The school of the count: complex examples

The count is the most important tool of play by logic. The author of "50 Shades of Red" explains why: without the count one cannot play classic Mafia well, because through it we not only find the blacks but also give arguments in our favour and gather hands for removing those we suspect. The author of "Mafia in the Style of Zen" adds that you do not need two higher educations to arrive at a simple truth: the method of the remainder is enough. The lesson goes through tasks from simple to complex. Solve each yourself on paper and only then read the analysis.

The method in one phrase

The count is a way of finding the mafia by the remainder, subtracting players. You count the reds at the table, including yourself. Among those left outside the count sit the blacks. It is pure logic proceeding from irrefutable data. What the count gives, in the author's words:

1. Not to lose the current round.
2. To find the blacks by the remainder, which insures against a mistake.
3. To determine who is mafia and, from his position, to determine correctly the other colours at the table.

If you have not yet settled on a line of play, vote guided by the count and by the continuation of the game.

Task 1. A guessing round of three

There are three of you: nos. 1, 2, 3. You are a civilian (no. 1), at the table is a live red check (no. 2) of the only sheriff. Who is mafia? We apply subtraction: three players, take away yourself and the red check, by the remainder no. 3 is black. What remains is to persuade the red check to vote together with you. The author asks not to laugh at the example: it really is the simplest task possible, and the habit of always counting begins with it.

Task 2. The critical round at five

Players nos. 1, 2, 3, 4, 5, you are red no. 5. There are two sheriffs in the town, nos. 1 and 2. There is a pair of blacks at the table; you know this from two votes "by the balance" (first the black check of one sheriff was removed, then of the other). Besides the sheriffs there is a double red check, player no. 4. Who is mafia? Whom to remove in this round?

Solution (according to the author): there are five players in the town, two of them mafia. Among the sheriffs there is definitely one red and one black. We count: 5 minus 1 (the doubly checked red) minus 1 (your red colour) minus 1 (the red among the sheriffs) minus 1 (the black among the sheriffs) equals 1. What remains is a player outside the sheriffs and outside the check, no. 3: he is mafia. By removing yourself, the checks and the link in which there are not two blacks, by the method of the remainder we found a mafia player. The author reserves in a footnote: "actually not exact", because the mafia could have given its own player as black, but that will become clear in the guessing round, when the sheriffs check those already removed.

The second variant of the same situation is given by "Mafia in the Style of Zen": a balance was played, at the table besides the two sheriffs there is a double red check (no. 3), whom both sheriffs visited on the last night. There are five of you, you are a civilian (no. 5). Here at the table there are two blacks and three reds: among the two sheriffs there is one red, and you and the double check are civilians. The black player no. 4 is together with one of the sheriffs. Without counting "magic" the player will choose one of the teams or play by gut feeling and lose two against two.

Task 3. The critical round at seven, three blacks

The author's condition: no. 10 was killed, in the round at nine players nos. 2, 3, 4 removed no. 9, no. 5 died, and at seven two sheriffs revealed themselves, nos. 7 and 8. Both have two red checks, these are players nos. 4 and 6. You are red no. 1. Who is mafia? Whom to remove?

Solution: there are seven people in the town. Two players are red on all versions (the checks nos. 4 and 6), you are red, and among the link of two sheriffs there is no pair of blacks, that is, one of them is red. We count: 7 minus 2 minus 1 minus 1 minus 1 equals 2. Outside the checks there are two players, and minus the reds they are mafia. If the town removes either of them, the game will not end with a black victory, and the sheriffs will make another check each. The author adds an assessment along the way: no. 9, who left at nine, was apparently a civilian, since one mafia, and possibly two, voted for his removal; then the trio of blacks is nos. 2, 3 and one of the sheriffs.

Task 4. Seven players, two live red checks and two live black checks

There are seven of you, you are red no. 1. One sheriff (no. 2) has two checked reds at the table (nos. 3 and 4), you are not counted among them. The other sheriff (no. 6) has two black checks: one is his opponent (sheriff no. 2), the other is a player who played against him (no. 7). On whose line do you play this round, and why?

The author's solution: on the first line the sheriff no. 2 himself is red, his two checks, and you, because you know your own colour. Four of the seven are found civilians, all the others are mafia. You vote for no. 7, because he is mafia on all versions of the game. Or the town does not have a critical round (then there will be more checks and information). This task shows: sometimes the count alone is enough, with no other information about the game.

The most frequent cause of "three against three" defeats

The author of "Mafia in the Style of Zen" says: without the skill of counting the game, games are lost three against three, when at seven players five vote for the real sheriff or for a "black check" on one of the versions. How did that happen? There are seven players, three mafia. It is logical that on the other version too there are three mafia at the table. Where do the five votes come from, then? Simply one of the players did not count that he is black on the other version of the game. Had he had the count behind him, he would have worked it out and possibly led the reds to victory. The author's rule: if by the count of the game you are mafia on the other version, vote against this team. Mathematical principles are unchanging and stand above evaluative judgments.

Why the count is especially needed in the early rounds

The count helps to find the mafia not in the guessing round but in the early and first critical rounds. The earlier conflicts are revealed, the more information. The more information, the better the decisions. The author of "Zen" concludes: if you have the count and do not make the mistakes described in the chapter for inexperienced players, this automatically makes you a player of average strength.

For the black

For the blacks the very first and quite productive tactic, according to the author: keep neutrality, wait for discord and "catch" the civilian's deciding hand in conflicts. The mafia team needs only one vote to win, and it is not always clear who it will be. Often the town has almost united, but a conflict arises, and the mafia wins. The count and a look at the layout of the game through the prism of three mafia at the table help the reds avoid this, so the black needs to upset the count: doubt checks aloud, throw in versions in which the remainder is different, and see that the reds do not count themselves black on someone else's line.

The author's self-check questions

1. How else can you tell who is a civilian at the table without checks?
2. The count can be upset when you are mafia. How, and can it be done in the tasks above?
3. There are nine of you, two sheriffs at the table, no votes or removals have happened. How many blacks are at the table? What is the probability that there are not three? If there are two, why?

The Academy's note on the third question: if nobody has been removed and only one player dropped out at night, there are usually three blacks. There would be fewer than three if the mafia removed its own itself (for example, for a show with two sheriffs), and this must be checked by the moves.

Practice

Compose five tasks of your own on the count with two sheriffs and double checks and swap them with a partner. Solve them in writing, then check where you lost a player. Then switch on a recording of a game and in every round at seven and nine count the reds on each version before you hear the result.

Typical mistakes

• Forgetting to subtract yourself.
• Believing any check without accounting for the second version.
• Not counting that you are black on someone else's line and putting a hand against the correct team.
• Treating the remainder as exact without remembering that the mafia could have given its own as black.

Source of the idea. The chapters "The most reliable count of the game" from Dmitry Novik's book "50 Shades of Red" (2021) and "Play by logic: the count" from "Mafia in the Style of Zen" (2020). The conditions of tasks one to four belong to the authors and are retold in our own words; the note on the third question, the practice and the advice for the black were written for the Academy.