## GCHQ Resistor Puzzle Solution

I've spent the last day doing this GCHQ puzzle. I waited until I got to work on Monday so my students could help me with an issue I was having yesterday.

The Problem I had yesterday was with the third cluster to the left of the cell:

Drawing it out I reduced it a bit by combining two of the nodes to this:

At this point this second diagram illustrated the minimum case of networks that I didn't know how to solve. After going down a dead end of looking at a similar configuration called a Wheatstone Bridge, but I then found out about something called a Δ-Y transformation which can turn a triangle into a Y shape.

This turns our problem into one we can reduce using the normal series and parallel arguments. In fact it looks like being able to switch back and forth between Δ and Y, plus reduce the parallel and series resistors are universal: by which we mean we can solve any system. I haven't proved this yet but I'm going to have a think about it.

In general if I have a problem of the form:

This is hideous, but once we had this formula we were able to plug in numbers for any of that form. Since we had a class of people we split up the resistor clumps and crowd sourced the solution. Here are some photos of the process:

The two messages say: "Puzzled? Try looking at the unused letters" and "ever wondered what a puzzler does all day?" Crossing out the unused letters we get the remaining letters below:

These lettrs give: Rvxdpxorhaxepzzoamxwaxigqaxgxwdlotxdvxniafxndxmdoqaxvrbtxdpnxfdlaxgnxksijxtgmixsglaalmxtdnxsdxtdnxph which has xs in what look like sensible places for spaces. Here we can put them in:

rv ydp orha epzzoam wa igqa g wdlot dv niaf nd mdoqa vrbt dpn fdla gn ksij tgmi sglaalm tdn sd tdn ph

This is a substitution cipher and putting g as a (single letter word) gives you a way in. After some work you get:

IF YOU LIKE PUZZLES WE HAVE A WORLD OF THEM TO SOLVE FIND OUT MORE AT GCHQ DASH CAREERS DOT CO DOT UK

An advert for working at GCHQ. They've earned it I suppose.