The Washroom Game

This article analyses a game where players sequentially choose either to become insiders and pick one of finitely many locations or to remain outsiders. They will only become insiders if a minimum distance to the next player can be assured; their secondary objective is to maximise the minimal distance to other players. This is illustrated by considering the strategic behaviour of men choosing from a set of urinals in a public lavatory. However, besides very similar situations (e.g. settling of residents in a newly developed area, the selection of food patches by foraging animals, choosing seats in waiting rooms or lines in a swimming pool), the game might also relevant to the problem of placing billboards attempting to catch the attention of passers-by or similar economic situations. In the non-cooperative equilibrium, all insiders behave as if they cooperated with each other and minimised the total number of insiders. It is shown that strategic behaviour leads to an equilibrium with substantial underutilization of available locations. Increasing the number of locations tends to decrease utilization. The removal of some locations which leads to gaps can not only increase relative utilization but even absolute maximum capacity.Efficient design of facilities; location games; privacy concerns; strategic entry prevention; unfriendly seating arrangement; urinal problem

Can large language models create new knowledge for spatial reasoning tasks?

The potential for Large Language Models (LLMs) to generate new information offers a potential step change for research and innovation. This is challenging to assert as it can be difficult to determine what an LLM has previously seen during training, making "newness" difficult to substantiate. In this paper we observe that LLMs are able to perform sophisticated reasoning on problems with a spatial dimension, that they are unlikely to have previously directly encountered. While not perfect, this points to a significant level of understanding that state-of-the-art LLMs can now achieve, supporting the proposition that LLMs are able to yield significant emergent properties. In particular, Claude 3 is found to perform well in this regard

About a combinatorial problem with nn seats and nn people

If you want to fill $n \in \mathbb{N}$ seats in succession with $n$ people and the rule that each person chooses one of the seats with the maximum distance to an occupied seat, then you can ask yourself how many possibilities there are for this. In this paper, based on initially mentioned ideas, a formula for the number of these possibilities will be found. In addition, a lower and upper bound for this formula will be given. Finally, formulas for the OEIS sequences A166079, A095236, A095240 and A095912 and an extension of the initial problem are derived.Comment: 21 pages, in German language, 3 figure

The Urinal Problem

Abstract. A man walks into a men’s room and observes n empty urinals. Which urinal should he pick so as to maximize his chances of maintaining privacy, i.e., minimize the chance that someone will occupy a urinal beside him? In this paper, we attempt to answer this question under a variety of models for standard men’s room behavior. Our results suggest that for the most part one should probably choose the urinal furthest from the door (with some interesting exceptions). We also suggest a number of variations on the problem that lead to many open problems.