Some simple dice games

J3 and I played several simple games recently that I want to record. One of them, race to the top, is a variation of a more sophisticated game that can be used more generally.

Digits three in a row

Materials: a 100-board, 2d10, colored tiles
Players: 2
Goal: mark three spaces in a row
Basic play:
Two players take turns rolling the two dice. On a player's turn, they form a 2 digit number using the two digits and then claim that space on the 100 board. If the space has previously been claimed, they lose their turn.

The first player to claim three adjacent spaces in a line (horizontal, vertical, or diagonal) is the winner.

Variations:
(1) we used one dice marked 0-9 and another marked 00 - 90 (all multiples of 10) and then added the values to get our two digit numbers. This eliminated any player choice, but helped reinforce the idea that the value in the 10's digit is the number of tens.

(2) the winning condition can be increased to require a line of 4 spaces

(3) the winning condition can be changed to allow any three (or four) spaces that are colinear; these spaces would not have to be adjacent.

Notes:
This is a very simple game, especially the variation we played, but J3 found it fun. It was a useful exercise to practice locating the numbers on the 100 board.

Race to the top

Materials: An 11x 6 grid with one side labelled 2 to 12, 4d6, 3 tokens. Optional: 11 distinct tokens/small objects per player.
Players: 2 - 4
Goal: capture 3 columns.
Basic play:
This game is fairly simple, but has resisted our attempts to succinctly summarize the rules. Here is an explanation as we play two turns.

Here's our playing material, the three green squares are temporary markers:

First player, J1 rolls two ones and two fours. With this roll, J1 could group them into two fives or a two and an eight:

J1 decides on two fives and puts a temporary marker on the second level of the 5's column. After you understand the rules consider whether this choice is better or worse than the 2 and 8.

J1 chooses to continue rolling and gets 1, 1, 4, and 6. J1 groups these as a 5 and a 7, then moves the temporary marker in the 5's column up one level and adds a marker at the bottom of the 7's column.

J1 rolls a third time, getting 1, 1, 2, and 6. The only option is to group these as 3 and 7, so J1 places a temporary marker in the 3's column and advances the marker in the 7's column.

At this point, J1 ends his turn and marks his progress. On his next turn, if he gets a 5, for example, the temporary marker will start at the fourth level of the 5's column (building on his consolidated progress).

D has the next turn. He gets 1, 4, 4, and 5, which he chooses to group as 5 and 9:

D chooses to roll again, getting 4, 5, 5, and 6. This has to be grouped as 9 (advancing in that column) and 11:

D chooses to roll again and gets 2, 3, 5, and 6. This is a lucky roll that can be grouped as 5 and 11, allowing two tokens to advance:

D presses his luck and rolls a fourth time, getting 1, 3, 3 and 5. The dice can't be paired to get a 5, 9 or 11 and there are no more temporary markers available to place, so D loses his progress. J1 will have the next turn.

To be clear about the failure condition: the player must be able to place or advance a temporary token for both pairs of dice. For example, if D had rolled 1, 3, 3, and 6, he still would have lost his progress.

Further rules:

• the first person to end their turn on the sixth level of a column "claims" that column.
• the first person to claim 3 columns wins the game.
• columns that have been claimed by any player are safe values for all players. Players do not need to allocate a temporary token to those columns.
• Players can occupy a square that an other player has marked.

Variations:
(1) Change the winning condition so that the first player to capture a column wins
(2) Change the height of the columns, either fewer than 6 for a faster game or more than 6 for a slower game
(3) Change the failure condition so that only one pair of dice needs to be playable and reduce the temporary tokens to 2.
(4) only allow each player to roll one time. This eliminates the "press-your-luck" aspect of the game and is much more basic.
(5) allow players to jump over a square that has been occupied by another player. This rule particularly fits well if you use objects to record your consolidated progress (which also makes the grid re-usable).

Notes:

I was originally taught this game by Mark Nowacki of Logic Mills.

J3 and I played the variation where each player only rolled one time on their turn.

Evens/odds and a quick update

Early years math seems to put a strange emphasis on even and odd numbers. Recently, a friend asked whether there was a point to this. Maybe it is just one of those little bits of terminology that we are asked to memorize for no reason?

By chance, this was something I had started considering about a month ago. It did seem strange that we spend so much time on this simple way of splitting integers. I wondered if it was worth the attention. From that point, my awareness was raised and I started noticing where it occurs and ways it links with more advanced concepts and future learning. My conclusion is that even/odd is surprisingly deep.

