Hot Wire

Copyright 2010, Phil Leduc

Hot Wire is an electrifying misère combinatorial game for two players. Players take turns connecting wires to pins on a prototype circuit board, which for safety reasons, is simulated using graph paper, pencils and a straightedge. Each connection reduces the number of possible connections until only the hot wire connection remains. The player who forces his opponent to connect the last wire wins the game.

Game Play

Starting with the red player, on each turn a player must connect a wire between two pins using the following restrictions.

Wire placement may not create an electrical short. That is,

• Only one wire may be connected to a pin.

• Wires may not cross directly over other pins but may pass close to pins.

• Wires may not cross other wires.

In addition, to encourage “open” games and conserve wire,

• Wire length is limited by the sum of vertical steps plus horizontal steps needed to move from the starting pin to the ending pin. In Hot Wire, wires can have lengths of 1, 2, 3, 4, 5 or 6.

Refer to Figure 2 for legal wire lengths. Aside from rotations, these are the only wires used in the standard game of Hot Wire. Figure 3, shows three illegal moves made by Black.

Winning the Game

Play continues until there are no more legal moves left on the board. The player who makes the last connection loses the game. Ties are not possible.

Strategy Tips

There is a lot more to this game than meeting the eye. Pay close attention to isolated groups of unused pins. Pins in these groups can only connect to other pins in the same group but not necessarily all the pins in the group. In Figure 4, the isolated groups of pins are indicated in gray. Note the subtle potential connection d5 to g3. Finding these "hidden" moves is part of the challenge of Hot Wire.

Isolated groups of two or three pins can only support one wire. If, following your move, you leave your opponent with an odd number of these one-wire groups, you will win. For example, if you leave your opponent with only three one-wire groups, your opponent connects one wire in one of the groups, you connect one wire in either of the remaining two groups and your opponent is shocked into connecting the hot wire in the last group. In figure 4, the pins b7-c7, f1-g1, and a3-a4-a5 form one-wire groups.

Isolated groups of four pins can support either one or two more wires and complicate the endgame. These groups can be thought of as switches since a player has the option to change the number of remaining moves from even to odd or odd to even or to simply leave the parity as is! In Figure 4, pins b2-c2-c3-d3 form a 4-pin switch. If you connect b2 to c2, you change the parity of one-wire groups. If you connect c2 to c3, you are in effect removing two potential moves at once and the parity of one-wire groups stays the same.

Have a look at Puzzle 1. Assuming Red played first, whose play is it and who wins the game? The answers follow the Variants section.

Variants

The recommended board size is 7 x 7 with wire lengths 1, 2, 3, 4, 5, and 6. For variety, players can agree to use a different size board (including rectangular boards) or a different set of lengths, such as 1, 3, and 5, or just one length, or even unrestricted wire lengths. The analysis of the game changes significantly when the allowable wire lengths change. But the underlying principles of one-wire groups and switches still apply!

Solution to Puzzle 1

Since Red went first and there are seven red and seven black wires, it is Red’s turn. There are four one-wire groups and two four-pin switches. If Red connects a wire in either of the switch groups, Black can use the last switch to create an odd number of remaining one-wire groups and will win. If Red uses one of the one-wire groups, Black should use another one-wire group. This whittles down the one-wire groups to an even number again and Red faces the same dilemma. Eventually, Red will be forced to use one of the switch connections and Black will leave the last hot wire connection using the remaining switch. Black wins.

Download Extras

The following Hot Wire files are available for download from my Game Files page:

• Hotwire.pdf - Hot Wire rules

• Hotwire.swf - Playable Flash app!!