A puzzle based on the icebreaker game.

Two truths and one lie:

1. I have a twin.
2. I used a random number generator to decide whether to place a true or false statement in #1.
3. I do not understand the rules of “two truths and one lie.”

Solution inside.

Hint

The solution to the puzzle consists of assigning a truth value (true or false) to each of the three statements.

Solution and explanation

Start by taking cases; there are eight:

• Suppose I used the RNG. Then we have `2: true` and
• `1: false` and `3: false` is an inconsistent assignment, because I claimed to understand the rules but broke them by writing two lies and one truth.
• `1: false` and `3: true` is an inconsistent assignment, because I denied understanding the rules but followed them.
• `1: true` and `3: false` is a consistent assignment, because I claimed to understand the rules and followed them.
• `1: true` and `3: true` is a consistent assignment, because I denied understanding the rules and broke them.
• Suppose I didn’t flip a coin. Then we have `2: false` and (applying similar logic)
• `1: false` and `3: false` is inconsistent.
• `1: false` and `3: true` is consistent.
• `1: true` and `3: false` is inconsistent.
• `1: true` and `3: true` is inconsistent.

The consistent assignments are as follows.

Statement: 1 2 3
Assignment A true true false
Assignment B true true true
Assignment C false false true

It appears that any of the statements could be true or false under a consistent assignment. However, since I could not know the result of the RNG in advance, I could only have used it if I knew that the three statements would admit at least one consistent assignment whether the RNG returned `1: false` or `1: true`.

But this is not the case: There is no consistent assignment in which I used the RNG and got the result `1: false`. Therefore, I must not have used the RNG at all, which rules out assignments A and B and leaves assignment C, namely `1: false`, `2: false`, and `3: true`, as the only option.

See Wikipedia, “Boolean satisfiability problem.”