# Sunday Times Teaser 3033 – Goldilocks and the Bear Commune

*by Susan Bricket*

#### Published Sunday November 08 2020 (link)

In the bears’ villa there are three floors, each with 14 rooms. The one switch in each room bizarrely toggles (on <—> off) not only the single light in the room but also precisely two other lights on the same floor; moreover, whenever A toggles B, then B toggles A.

As Goldilocks moved from room to room testing various combinations of switches, she discovered that on each floor there were at least two separate circuits and no two circuits on a floor had the same number of lights. Furthermore, she found a combination of 30 switches that turned all 42 lights from “off” to “on”, and on one floor she was able turn each light on by itself.

(a) How many circuits are there?

(b) How many lights are in the longest circuit?

This teaser appears to have a flaw since it produces three solutions. But if we require the configuration of the circuits on the three floors to all be different we then obtain a unique solution.