## Logicians Walk into a Cafe

My former IMO coach, Gregory Galperin, converted a famous joke into a logic puzzle, adding a variation.

Puzzle. Ten logicians walked into a cafe. Each knew whether they wanted tea or coffee, but no one knew each other’s preferences. When they sat at a table, the waiter asked loudly, “Will everyone be having coffee?” Then the waiter went around the table, writing down each person’s answer.
There were three possible answers: “I don’t know”, “Yes”, and “No”. All answers were truthful and spoken loudly so that all group members heard them.

1. Suppose the first nine people said, “I don’t know”, and the tenth person said, “Yes”. How many of them wanted coffee?
2. Suppose the sixth and the seventh answers were not the same. How many people said, “I don’t know”, how many said, “No”, and how many said, “Yes”. Find the smallest number of people who for sure would have ordered coffee and the smallest number who for sure would have wanted tea?
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1. #### Leo B.:

1. All of them, obviously.

2. 6 people said “I don’t know” (they all want coffee), the remaining 4 said “No” (one definitely wants tea, nothing can be said about the remaining 3).

Nobody but the last person is able to say “Yes”, and only if the previous 9 had said “I don’t know”. As soon as anyone says “No’, the subsequent people are unable to say anything but “No”.

2. #### NP:

1.All ten of them wanted coffee.
2.Six people said “I don’t know”. Four people said “Yes”. Six people for sure wanted coffee. One person for sure wanted tea.

3. #### tanyakh:

Leo, nice analysis.