# Melbourne University Mathematics and Statistics Society

 Solution for 5.1 Power Circuits (Answer = ZOLDER)

The first thing to notice is that the KEY has 3 graphs, indicating that each graph corresponds to a single letter. Furthermore, with all the pluses, minuses and zeros floating around, it's safe to assume that each graph gives a number between 1 and 26, and a standard alpha-numeric substitution is used (1 = A, 2 = B, 3 = C) etc. So the question remains, how could anyone possibly get a number out of these cursed graphs, in a way so the bottom three correspond to 11 = K, 5 = E and 25 = Y respectively?

The vertices of each graph can be assigned a number from the bottom up as follows. The number is a power of 2, where the power is given by the signed sum of the vertices being pointed to by the vertex in question. The sign is given by the sign of the arrow pointing to that vertex. This is all a mouthful so for example, if a vertex only has one arrow pointing away from it with a +, and that arrow points to a 0, then the vertex is assigned 2^0 = 1. If a vertex points to 2 other vertices, one labelled 2 by a - arrow, and one labelled 8 with a + arrow, then that vertex is assigned 2^(8 - 2) = 2^6 = 64.

Here is the puzzle with numbers assigned to each vertex. The signs on the pink vertices have been moved outside the circles, we'll get to those in a moment!

Finally, we use the pink signed vertices to extract a single number, by simply taking the signed sum. So, from left to right we have

NumberLetter
32 - 8 + 2 = 26 Z
16 - 1 = 15 O
4 + 8 = 12 L
4 D
4 + 1 = 5 E
16 + 2 = 18 R

This gives the solution ZOLDER, which is a motorsport circuit in Belgium.

Author's Notes:
Power circuits were introduced in 2010 by Alexei G. Myasnikov, Alexaner Ushakov, and Dong Wook Won in this paper. For those playing along at home, they use power circuits to prove that some surprising things can be computed in polynomial time! It's pretty cool.

When I wrote this puzzle, my goal was to separate the Plughs from the Plughs. It only partially succeeded, and it caught a few too many teams as collateral! Sorry about that.