March 21, 2026

Math Puzzle: March Math-ness

Alice, Bob, Celia and Dylan are playing in a one-on-one basketball tournament. Alice will play Bob, Celia will play Dylan, and the winners of those games will play each other to determine the grand champion.

You and two friends have made brackets predicting who will win each game. A bracket scores one point for each correct prediction of a game’s winner. You and your friends made the following brackets:

You:

Your bracket shows A and B playing a game, with B winning; C and D playing a game, with D winning; and B and D playing the championship game, with B winning.

Friend 1:

Friend 2:

Although you all miss the tournament, someone who does watch it scores your brackets. You can see each other’s brackets, but each of you only knows your own score. The following conversation ensues:

You: I don’t know the results of the tournament.
Friend 1: Neither do I.
Friend 2: I didn’t know before you just said that, Friend 1, but I do now.

You and your friends are great at logic and were able to draw conclusions from one another’s statements over the course of the conversation.

Who won the tournament?

We can make a table to represent the possibilities after each statement.

Here are what all the possible results of the tournament are (in order of first two game winners and then the final champion) and how you and your friends would have scored:

A table lays out how many points you and your friends would have scored with each possible tournament result.

You and your friends have all seen one another’s brackets, so you know all the possibilities. If you had scored 3, you would know your bracket was correct—so when you say, “I don't know the results of the tournament,” everyone can count out possibility BDB.

A table of all the points and tournament results strikes out the possibility in which you correctly predicted the entire tournament, scoring three points (result BDB).

Then Friend 1 says, “I don’t know either.” That means Friend 1 cannot have a correct bracket, which means BCB is ruled out. If Friend 1 had scored 2 points, the only possible solution would be BCC. But because Friend 1 still doesn’t know the solution, everyone can count that one out as well.

A table of all the points and tournament results additionally strikes out the possibility in which Friend 1 correctly predicted the entire tournament, scoring three points (result BCB), as well as the remaining one where Friend 1 scored two points (result BCC).

Finally, Friend 2 says, “I didn’t know before you just said that, Friend 1, but I do now.” We know that Friend 2 did not know at the beginning of the conversation, so we can count out the possibility that they got 3 points. We also know that they do know the results now. So they couldn’t have scored 2 points because, if they had, there would still be two possible outcomes for the tournament. We are left with ADD (1 point) or BDD (0 points). But if Friend 2 had gotten 0 points, there would have been only one possibility left after your statement. Because we know they didn’t know the results until after Friend 1’s statement, ADD must be the correct tournament result.

A table of all the points and tournament results additionally strikes out the possibility in which Friend 2 correctly predicted the entire tournament, scoring 3 points (result ACA), the remaining one in which Friend 2 scored 2 points (ADA) and the one in which Friend 2 scored 0 points (ACC). This leaves only one possibility: the tournament result was ADD, for which you scored one point, Friend 1 scored zero points and Friend 2 scored one point. — Jen Christiansen

Bonus challenge: There is only one pair of people among the three who would be able to reconstruct the tournament together no matter what they scored. Which pair is it?

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