Problem of the Week 970

A Four-Square Square Arrangement

The point of this problem is to learn about an array of nonoverlapping congruent squares such that each one touches exactly four others.

Take a regular n-gon P with unit sides. From each side of P, construct a unit square exterior to P, resulting in n squares. These are the "inner" squares. Take n more unit squares, the "outer" squares, and place each into the wedge shaped gap between adjacent inner squares so that it touches both of them. Then each inner square touches four others (two inner and two outer) and each outer square touches two inners. Find the least n so that there is such an arrangement so that each outer square touches its two neighboring outer squares as well. This yields a set of 2n squares so that each one touches exactly four others (the touching graph is 4-regular).

Source: Squares touching a constant number of other squares, by Erich Friedman in Geombinatorics, XII, 55-58. October 2002

© Copyright 2002 Stan Wagon. Reproduced with permission.

5 November 2002