Numbers Brain Teaser

Question

Alice and Bob have two positive integers, x and y respectively, glued to their foreheads, so that each can read the other’s number but not their own. They also know that |x − y| = 1. The following conversation is overheard between Alice and Bob:

Alice: I don’t know my number.

Bob: I don’t know my number.

Alice: I don’t know my number.

Bob: I don’t know my number.

...

They say this to each other 200 times. And then we overhear them say: Alice: I know my number! Bob: I know my number too!

Can you explain the conversation and figure out the numbers x and y. You may assume that Alice and Bob each know that the other is extremely smart — so each is confident that if the other has sufficient information to deduce the answer then he/she definitely will.

Answer 1

Hint

Alice's first turn.

• Suppose Alice sees $y=1$. Then Alice knows that $(x,y)=(2,1)$ is the only possibility and says she knows.
• Suppose Alice sees $y\ge 2$. Then Alice says she doesn't know.

Now it's Bob's turn.

• If Alice said she knows, Bob knows that $(x,y)=(2,1)$ so we are finished.
• If $x=1$ then Bob knows that $(x,y)=(1,2)$ and says so.
• If $x=2$ then Bob knows that $(x,y)=(2,3)$ because otherwise Alice would have seen $y=1$ and said so. He announces that he knows.
• If $x \ge 3$ Bob says he doesn't know.

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Now it's Alice's second turn.

• If Bob said he knows, Alice can work out from what she sees what the pair is. (Either $(1,2)$ or $(2,3)$.)
• Otherwise, $x\ge 3$ and $y\ge 2$. If $y=2$ Alice announces she knows. Otherwise, we have a situation where it is Alice's turn and she knows $x,y\ge 3$. This bears some similarity to the original problem.

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This reminds me very much of the blue-eyed islanders.

Answer 2

Let us denote Alice's number as $a$ and Bob's as $b$. So, we have:

• Alice: I don’t know my number. $\Rightarrow$ Bob concludes what Alice already knows: $b > 1$, because if $b = 1$, Alice would deduce hers was $2$, i.e., she'd know the answer.

• Bob: I don’t know my number. $\Rightarrow$ Alice concludes what Bob already knows: $a > 1$, because if $a = 1$, Bob would deduce his was $2$, i.e., he'd know the answer.

Now both of them know that their numbers are greater than $1$.

• Alice: I don’t know my number. $\Rightarrow$ Bob concludes what Alice already knows: $b > 2$, because if $b = 2$, Alice would deduce hers was $1$ or $3$, but she already knows it's not $1$, i.e., she'd know the answer.

• Bob: I don’t know my number. $\Rightarrow$ Alice concludes what Bob already knows: $a > 2$, because if $a = 2$, Bob would deduce his was $1$ or $3$, but he already knows it's not $1$, i.e., he'd know the answer.

Now both of them know that their numbers are greater than $2$.

And so on.