The Urinal Problem

The unwritten rule of the men's room, checked against the numbers. Pick the urinal that is furthest from everyone else. Nobody taught it to you, yet everybody follows it. So what actually happens when nine people walk into a row of five urinals? WHAT YOU THINK YOU KNOW, MEASURED Call a spot comfortable when the nearest occupied urinal is at least two away. The ceiling for n urinals is then ceil(n/2) people — three out of five, a 60% occupancy rate. The fourth arrival always ends up next to somebody. THE FIRST PERSON DECIDES EVERYTHING If the first arrival takes an odd-numbered urinal (1, 3 or 5), three people can stay comfortable. If they take an even one (2 or 4), the ceiling drops to two — before anyone else has even walked in. One opening move fixes the throughput for everyone who follows. The rule of thumb: take an odd-numbered urinal first. WHAT YOU CAN DO - Simulate 2 to 12 urinals and 1 to 30 people - Watch the whole thing play out in pictograms at 1x / 2x / 4x, with a scrub bar - See comfort rate, peak comfortable count and the theoretical maximum - Read the timeline as a Gantt chart; cramped stretches are hatched - See why each person chose their spot — far-end seekers, wall huggers, germaphobes and people who ignore the rule entirely - Export the result as a vertical card image and share it - Every run is fully reproducible from its seed, and kept in your history - A short explainer page covers the theory NOTES - Everyone is drawn as a pictogram, the way restroom signage does it - Completely free. No ads, no in-app purchases - No network access and no permissions. Everything runs on your device - English and Japanese