Beguiling Games II: Nuh-Uh!

First, we’ll look at the game I left you with in the last installment of Beguiling Games.  These were the rules of yet another variation of Tic-Tac-Toe:  if during the game either play gets three-in-a-row, then X wins.  If at the end, no one has three in a row, then O wins.  Does X have a winning strategy?  Does O?  Why?

We’ll show that X has a winning strategy.  As with usual Tic-Tac-Toe, X starts at the center.  If O goes in any square which is not a corner, then X will win as a result of the winning strategy in Tic-Tac-Toe.  So all that’s left to do is look at the case when O goes in a corner.  Let’s suppose it’s the upper right corner.

Then X goes in the lower right corner.  O must block, or else X will get three-in-a-row and win (since X wins if either player gets three-in-a-row).  So now, the board looks like this:

Take a moment to see if you can figure out X’s winning strategy.  Do you see it?  All X needs to do is place in the center of the bottom row.

Look at the top row.  Someone eventually has to place an X or O in this location.  No matter which one, a three-in-a-row will be created, and X will win!  So in this version of Tic-Tac-Toe, X has a winning strategy.

Today’s game will involve a different dynamic — cards and logic.  But first, let me give a little bit of context.  I have often used the book Problem Solving in Recreational Mathematics by Averbach and Chien when I’ve taught an introduction to proofs course.  There are many interesting problems in this book, and the chapter on logic has several problems involving Truthtellers and Liars.

So you have to solve problems by figuring out who are the Truthtellers and who are the Liars.  I like to use these problems to get students writing arguments in complete sentences since not only are they fun to work out, but they don’t involve notation.  Using notation correctly is another issue entirely, and I prefer to deal with that later on in the course.

As before, there’s a story to go along with the game….

Lucas, Mordecai, Nancy, and Ophelia decided that enough was enough, and so ended their Saturday afternoon’s cramming for the National Spelling Bee.  Wondering what they might do to pass the time, Lucas suggested, “Let’s play Nuh-Uh!”  Everyone enthusiastically agreed.

And so a game of Nuh-Uh! — whose creators, incidentally claim that it is the only game on the market which “Tortures your Mind, Warps your Character, and Impoverishes your Soul — all at the Same Time!” — commenced.

So Lucas shuffled the deck of eight cards; four of the cards had the word Truthteller on them, and four had the word Liar.  Dealing from the left, Lucas first dealt a card to Mordecai, then one to Nancy, one to Ophelia, and then finally one to himself.  Cards are dealt face down so a player can only see his or her card.

Per the rules of Nuh-Uh!, Lucas, then Mordecai, Nancy, and finally Ophelia made the statement “I am a Truthteller.”  Of course, such a statement was consistent with each player’s card, regardless of what was written on it.  Thus ended the first round.

In subsequent rounds, each player passes his or her card to the player on the left, and makes a statement consistent with the new card.  Thus, if Lucas passed Mordecai a card with Liar written on it, Mordecai would have to make a statement which is false.  The statement a player makes is based on the knowledge of the cards he or she has seen, and any other information which may be deduced from the statements of the previous players.

The same player begins each round of making statements.  So Lucas began each round in this game.  Once the game is over, the deal passes to the left, so Mordecai would deal and begin each round with a statement.

After a player makes a statement, players may, beginning with the player on the left of the one who just spoke, “Declare” — that is, say what card each of the players held during the first round.  The declaring player then looks at the cards held by the players to decide if he or she has won.  If an error is made in declaring, the player drops out and play continues; otherwise the cards are turned over and the deal passes to the next player on the left.

So the players passed their cards to the left, and thus the second round of making statements began.  Lucas started off with “Ophelia told the truth in the first round.”  After a polite few seconds of pause to give someone a chance to declare, Mordecai said, “Lucas also told the truth in the first round.”

Another brief pause ensued before Nancy stated, “There is at least one liar at the table.”  Note:  there are four Truthteller and four Liar cards, so it is possible that all players were dealt a Truthteller card.

After Nancy made her statement, one of the players had enough information to declare and win the game.  Which player declared?

Yes, there are enough clues to solve this!  But a word of caution — the solution to the puzzle only involves answering the question, “Which player declared?”  There may not be enough information to give a more complete answer….

Good luck!  In the next installment of Beguiling Games, I’ll give the solution to this logic puzzle, and give a geometrical two-player game to analyze as well.  Happy solving!