: every floor field gets some dirt, the robot gets a bag that can fill up, and then a battery that runs out. It turns a textbook traversal into a real optimisation problem. Part 3 asks for the shortest route, takes away the map, adds a dock where the bag empties, and brings in a second robot. Parts 4 to 7 keep the square fields but stop the floor being flat: a room that wraps around, a strip with a twist, the walls of a box, and a floor with five squares at every corner. Parts 8 to 11 go back to the flat floor and take its optimisation problems to the tools of operations research, starting with a linear program that bounds the shortest route from below.