Counting and the remainder method

Counting is the most important tool of logical play. It does not depend on emotions, does not require reading faces and obeys simple arithmetic. A player who masters counting finds at least one mafia player at the table for certain and, from that player's position, works out who the real leader of the reds is. A player without counting picks one of the teams at random, "by gut", and often loses "two on two". Once you have mastered this lesson and the thirteen basic rules, you automatically become a player of average strength. More complex topics begin after that.

Basic arithmetic

We start with ten players: seven reds (six civilians and the sheriff) and three blacks (two mafia and the don). Every round the number of players shrinks: at night the blacks kill a red, by day the table votes someone out. When the blacks are as many as the reds, the mafia wins. When there are no blacks, the reds win.

An important view of any game follows from this. At any moment at the table, from the point of view of any version of the game, there are exactly three blacks until one of them is removed. If there are two lines (two sheriffs), each has its own trio of blacks, and they partly overlap. The sum of reds and blacks on any line always gives the whole number of the living at the table.

What "by a line" means

A line, or version of the game, is the set of claims of one of the sheriffs: he is red, his checks are of such-and-such colour, so the blacks are such-and-such. If there are two sheriffs at the table, then on the first line the first sheriff and his red checks are red, and the second sheriff on this same version is black. If you are red, you know that you are red, and this is your exact knowledge. Everything else depends on the chosen line.

The remainder method

The idea: there are always exactly three blacks at the start. If you have found all the reds on a line, the rest turn out to be black. We subtract the found reds from the number of living players.

Tasks:

Task 1. A balance was played. Five are at the table: two sheriffs (No. 1 and No. 2), a double red check (No. 3) to whom both sheriffs went on the last night, player No. 4 and you (No. 5), a civilian. Whom do you vote against and who is mafia?

Solution: at the table there are two blacks and three reds. We count the reds. One of the sheriffs is red, you and the double check are civilians. Then the black player is No. 4 together with one of the sheriffs. So by the remainder method you have found one mafia player for certain (No. 4) and from its position can work out who the real sheriff is.

Task 2. There are seven of you, you are red (No. 1). One sheriff (No. 2) has two red checks (No. 3 and No. 4), among whom you are not counted. The other sheriff (No. 6) has two living black checks: one is the opponent who gives out the same role (No. 2), the other is a player who played against him, not from the number of the opponent's red checks (No. 7). Whose line will you play on this round and why?

Solution: on the line of sheriff No. 2 there are four reds: he himself, his two checks and you (you know your colour). Four of the seven are found to be civilians, so the other three are mafia, and No. 7 is among them. You vote for No. 7 because he is mafia on all versions of the game, or else the town is not in a critical round.

What it is all for: a typical catastrophe

For victory the black team needs just one vote, and who will become it is not always clear. Often the town has almost fully united, but a conflict arises that lets the mafia win. So the first and quite effective tactic of the blacks is to keep neutrality, wait for a quarrel and then "catch" the deciding hand of a civilian.

Counting, and a view of the situation through the prism of three mafia players, help the reds avoid this. Here is a typical catastrophe without counting. Seven players, three mafia. At seven, five vote for the real sheriff or for a "black check" on one of the versions, and the town loses "three on three". Where do five votes come from? Logically, on the other version there are also three mafia at the table. It is just that some of the players did not count that he himself is black on the other version. Had he known counting, he would have worked it out and possibly led the red team to victory.

If by the count of the game you turn out to be mafia on some line, even without anyone's check, vote against that team. This skill is key.

Important end states

Eight players are at the table, an unchecked player left at nine. At the next vote there will be seven. A mistake at seven, and after the night it is three on three. The game is over.
At six or five, if a red left earlier: two on two.
At four and three any removal decides the game.

That is why mistakes at the start of the game are forgiven, and in late rounds they are not.

The count can be broken. A black can give "his" black away as red (give the mafia a red check), present himself as red on both lines or create a third line of the game. The mafia's task is to make the reds count differently and split over hands. So the reds should not trust the count blindly: before voting, check it against the players who have left and against the moves.

Application

Mathematical principles are unchanging and stand above evaluative judgments. If, having counted the checks, you understand that you are mafia on one of the lines, vote against your opponents and persuade the rest that you are right.
Counting helps to find the mafia not in the guessing round but in early and first critical rounds. The sooner conflicts are revealed, the more information. The more information, the better the decisions.

Questions for self-check

How else can you tell who is a civilian at the game table without checks?
The count can be broken by being mafia. In what way, and can it be done in the tasks above?
There are nine players, two sheriffs at the table, no votes and no removals have taken place. How many blacks are at the table? What is the probability that there are not three? And if there are two, what for and why?

Short tasks (10 to 15 seconds each)

Nine players. You are red. Two blacks have been found. How many blacks are still unfound? Answer: one.
Seven players, two definitely red, you are red. How many players are outside the checks? Answer: four, and three blacks are among them.
Five players, the sheriff is dead, there are two red checks, you are red. How many blacks are among the others? Answer: both remaining, if the checks are right.

Practice

Take a recording of a game for five or seven players, stop at the moment of the vote and count the reds on each line. Write down: who is black on the first line, who on the second, who coincided. Compare with the real roles.

Typical mistakes

Not counting yourself in the calculation.
Believing a check without accounting for a false sheriff.
Counting in the last round from memory without writing it down.
Forgetting that a third line is possible if the blacks break the count.

Source of the idea. The chapter "Playing by logic: counting" of Dmitry Novik's book "Mafia in the Style of Zen" (2020) and the sections on counting in "50 Shades of Red" (2021). Tasks 1 and 2 are based on the author's examples, the other tasks were written for the Academy.