Greed
There is a healthy number of turn-based dice games with a press-your-luck element such as 10,000, Farkle, and Greed. Here is a variant that keeps the numbers smaller and probabilities simpler in order to make the arithmetic and analysis more tractable for younger audiences.
Setup
Split students into groups of 2-5 and give each group 4 standard 6-sided dice. They will also need some paper to keep score. Students can choose who goes first or roll for it, from which point turns proceed clockwise. All players start with a game score of 0 points.A turn goes as follows:
- A player starts their turn with a turn score of 0.
- The player starts by rolling all four dice hoping for a combo. If no combos are produced, their turn ends. Otherwise, the player must choose one or more combos to set aside. Combos are as follows:
- 1 and 2 are worth 1 and 2 points, on their own.
- Pairs of summing to 7 are worth 7 as a pair. (For example, a 2 and 5 are together worth 7.)
- Doubles, triples, and quadruples are worth their sum. (For example, three 4's are together worth 12.)
- When a player sets a combo aside, its point value is added to their turn score. The player then has the option of ending their turn or rolling again. Their turn continues until either:
- The player rolls again and fails to get a combo. They earn no points in this case.
- The player quits after rolling a combo. Their turn score is added to their game score.
- When rolling again, only the dice not currently set aside in a combo are rerolled and new combos must come from this subset of dice. If all four dice are set aside in combos, then all four are rerolled anew.
When one of the players reaches a game score of 100 or more, each other player gets one more turn to try to beat it. The player with the highest game score after all others have had their turn wins.
Example
Here are the first three turns of an example two-player game:
- Player 1 starts turn with game score 0:
- Rolls 1, 1, 2, 3 and keeps 1+1+2. Turn score now 4.
- Rerolls one die, gets 5, busting. Turn ends with game score still 0.
- Player 2 starts turn with game score 0:
- Rolls 3, 3, 4, 6 and keeps 3+4. Turn score now 7.
- Rerolls two dice, gets 1, 5 and keeps 1. Turn score now 8.
- Rerolls one die, gets 2 and keeps 2. Turn score now 10.
- Rerolls four dice, gets 3, 3, 3, 4, and keeps 3+3+3. Turn score now 19.
- Ends turn. Game score now 0 + 19 = 19.
- Player 1 starts turn with game score 0:
- Rolls 4, 5, 5, 6 and keeps 5+5. Turn score now 10.
- Rerolls two dice, gets 1, 6, and keeps 1+6. Turn score now 17.
- Rerolls four dice, gets 3, 3, 3, 5, and keeps 3+3 (choosing to leave one 3). Turn score now 23.
- Rerolls two dice, gets 1, 6, and keeps 1+6. Turn score now 30.
- Rerolls four dice, gets 1, 2, 3, 4 and keeps 1, 2, 3+4. Turn score now 40.
- Rerolls four dice, gets 2, 2, 5, 6 and keeps 2+5, 2. Turn score now 49.
- Ends turn. Game score now 0 + 49 = 49.
- Player 2 starts turn with game score 19:
- Rolls 1, 1, 1, 6 and keeps 1+6, 1, 1. Turn score now 9.
- Rerolls four dice, gets 1, 3, 4, 4, and keeps 4+4 (choosing to leave a 1). Turn score now 17.
- Rerolls two die, gets 1, 3, and keeps 1. Turn score now 18.
- Ends turn. Game score now 19 + 18 = 37.
- Player 1 starts turn with game score 49:
- Rolls 3, 4, 5, 6 and keeps 3+4. Turn score now 7.
- Rerolls two dice and gets 5, 6, busting. Turn ends with game score still 49.
- Player 2 starts turn with game score 37:
- Rolls 1, 5, 5, 5 and keeps 1, 5+5+5. Turn score now 16.
- Rerolls four dice, gets 3, 6, 6, 6, and keeps 6+6+6. Turn score now 34.
- Ends turn. Game score now 37+34 = 71.
Initial Questions
- What are good circumstances to keep rolling versus quit and keep the points? For example:
- When is it a good or bad idea to roll all four dice? Three? Two? One?
- Is there a "sweet spot" for the number of points to earn before stopping? Or is there a number of points that is too large to risk?
- How does each roll depend on the previous rolls?If you have had a string of several lucky rolls, do you think your luck will continue? Or that you're "due" a loss?
- Is it best to set aside all combos before rolling again or is it sometimes useful to have the extra dice to reroll?
- How does each roll depend on the previous rolls? For example:
- If you have had a string of several lucky rolls, is that evidence your luck will continue?
- Are you ever "due" a loss?
