Eight players and twenty-eight decisive games. The challenge is not to identify the best player, but to chain the victories.

The Winning Line

Strategist

Riddle statement

Eight chess players play a round-robin tournament. Each pair plays once, and no game ends in a draw.

After the tournament, you want to arrange them in a line so that every player has beaten the player immediately behind them.

Is this always possible, whatever the results, or can some tournament make such a line impossible?

Show solution

Answer: it is always possible.

We build the line by adding the players one at a time.

Suppose we already have a valid line

$ P_1\to P_2\to\cdots\to P_k, $

where each arrow means that the player on the left beat the player on the right. We want to add a new player $X$.

Move along the line from the front and find the first player $P_j$ whom $X$ beat.

If that player is $P_1$, place $X$ at the front:

$ X\to P_1\to\cdots\to P_k. $

If $j>1$, then $P_j$ is the first player whom $X$ beat, so $X$ did not beat $P_{j-1}$. Since there are no draws,

$ P_{j-1}\to X. $

We also know that

$ X\to P_j. $

Therefore, $X$ can be inserted between them:

$ P_1\to\cdots\to P_{j-1}\to X\to P_j\to\cdots\to P_k. $

If $X$ beat nobody in the line, every player beat $X$, including $P_k$. We simply place $X$ at the end:

$ P_1\to\cdots\to P_k\to X. $

Every case produces a valid line with one additional player. Starting with a single player and repeating the insertion eventually places all eight.

The argument does not depend on the number 8: it works for every tournament without draws.

Key idea: insert each new player immediately before the first player they beat.

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