# A basic game to make arguments about

I think a significant start to my development as a mathematician was playing card games (mostly Euchre) with my parents in my youth. After a particular round, my father would tell me, "Well, with your hand, you should have done this. It would have forced me to do that. And then you could've come back with this, thus winning the round." Basically, I feel like I was taught early on how to reason and make an argument in the context of a game.

Fast forward, next semester I'm teaching an "intro to proofs" course (not for the first time) and I'd really like to start the semester off with having the students play a game in pairs such that they can explain after each round who should have won. Unfortunately, most of the games I know and enjoy playing are too complicated or too long to achieve this efficiently.

So my question is: Does anyone know of a game that satisfies some or all of

1. Easy to learn (so it doesn't take all class to explain it).
2. Rounds are short (so multiple rounds can be played and students can get a sense of how it works).
3. Easy to move back and forth in time (so it's easy to replay the game from various points, so probably only a handful of moves in each round).
4. Sometimes, if the opponent plays a certain way, your choice of moves may be reduced to only one option.
5. Conducive to basic logical analysis.

The only idea I've had so far is Tic-Tac-Toe, but I thought I'd ask to see if there's maybe a lesser-known game that also fits the bill.

• – Benjamin Dickman Nov 12 '14 at 21:02
• Mancala: en.wikipedia.org/wiki/Mancala – Andrew Sanfratello Nov 13 '14 at 2:02
• Tic Tac Toe is a relatively complicated game. I would avoid it. – Jack M Nov 18 '14 at 1:03
• @Jack M: Really? There's only 3 nonisomorphic first moves. The second player has at most 5 moves and after that most of the moves are either required (to block a win) or obvious (setting up to win in maybe 1 of 2 places). – Aeryk Nov 18 '14 at 14:18
• Definitely Nim. Great logic game. – Dave Chamberlain Jan 3 '15 at 2:04

Sprouts is a very excellent game that has a lot of scope for proving things, not just for strategy but also for other aspects of the game too. http://en.wikipedia.org/wiki/Sprouts_%28game%29

The game begins with a number of dots drawn on the page (5 is good for a short game). On your turn you can join two dots or a dot to itself as long as any line you draw doesn't cross other existing lines or pass through other dots. Then you draw a new dot on the line you just drew. Also, if a dot has three paths exiting from it, then it can no longer be used. Note that this means any dot drawn on a line has only one more line that can go from it.

The winner is the last person who draws a line.

Good strategies involve enclosing dots inside loops so that even though they have a free path you can't draw it without crossing another line.

I think you will like Eleusis, a card game where one player invents a rule for valid sequences of cards, and the rest of the players "experiment" by proposing a sequence and being told whether it is acceptable or not.

The game rules are a bit involved, but there is a version called "Eleusis express" that may be suitable. Quoting the Wikipedia link above, "In 2006, John Golden developed a streamlined version of the game, intended to assist elementary school teachers in explaining the scientific method to students."

The most basic one would possibly be the following classical game-theoretic example (also appearing in the movie "Last Year at Marienbad").

The game starts with a number of matches on the table (replaceable by any collection of similar object if matches are unsuitable for class, of course). The two players take rounds; at each round the player picks $1$, $2$ or $3$ matches from the table. The player picking the last match loses.

The interest of this game is that it is easy to analyse it completely, determining which player has a winning strategy given the number of matches. It is usually played with 13 matches, in which case the second player has a winning strategy. This produces the following dialogue in "Last year at Marienbad" (approximately remembered and translated):

A: "This is a game I always win.

B: If you can't loose, it is not a game.

A: I can loose, but I always win. I'll let you play first"

• The game played in Last year at Marienbad was a version of Nim with heap sizes $1, 3, 5, 7$. – Dag Oskar Madsen Jan 3 '15 at 15:44

You might also consider the game Set. https://en.wikipedia.org/wiki/Set_(game) shows some of the basic games with the cards and some associated combinatorics. I also have some more materials for this game if you are interested.

• Incidentally, I have some remarks about the game SET in my answer to MESE 2528. – Benjamin Dickman Jan 16 '15 at 21:31

To some, chess is. It's basic in the sense that it is perhaps the most popular game, and it provides many fruitful visual aids containing TONS of information!

There is a simple game that I like a lot as it illustrates in an interesting fashion the theorem that in deterministic two-players games without draws nor hidden information, either the first or the second player has a winning strategy.

The game starts with a chocolate bar, of size $n$ by $m$ say, whose uppest rightmost piece is called the poisoned piece. At one's turn, one chooses a piece of chocolate and removes all pieces lying in the smallest lower-left semi-infinite rectangle containing the chosen piece (so , choosing the poisoned piece would remove all pieces, while choosing the lowest leftmost piece on the first move would only remove one piece). One says the player "eats" the removed pieces, and the player eating the poisoned piece looses.

Now, by the above-mentioned theorem, either the first or second player has a winning strategy. Claim: for this game, we can prove that the first player has a winning strategy regardless of $n$ and $m$ (except for $n=m=1$, that is); but as far as I know, the winning strategy is unknown except for very small values of $n$ or $m$.

Proof of the claim: assume the second player has a winning strategy. Then if the first player first picks the lowest leftmost piece, the second player has a follow-up move consisting in picking some piece $P$. But then, for the first player to pick $P$ as its first move would be a winning strategy, as it leads to exactly the same position as the above scenario, except with player two to take her turn.

• aka Chomp – Benjamin Dickman Jan 12 '15 at 22:08
• @BenjaminDickman: thanks, I never new the name of this game! The link you provide is great, with many variations, I advise anyone interested by my answer to read it. – Benoît Kloeckner Jan 13 '15 at 12:47