Explorations
After considering the different ways four dice can roll, students might note that four dice will always produce at least one combo. One way to see this is to note that two of any number is a double, so the dice would have to be distinct to not form a combo. Since 1 and 2 are combos on their own, the four distinct numbers would have to be 3, 4, 5, 6. However, 3+4 = 7 is a combo. You can work out the probabilities of getting at least one combo when rolling any number of dice:
| Number of Dice: | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Combo Probability: | 1/3 | 13/18 | 17/18 | 1 |
A sense for these numbers can be useful for answering questions like whether it's worth rolling at a certain point in the game.
If a player rolls 3, 5, 5, 5, for example, if they keep 5+5+5 and roll again with one die, they have a 2/3 chance of busting. On the other hand, if they keep 5+5 and roll again with two dice, they only have a 5/18 chance of busting. So if they're determined to continue their run, it seems best to only keep two of these numbers.
If students have a more advanced sense of probability, concepts like expected value can be used. In the previous example, for instance, we can compute the expected value after two rolls based on keeping 5+5+5 versus keeping just 5+5. The expected value of a roll with one die is
\[ \frac{1+2+0+0+0+0}{6} = \frac{3}{6} = \frac{1}{2} \]so the expected value of the 5+5+5 and a single follow-up roll is 5+5+5+0.5 = 15.5. On the other hand, the expected value of a roll with two dice, keeping all combos, is
\[ \frac{12(1 + 2) + 4(3 + 4 + 5 + 6)}{36} = \frac{108}{36} = 3 \]so the expected value of 5+5 and a single follow-up roll is only 5+5+3 = 13. (You might also consider the chance of being able to roll all four dice again in each case, since that would guarantee extra points in each case!)
This activity can be a good springboard for discussing common errors in probabilistic thinking such as the gambler's fallacy and the hot hand fallacy.
Another interesting direction with this activity is Monte Carlo simulation. For example, it is tricky to figure out how many rolls are in an average run. However, while the students are playing, they can keep track of the number of rolls they make and take the average after a large number of turns. This is very strategy-dependent, of course. Suppose a player never quits of their own accord, keeping all available combos each roll and rolling until they bust. Then they will average about 4.4 rolls and 28.3 points before busting.
Wrap-Up Questions
- Were there any strategies that seemed like good ideas before you started but seemed less good after playing with them for a bit?
- When is it that you tend to stop your turn after having played for a bit? Is there a minimum number of rolls or a score you're trying to get to?
- When you won, did you end your turn immediately after reaching 100 or roll a bit further? How did you decide?
Extensions
There are many other versions of this game that could be played. I picked this one because the player gets at least two rolls per turn and it seemed like each player would typically get a few turns in before the game ended, but some variants are more exciting and some variants are easier to analyze. Here are some to consider:
- Use the same rule set, but only roll with two dice. This pared down version makes it much easier to think about questions like, "What's the probability of rolling three times without busting?"
- If you want fewer moving pieces, you can play with four dice and throw out either the 7-sum combos or the tuple combos. Both of these variants allow the player to bust on the first roll but are easier to play for younger students and easier to analyze for more mature students.
- A version I grew up playing involves larger numbers and more mechanics, but it makes for a more exciting game with occasional huge swings in scores. In this variant, players play to 10,000 using six dice. The combos are
- 1 is worth 100 on its own
- 5 is worth 50 on its own
- A triple is worth 100*x if x is 2, 3, 4, 5, or 6 and 1000 if x is 1. Every single additional x rolled is worth 100*x or 1000, respectively, until a roll fails to produce an x, even after rerolling all six dice. Multiple numbers can be "hot" in this way, as long as at least one is in every roll. (For example, if the player rolls 3, 3, 3, 4, 4, 4, then that is 300+400 = 700 points, with 3's and 4's hot into the next roll. If the player then rolls 2, 2, 2, 2, 3, 6, then they could tally up 200+200+300 = 700 more points for a running total of 1400 and choose whether to reroll the last die with 2's and 3's hot but 4's not.)
- A straight (1, 2, 3, 4, 5, 6) is worth 3000 points, but the player must roll all six dice again to keep them. (Alternatively, they could keep the 1 and 5 for 150 and not risk another roll.)
- Three pairs (e.g., 2, 2, 5, 5, 6, 6) is worth 1500 points, but the player must similarly roll all six dice again to keep them. (Four of a kind and two of another can be counted for this if the player would rather take 1500 and roll again rather than keep a triple or more and continue rolling with some numbers hot.)
- No usable numbers with six dice (e.g., 2, 3, 3, 4, 4, 6) does not only end the player's turn, but subtracts 1000 from their current score -- possibly bringing them into negative territory!
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