First, it is a simple version of concepts that will be developed further. For example, the alternating (starting with 0) even, odd, even,odd, even... is an illustration of a pattern. They will soon see other alternating patterns, then more complicated patterns and 2d or 3d patterns.

For another example, evens are multiples of 2, odds are numbers with a non-zero remainder when dividing by 2. This leads to understanding other multiple families, division, and division with remainder.

Second, the even/odd distinction is helpful for improved understanding of different calculations. For example, the observations that even+even = even, while odd + odd = even, etc. These can be used to help self-check their calculation and also will form early experiences with algebra. Similarly,
even x odd vs odd x odd reinforce understanding of multiplication. Again, this gets broadened for multiples of 3, 4, 5, etc.

Third, there are a lot of more advanced results that are easiest to prove by parity arguments. Sometimes we are working with a set of things that are even and the key observation is we can pair them up. Other times, we have a set that is odd and the key insight is that, when we pair them, one must be left over.

Recently, with the J1 and J2, we were looking at some constrained ways to put the numbers 1 to 25 on a 5x5 checkerboard. They were able to prove that some versions were impossible simply because 25 is odd, so there are more odd integers in 1 to 25 than even integers.

Lastly, there are techniques in computer science that involve even vs odd. This comes up pretty naturally because of the essential use of binary.

Some games

Recently, J1 and J2 have gotten hooked on some classic card games. In particular, we've been playing a lot of 3-handed cribbage. It is a nice way to do some simple addition practice and build intuition about probability. Probability is now getting even more share of mind: in the last couple of days we started playing poker together. This was actually inspired by some of our reading together.

We are reading the Pushcart War.

In one scene, there is a poker game. Of course, the J's insisted that I explain the game and were eager to try it out. J3 was the huge winner tonight, while I busted out. Oh well.

Logic Puzzle collection and Dropping Phones

Recently, Mathbabe put out a request for riddles. There are some good links in the comments:

• The Riddler on FiveThirtyEight
• Logic Puzzles on Math is Fun
• Alex Bellos at The Guardian
• Riddles.com (naturally)
• Puzzling.stackexchange.com (is there a stack exchange for everything now?!)
• Authors cited: Dudeney, Gardner, Polya, Smullyan, Vaderlind. 

We've played with puzzles from almost all of these sources in the past, but this was a good opportunity to put together a nice list.

The one that stood out to me was on FiveThirtyEight. That's a site I read frequently, especially during this US election season ... and I was totally unaware of The Riddler feature. We kicked off with the oldest puzzle from their archive: Best way to drop a smartphone.

Breaking stuff

Right away, J1 and J2 loved the theme and were into questions about whether they could somehow keep the phones, if they weren't broken, or utterly destroy them, if they were broken. We talked about setting an upper bound with a very simple strategy of starting on the 1st floor and working up each floor. That's not a great answer, but it got them into modifications and improving strategies.

Through the conversation, it was interesting to see them start with the idea that the drops for one of the phones would be de minimis and could be ignored, but then start to pay attention to that aspect. Also, they had to grapple with the idea of balancing the number of drops that would be required in different cases.

While we didn't get to the optimal strategy, but the kids managed to get a version that, at worst, would take 19 drops for the 100 story building. Their intuition was based around taking the square root of 100. They could see that this probably wasn't the best answer, since there were still cases that, at worst, would take 10 drops and others that, at worst, would take 19.

Can you do better?

Smaller before bigger

The puzzle page poses the same challenge for a 1000 story building. However, we found something interesting when working on the version for a 10 story building: there is  strategy where all worst cases take the same number of drops, but it is not the optimal strategy!

It is a little hard to write about this without disclosing the strategy, but here's a hint:

• 4 + 3 + 2 + 1 = 10 and 5 + 4 = 9
• We are always allowed to assume that the phones will break if dropped from the top floor of the building

This led to another extension: what size buildings will have the same issues as for a 10 story building?

Probability comes in

Another extension is to think about the expected number of drops required and strategies that minimize this. Crucially, this extension introduces the idea about our prior beliefs about the sturdiness of the phones: where do we think the phones are likely to break, what is our confidence?

We didn't pursue this extension very far, but it did lead to some interesting conversations about terminal velocities. For those who want to follow that thread, this (other) stack exchange thread might suit you: How to figure out height to achieve terminal velocity.

