# Liar's Dice

Players: 3-6
Ages: 7 and up
Cost: Free! (although you need 5 dice per person)
Math Ideas: Probability, expected value
If we roll six dice right now, how many do you expect will roll a 4? What if we roll 12 dice? 20?
Based on my bet, what dice do you think I rolled?
Is my bet likely or unlikely? How do you know?

If you are a reader of Games for Young Minds, then you are probably either:

a) A parent of kids who, after a long Winter Break, need something to keep them entertained so they will Stop. Screaming. At.  Each. Other.

b) A teacher who is trying to think of a fun activity for the first day of the new semester.

c) All of the above

I fall into that third category, so I thought I'd share a game that is always one of my biggest hits of the school year: Liar's Dice. This is also a perfect home-for-the-holidays game, since you can easily accommodate three, four, or more players. As long as you have enough dice for everyone, everyone in the family can join!

## How to Play

To play, each person needs five dice and a cup. If you've got a set of Tenzi dice lying around, then you've got enough dice for eight players! I think the game is best with 4-6 players, but I bet bigger games can also be fun.

All players roll their dice at the same time by flipping over their cups, keeping the dice hidden underneath. Look at your dice, but keep them hidden from your opponents.

Once each player has looked, the betting starts. Each person takes a turn betting about how many dice of each number are on the table. The lowest bet is "one 1," which means that you are betting there is at least one die that rolled a 1, somewhere on the table.

So let's say the first player bet "one 2," meaning that she thinks there is at least one die that rolled a 2 on the board. The next player must raise the bet, either by raising the value on the dice to "one 3" or "one 5," or by raising the number of dice to "two 2s" or "three 1s."

One important point: You can never decrease the number of dice in your betting. As a result, the bets will become less and less likely, until one player just doesn't feel comfortable raising the bet.

Let's say that's you: the person right before you bet "six 4s," and you just don't think there are six 5s or 6s, much less seven of any particular dice. Instead of raising, you call "bluff!"

At this point, everyone at the table raises their cups, and you count the total number of 4s on the table. If there are six (or more) 4s on the table, your opponent was right! You lose a die. But if there are fewer than six 4s on the table, you win and your opponent has to forfeit a die. Everyone puts their remaining dice in their cups and rolls again. You continue playing, calling bluffs, and forfeiting dice, and the last player standing is the winner!

Since this game is a bit complex, here is a great, one-minute explanation from Youtube.

## Where's the Math?

Liar's Dice is all about probability and using limited information to make inferences. Let's say you are playing a game with four players. On the very first roll, you look down to see the dice pictured: two 1s, a 2, a 3, and a 5.

You decide to start slowly and bet one 1 (which you know is a safe bet, since you have two of them).

• Player B: One 3

• Player C: One 6

• Player D: Two 1s

Now it's your turn again. What information did you learn?

Well, Player B most likely has a 3, since she made that initial bet. Since you also have a 3, you could probably feel safe betting "two 3s" on this turn. But what else do you know? Player C probably has a 6, and Player D likely has at least one 1. Since you have two 1s, then you know that betting on 1s is probably your safest course of action in the next round.

So you bet "two 3s" and wait to see what your opponents do, knowing that they are gleaning information from your bets in the exact same way that you're gleaning information from theirs.

You can also use math to determine when to call your opponent's bluff. Let's say your opponent in this same game bets "seven 2s."

Out of the twenty dice on the board, you can see five of them. Only one of them is a 2, so that means that six of the remaining fifteen dice must be 2s in order for your opponent to win.

The odds of rolling a specific number are 1 in 6, so in a random group of fifteen dice you'd expect two or three of the dice to be 2s. Based on that information, it seems unlikely that your opponent is correct. Even if she has a couple of 2s hidden under her cup, she's probably counting on some lucky rolls among her opponents. It's probably safe to call her bluff, especially since your other option is to raise the bet even higher!

The tricky thing about Liar's Dice is that you are purposefully playing with limited information. If you give your opponents too much information, they have the ability to beat you.

With that in mind, the best questions are probably the sorts of questions that help your kids understand the basics of probability.

A great introductory question is "If I roll this die, what are the odds of rolling a 4?" You can begin to build a common vocabulary of probability with your kids.

If your kids feel comfortable with the idea of 1 in 6, then you can ask this question: "If I roll six dice at the same time, how many of them do you expect to roll a 4?" Just like the last question, the answer to this is that one out of the six dice should roll a 4. But by framing the probability in terms of multiple dice, you are helping your child build a method of thinking about probability in Liar's Dice.

The last question in this sequence directly ties to the game: "If we roll twenty dice at the same time, how many would you expect to roll a 4?" If your child could understand the last question, then they have a strategy for answering this one: Group the dice into sets of six, since we expect to roll a 4 in each set of six dice. You end up with three full sets, plus a couple of extra dice. Therefore, it's safe to assume that between three and four of the dice will roll a 4.

The most fun thing about probability is that you can test your predictions! They won't always work out, but it might be fun to try! Roll your twenty dice a bunch of times, counting the number of 4s you see with each roll. Sometimes there will be 3, sometimes there will be 4, and sometimes there will be fewer or more. While you roll, you can ask questions such as "Do you think we'll ever roll all twenty dice and not roll a single 4? Why or why not?"

This sort of scenario, which is plausible yet very unlikely (~3% chance) will further help your child build an intuition for the likelihood of events in a game like Liar's Dice.

Ok, I am stopping now simply because this is the longest newsletter I have ever written! I have so many more ideas for Liar's Dice, and for explorations of probability. I'll have to find another dice game so I have an excuse to write more!