# Hare & Hounds

## Hare & Hounds

This classic two-player strategy game was popular in medieval northern Europe until the 19th century. It’s also known by many other names, including the Soldier’s Game, the French Military Game, and The Devil Among Tailors, to name just a few[1].  If you’re interested in combinatorics or graph theory, this might be the game for you! Play against the computer at the NeoK12 website, or play it with a friend!

## Rules of the Game

To start, the board is set up as shown below:

To begin play, you first have to pick a side: hare or hounds.

Hounds:

• Movement: The hounds are restricted and can only move in one direction (never backwards), from one space to another adjacent space. We’ll be starting the hounds on the left side, which means the hounds can only move diagonally or directly to the right, or up and down in one column (never to the left).
• Goal: Trap the hare so it can’t make any moves (i.e., on a space that can be surrounded by 3 hounds with no adjacent open spaces).
• Note: If the hounds move vertically 10 moves in a row, it’s considered “stalling” and the hare wins.
Hare:
• Movement: Hares can move in any way they want – even backwards! However, the hare (ironically) can’t “hop” over the hounds. It must move from one space to another adjacent space just like the hounds.
• Goal: Escape the hounds and reach the leftmost space on the board. If the hare passes the leftmost hound on the board, it automatically wins since the hounds can’t move backward to trap it!

## Strategy: Keeping the Opposition

One strategy – for either the hare or the hounds – is to try to keep all the animals on the board in a certain formation. Look at the board below.

The strategy is, at the end of your turn (i.e. when the other player’s turn starts), to try to ensure that one animal is on a purple space, one is on an orange space, and two are on white spaces. We call this formation “keeping the opposition.” Here are a few examples:

At the end of your turn, you always want to keep the opposition.  If your opponents move in such a way that you can’t make that formation, try to move so that it’s impossible for your opponent to take the opposition on their turn. Keep reading to see these formations in action.

FOR THE HARE

As the hare, one good strategy is to make it seem like you’re cornered, but then force a win! Trying to move the hare forward isn’t always the best idea, since it’s easy for the hounds to block your every move. Sometimes, letting the hounds come to you can be advantageous. Look at the formation below:

If you can get the hounds into this position (where it’s their turn – this is important), you’ve already won! Because the hounds have to make a move, they are forced to move either the hound above or below you to the rightmost space. Then, you can make your escape through the open spot they leave behind – the leftmost hound has no way to stop you, no matter where it moves.

Note: this formation has the desired pattern for keeping the opposition!

However, as the hare, actually getting the hounds into the desired formations is the hard part. Because you can’t anticipate your opponent’s every move, not everything will work out perfectly. You may not be able to make a move that stays on pattern. Your best bet is to try to get the hounds to trip up and leave an open space that can’t be blocked – you don’t necessarily need to keep pushing forward if the hounds keep blocking your path!

FOR THE HOUNDS

As the hounds, if you can start your turn on the formation shown below, you’ve already won! This might be a little harder to see than the last example. Let me explain.

This starting formation directly results in the formation shown below for the hare’s strategy, after a couple of smart moves:

1. Move the bottom hound one space to the right. Notice that at the end of this turn, you now have a formation that’s keeping the opposition, just like we wanted! It also forces the hare to make the only move it can: to the left one space.
2. To keep the opposition, move the leftmost hound to the right one space. Now we have the same formation as we had for the hare’s strategy.

But this time, it’s the hare’s turn, and their only move is one step to the right. Finally, move the leftmost hound to the right one space, and the hare is trapped! You win!

But this is just one example, and not necessarily the easiest pattern to form. Let me show you a way the hounds can win every time!

A (MOSTLY) FOOLPROOF WINNING STRATEGY

As the hounds, you actually have a competitive advantage: you go first. If you move the leftmost hound right one space, you already have the opposition (two hounds on white, one on purple, and the hare on orange) at the end of your first turn. Then, no matter where the hare moves (orange, purple, or white), you can always counter their move by moving one of your hounds to keep on the opposition.

It’s not just that simple, though. Sometimes, you can be on pattern, but still leave room for the hare to get by. Here’s an example of such a formation:

Because the hounds can’t move backwards, this formation, although it’s on pattern (supposedly to the hounds’ advantage), the hare will win. So, aside from keeping the board on pattern at the end of each of your turns, you also have to make sure your move isn’t leaving any wiggle-room for the hare to get through. If you can do that, you’re guaranteed to win!

After learning everything you know now, you may want to choose your side strategically. Or, maybe you’re looking for a bit more of a challenge – it’s all up to you. If you decide on the hounds, happy winning! If you choose the hare, good luck outsmarting the hounds!

BUT WHERE DOES ALL THIS “OPPOSITION” STUFF COME FROM?

As it turns out, this game has been very popular with mathematicians.  In 1943, mathematician Frederick Schuh was able to reduce the game to 24 positions for the wolves which guarantee a win[2].

Schuh found that if the hounds go first, they are guaranteed a win wherever the hare starts.  The only exception is if the hare starts right in the middle of the three hounds, since on the hounds’ first turn they would then have to move past the hare and give it an easy win.

The game was later popularized by Martin Gardner in the 1960s, and Schuh’s winning positions would be built upon, resulting in the notion of the “keeping the opposition.”[3]

## References

[1] Elwyn Berlekamp, John Conway, R. K. Guy, Winning Ways for Your Mathematical Plays, Volume 3, 2018, 711 - 723.
[2] Frederick Schuh, The Master Book of Mathematical Recreations (transl. F. Gobel, ed. T.H. O'Beirne), Dover Publications, New York, 1968, 239-244.
[3] Martin Gardner, Sixth Book of Mathematical Games from Scientific American, W.H. Freeman, San Francisco, 1971, Ch. 5.