Dots and Boxes variation

In grades 1, 2, 3, we played this variation of dots & boxes: Mathify the Squares Game.

I'm enthusiastic about this game, but can't resist a quick comment about the "mathification." Dots & boxes is already a mathematical activity, it doesn't need to be "mathified." This term implies confusing arithmetic and calculation with math, something I've written about elsewhere and, I hope, is clearly not implied by our blog.

In any case, I'll use the shorthand MD&B to refer to this dots and boxes variation.

Notes from playing in class

In class, we first introduced the kids to vanilla Dots & Boxes with a pre-printed grid of dots. We knew that it would be too much to play on a lattice covering the whole A4 sheet, but we thought a quarter of the grid would work. That turned out to be too big and the game started to seem monotonous to the kids as there was too much time spent on the opening (playing on squares that don't yet have any filled sides.)

We rectified this problem in 3rd grade and played on much smaller grids, with sizes between 7x7 lattices (which yield 6x6 squares) and 10x10.

After they were comfortable with the vanilla game, we introduced the product version with dice. In our case, we just had the players take turns and didn't give an extra turn when someone completed a square.

Some rule variations
There are some simple variations depending on how you deal with completed squares:

• no extra turn (this was the version we played in class)
• player adds another side to a different square with the same value. For example, say the dice are 2 and 4 (product 8) and the player fills a square. They also must add a side to another square with value 8.
• player rolls the dice and adds a new side (a full extra turn)
• player adds a side to any square (dice and square values ignored)

Of course, you could also make the extra turn optional instead of compulsory. You might also have some ideas about different ways to handle cases where there are no more free sides on squares with the required number value.

Probability questions

Dice games naturally lead to probability questions. Here were two that I really liked, based on scenarios from a recent game play:

What question are we asking?

What is the chance a player will get both the 20 and 36 boxes:

One great answer was 1/2. The reasoning: we are going to play until all boxes are filled and each of us have an equal chance to fill this box, so 1/2. This is not quite right, since the person who is about to roll has an advantage, but I thought it was an interesting interpretation of the question.

A 2x2 square

In this configuration, what is the probability that the next throw will allow the player to complete a box in the 4-30-25-15 zone?

  

A bigger D&B family

One of the reasons I thought this D&B variation was so cool is because of our games matrix. Whenever we play games, J1 and I talk about some key characteristics of the game, particularly the amount of randomness and the strategic complexity. These are not entirely independent dimensions, since a larger amount of randomness reduces the number or importance of each player decision, thus the strategic depth.

The mechanics of this game gave us some ideas about how to dial up or down the amount of randomness in this family of games. Here is our list of members of this family, roughly ordered from least random to most random:

1. vanilla dots and boxes: no random element
2. mash-up with product game: squares are still labeled, but players control the two factors using selectors (like in the product game) instead of using dice. This can be played with different collections of factors and different size boards (including board variations where a product appears multiple times or only a single time, where values are ordered or randomly distributed).
3. half-way house: one factor is chosen by players moving a selector, the other is determined by a dice roll (either before or after the "free" factor is selected).
4. MD&B game: as played in class and described in the first link
5. MD&B game where each factor appears only once. This is a case of dialing up the randomness by reducing the strategic options of the players. 

Ideas for other games

I'm excited to see what other games we can modify use the underlying idea from the MD&B variation. To be clear: use numbers to label parts of the game and then constrain the players' actions based on a die roll to involve either the pieces with corresponding labels or board positions with those labels.

Three specific examples:

1. Hackenbush variation where segments of the picture are numbered. This could nicely incorporate probabilities by putting values of the least likely dice rolls closer to the ground.
2. Ultimate tic-tac-toe meets the product game: from Art of Math. This is an old post, but I just happened to see it when preparing this post.
3. Dice chess. Here's the wikipedia article. For some reason, I often forget about this variation, even though it is a nice way to reduce the strategic complexity of vanilla chess for beginners and has some nice links with probability.

If you have some favorites, I would love to hear in the comments!

*Update*
Playing through the MD&B version several times, we came up with these rule variations that are worth your consideration:

1. Game stops as soon as someone rolls a value that can't be played (alternatives are to let that person roll again or have them pass their turn)
2. Remove some of the randomness: (a) on your turn, you roll and play a side with the required value, but they opponent also plays a side with that value. As we played it, that means moves (without filled boxes) go: A, B, B, A, A, B, B, etc. (b) When a player fills a box, they can choose to re-roll both, either, or none of the dice for their extra turn.

