Fewer or More Knights:

Try 7 knights, or 11. Any odd number will work.

Even Knights:

Start with an odd number of knights, but the player who ends up with an even number of knights wins. (This is a useful game to play in trying to figure out a winning strategy for Odd Knights.)

Even and Odd Numbers:

Odd Knights can be used to explore questions relating to parity (whether a number is even or odd). First, play several rounds of the 9-knight game. Then ask how many of their games ended in a tie. Let kids explain why 9 can't be split into two even or two odd groups. Give them time to notice that, in fact, any odd number of knights would have this property (no ties!). If they haven't already speculated what will happen when starting with an even number of knights, play a few rounds of an 8-knight game. It shouldn't take long to notice why this is a boring game. It always ends in a tie! Can kids explain why an odd + odd = even, even + even = even, but even + odd = odd?

Another interesting question for younger players: After they have taken a single knight, ask them whether they should now try to get an additional even number of knights, or an additional odd number of knights. (Adding 1 to an even number equals an odd number, so after taking 1 knight, a player should try to get an even number of additional knights.)

Or notice that, after taking 2 knights to start the game, a player's goal is to take an additional odd number of knights. (Adding 2 to an odd number equals an odd number.) Since this was also the original goal, taking an even number of knights to start a game basically starts a new game of Odd Knights but with a different starting arrangement and the other player going "first." (This insight is important for the winning strategies described below.)

Problem-Solving:

Older kids can explore the question of which player can be guaranteed a win, and what strategy that player should use. First, play several rounds of the 9-knight game. As you play, ask at which point in the game they are able to tell who will win, and at what point a player is able to control whether they will win or lose.

One way to test your guesses is to use the common problem-solving technique of first looking for a simpler problem to solve. In this case, play several rounds of a 5-knight game (or even a 3-knight) game to determine which player can guarantee a win, Player 1 (P1) or Player 2 (P2).

3-knight game:
If P1 takes 1, P2 takes 1 also, forcing P1 to lose with 2 knights. If P1 takes 2, P2 takes 1 to win. Either way, P2 wins.

5-knight game:
P1 can take 2 to start the game leaving 3 knights, so the 3-knight game above plays out, but with the roles switched. P1 can guarantee a win. (Note that starting the game with 2 knights is basically the same as starting the game with no knights. 2 plus an odd number is still an odd number, so the goal is still to end with an odd number after the initial 2 knights.)

7-knight game:
If P1 takes 2, P2 can take 2 to leave 3 knights, so the 3-knight game plays out with P1 losing.

If P1 takes 1, P2 can take 1 leaving a 3-2 split of knights. Then for the rest of the game, P2 can mirror any moves P1 makes to win. So if P1 takes 1 knight, P2 takes 1 knight. If P1 takes 2, P2 takes 2. In any case, P2 can guarantee a win. (Note that, after each player has taken 1 knight, their goal from that point is to get an additional even number of knights, since 1 plus an even number equals an odd number.)

The game grows in complexity as more knights are added, but kids may notice some basic strategies that they can continue to employ in analyzing larger games: mirroring their opponents moves, or noticing when a game has reduced to a simpler game. In general Odd Knights is winnable by Player 1 with 1, 5, 9, 13, etc. knights, and winnable by Player 2 with 3, 7, 11, etc. knights (though the winning strategies are harder to determine).