I think our favorite was a combination of all three of these components. Mixing 2a and b, you have to be careful to keep track of whose turn it is, but it lead helped bring out elements of strategy and more thinking about probability.

*Update 2*
Game phases
Above, we talked about how A4 (or even 1/4 of an A4, which I guess is equivalent to A6) is too big for beginning players of Dots & Boxes. They found the game "boring." J1 and I talked about this experience and it led us to considerations around game phases: opening, middle game, and end game. These are terms we first learned in chess, and we found it useful to contrast the two games.
Here were some observations:

• Opening: a lot of choices, not obvious how most of those choices link with "scoring" or the winning objective of the game. At this stage, there seems to be little interaction between the players (there is enough space that most of their actions either don't bring opposing pieces together or there is a lot of open territory).
• Middle game: still many playing choices, increasingly direct conflict between players, interim objectives within the game become more clear and there are some chances for plays that either score or more clearly move closer to the overall game objective.
• End game: significantly fewer choices for each turn than the other phases, either because there are fewer pieces (chess) or most territory has been claimed. At this stage, players are able to focus on the overall game objective, rather than interim objectives.

What we realized is that the larger playing area for D&B significantly increases the length of the opening. Because this phase is the least connected with capturing boxes, it is the hardest for beginning players to see how their choices ultimate lead to scoring and it is the phase with the most available choices on a turn.

another race to 100 game

Today's game at the math classes was not particularly well liked, but we are including this note for completeness and future reference.

Race to 100

how many players: 2-5
material: 1d6, 100 board, position markers (the kids made their own out of play-dough)
start: all players start on 1 on the 100 board
turns: each player's takes a separate turn. They roll the dice, then move their piece up the 100 board some multiple of the dice value (up to 10x).
winning: first player to get exactly to 100 wins

This game practices multiplication, skip counting, and factoring. Here are some example questions to stimulate thinking about game strategy:

• Would you rather have your piece on 99, 98, or 96?
• What about 71 and 70?
• If you are on 88, what are your chances of winning on the next roll?

Game reception
The kids found this game fairly easy. In retrospect, perhaps we should have played this game before the Times Square factors game.

Potential extension
The game is nicely suited to analysis by working back from the higher positions and/or analyzing a simpler version of the game. This may be a nice exercise for our programming classes, especially as we have recently been working with arrays.

Half-time scores

This discussion was suggested by a commenter who was asked this question in a math class, presumably as a "real world" word problem. These are just some rough notes relating to a string of conversations we've been having around this idea.

The scene
Our friend calls excitedly to tell us about the football (aka soccer) game she just saw: "[favorite team] was leading 2-1 at halftime." Suddenly her phone runs out of batteries and we don't get the final result. Can we figure out what it was?

A math class answer

4 to 2 victory for our side, of course. This is the naive model where the scoring rate is constant over the course of the game, so 2x as much time means 2x the score. Nonsense, for anyone who has a passing familiarity with the real game.

In discussion, one of the J's offered 3 vs 3 as an alternative and explained his thinking was that they change ends at half-time. The underlying model was that the direction of play determined the outcome. Quite a strange model!

Some stats

Given a half-time score, what can we say about the final result?

1. scores go up, so the interim score for each team is the least they could have at the end
2. Intuitively, the team leading at half-time is likely to win the game

There is some intriguing data related to half-time and full-time scores on the OptaPro Blog and they have a further link to the Football Observatory. One thing we saw right away is that 2 -1 and 1-2 halftime scores are fairly uncommon (about 5% of the sample games, when taken together). Perhaps this is why 2-1 and 1-2 halftime scores weren't included in some of their conditional tables, though, together, that was still about 970 matches (17,656 * 5.5%).

Corroborating our intuition from point 2, we looked at the 1-0 and 0-1 lines in their table 6 to guess that 2-1/1-2 matches would also have roughly a 30% chance of a change in outcome. The table doesn't specify whether the change is to a tie or a change of winner, but we guessed that the latter was less than half the change cases.

Comparing with other sports

The last topic we discussed was to compare with other sports, for example professional basketball.
First, if we kept the scores unchanged (2 vs 1 at half-time), then we know that we are watching an extremely unusual basketball game. At that point, we are so far into the tail of the distribution that it is hard to know what is happening and very dangerous to make guesses about the rest of the game.

Second, let's say that we have a more reasonable half-time score, but one team leads by a single point. We aren't so familiar with NBA results and I couldn't find a great stats source, but we assumed one team had 51 points and the other 50. In contrast with football, we concluded that this was not likely to tell us much at all about which team would win at the final whistle.

What if, instead, we assume a 60-30 point split? Well, in this case, we reasoned that a guess of 120-60 was much more reasonable because each scoring event is much small and more frequent than in football. Also, we were much more sure that the leading team at half-time had demonstrated  statistically significant strength relative to the trailing team. We were pretty confident that they would win in the end.

However, we also recognized that confidence was not mathematical certainty. Even in basketball, scoring doesn't happen at a continuous rate. Also, it was easy for us to come up with events (player substitution, player injury, change of strategy, fatigue) that would create a different scoring rate in the second half.

Your turn

What about you? Have any favorite "real world" questions from math class that, when you use your own real world experience, are actually very silly? Any beloved sports which you think offer another point of comparison for our discussions? Maybe games like cricket, baseball, or tennis where the end of the game is not determined by time?

math games class

First Grade

Warm-up
As a very quick warm-up, I asked the kids about days of the week: for example, what day follows Monday? what day precedes Friday? This was more challenging than I expected and I think need to practice.

First Game: number line squeeze
We made a number line with integers marked from 0 to 20. To start the game, I think of an integer in this range. Then, the kids take turns guessing. With each incorrect guess, I will give them an extra clue: "my number is smaller than [the number they just guessed]" or "[guess] is larger than my number," etc. When they get this clue, I had them draw a bracket on their number line, ( when their guess was too small and ) when their guess was too large.

This is a quick game to help encourage their familiarity with the number line and some logical inferences. For example: if they already know that 5 is too small, what do they know about 3?

We also took turns where they picked the secret number.

There are a couple of extensions we can play in the future:

1. Allow them to ask about ranges, for example: is the number between 5 and 11?
2. Ask "or" questions: is the number smaller than 3 or larger than 15?
3. Ask calculations: is the number equal to 3+4? 

Don't make a triangle
Our second game was a fun one from Math4Love. Don't make a triangle. The idea is simple: draw a bunch of starting dots in black (we used 6). Each player has a chosen color takes turns connecting the original dots. You lose if you form a triangle with all sides of your own color.

This was a fun game which, as you can see, leads to some pretty pictures. Especially when we play this back to back with tic-tac-toe, it leads to an interesting question: can the game end in a tie? Why or why not? Subtle stuff for first grade.

2nd and 3rd grades

Our game this week was related to Dice Miner from last week and comes from friends at Logic Mills in Singapore. As with Dice Miner, players start with 11 tokens of the same color and we play on a grid with columns labeled 2 through 12. This time, though, the grid has 8 rows, 3 players use the same grid at one time, and we had three extra markers (we used loom bands).

On each player's turn, they throw 4 dice, then choose to group them into two sets of two. They take the sum of each set and place a temporary/common marker (the loom bands) in the columns. After marking their territory with the temporary markers, the player has a choice: throw again or claim their progress by placing one of their tokens on the spaces they've reached. If they throw and can't place a temporary marker (there are only 3 at a time!) then they lose all of their progress for the turn.

When a player has reached the top row of a column, they claim that number. The first player to claim 3 numbers wins.

For those who like this type of classification, the game has a territory conquest objective played through a race-against-each-other mechanic.

Here are some pictures, mid-play.

In this picture, orange has capture column 7. Now, any two dice that sum to 7 are safe for all the players and orange is 1/3rd of the way to victory.

A bit later, we see that blue is close to claiming 9 and pink is close to claiming 6, but orange is close behind there.

In a different group, white is close to claiming column 8, but a good turn from yellow could disrupt that plan.

Dice mining (grades 2 and 3 math)

Notes for Grade 1 will come later

This week, we continued our run of dice games. This is a simple game we found in Marilyn Burn's About Teaching Mathematics. She calls it Two Dice Sum Game, but I like Dice Miner better (to pair with Dice Farmer, of course). Here's how to play:

1. Students make a number line with slots for each integer 2 to 12
2. Everyone gets 11 blocks or other counters (we used unifix cubes which were nice for stacking).
3. Players put their counters on the numbers, distributing them in whatever way they want. In particular, they can put more than one counter on a number or none on a number.
4. Everyone takes turns rolling two dice (normal 6-sided). Players can take one counter off their board for the sum of the two dice. For example, if I have 3 counters on 7 and roll 6+1, then I can take one counter off and now have 2 counters remaining on 7. If I had no counters on 7, then I would not take any action.
5. The first player to have all counters removed is the winner.

Any activity with construction cubes is bound to create opportunities for other mini-conversations:

• using the cubes as a ruler to make the number lines
• comparing how many cubes one person has to another by measuring lengths (before we made sure everyone had exactly 11)
• using the cubes to form a ruler to measure other things (our whiteboard, a table leg, a friend)
• making a pattern with light and dark colors from the cubes

As to the main game, there is some basic arithmetic practice through all the addition, but the real interest lies in trying to figure out how best to arrange your counters at the start. All the kids had their own hypotheses, but it was interesting to see these highlights:

• One student realized that 7 was the single most likely result and put all his cubes on 7 (this did not win that round)
• Most students started with a uniform distribution, one cube on all numbers
• There was a surprising amount of enthusiasm for 2 and 12, with many students initially playing multiple cubes on those extremes
• One student tried to be tricky and put cubes half on 6 and 7, presumably planning to take them off if either number came up.

Toward the end of the class, there were two comments that were really interesting. First, one student suggested rolling the dice many times, recording the results, and seeing how often all the numbers came up. This would then inform his strategy for how to distribute the dice.

We asked the students how many times they would have to roll. Would 1 or 2 rolls be sufficient? No, everyone was sure that wasn't enough. What about 25 times? One student pointed out that there are 11 slots, so 25 times is only an average of 2 per slot, so that didn't feel like enough to get a good sense of the "true" answer. Most students were eager to see for themselves, so this became their homework.

Homework

1. test the distribution by rolling two dice 100 times (or more) and tallying how often each number comes up as the sum of the two dice.
2. Play Dice Miner 5 times with your friends/parents at home. Record your initial starting position for the cubes, how many rolls to take them all off and who won each round.

Pandemic: different types of randomness

who: J1
where: my home office
when: after lunch

---

We've been playing pandemic a lot recently.

For this post, you don't really need to know the game, just that there are a deck of playing cards from which each player draws on their turn. Usually, these are resources that are needed to win the game, but occasionally there are epidemic cards that really stink. Oh, also know that everyone is playing together, cooperatively, against the game itself.

J1 noticed something very interesting by varying the set-up:

• Normal set-up: split the playing cards into equal piles and put one epidemic card into each pile (number of piles is the number of epidemic cards you are using). Shuffle those piles and then stack them on top of each other.
• Modified set-up: put the epidemic cards you are using in the deck and then shuffle the whole deck.

Focusing just on the epidemic cards, these result in very different distributions. Here are some challenging questions (assume there are 5 epidemic cards and 50 non-epidemic playing cards):

1. For both distributions, what is the probability that the n-th card in the deck is an epidemic card?
2. For both distributions, what is the probability that two cards in a row are epidemic cards?
3. For both distributions, what is the probability that three cards in a row are epidemic cards?
4. For both distributions, what is the probability that all epidemic cards are in a row?
5. What have we learned about how these distributions compare?

Ok, these are hard questions, particularly if you work through them in order 1 - 5. Without giving away the answers, I think it is striking that:

1. the answer is the same for both distributions, but what follows is very different!
2. left for reader
3. much higher for modified set-up
4. actually not too hard to calculate for each distribution; again, much higher for modified set-up
5. very different for potential impact on game play

Conditional probabilities
I'm open to other suggestions, but currently think that the key way to see the difference in these two ways of shuffling is to focus on conditional probabilities. For method 1, say p1(n|i) is the probability that the nth card will be an epidemic card, given that there have been i epidemic cards already drawn up to that point and similar for p2(n|i). For a fixed i, even the domains of definition of p1 and p2 aren't the same!

What happened when we played?
There's one key fact you need to know to understand our actual game: J1 hasn't learned how to really shuffle yet. That's right, all the epidemic cards were in one cluster!

J1 asked me to include some other information about our game play:

• J1 really likes to be the medic. At the start of the game, it is the most helpful piece
• Dispatcher plus medic is a very powerful combination. The dispatcher becomes even more powerful than the medic once cures have been found.
• We don't really follow the hand-limit rule.  This feels like an unnatural restriction that doesn't fit with the story of the game.