2.Six people said “I don’t know”. Four people said “Yes”. Six people for sure wanted coffee. One person for sure wanted tea. ]]>

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”.

]]>7a: Bob can win by symmetrical play. Define a mapping of suits (e.g. spadeshearts, clubsdiamonds), and always play the card of same rank, and corresponding suit, as Alice’s last move.

9: Two things contribute to the 230 sums: pairs of multiples of three, and pairs of a 3n+1 and a 3n+2. That works out to n0(n0-1)/2 + n1n2 = 230. The problem is symmetric in n1 and n2, and we only have to figure out n0, so let’s start with n1n2 = 230 – n0(n0-1)/2, and n1+n2 = 30-n0. Requiring n1n2 to be positive leads to a quadratic equation implying n0<22, and more algebra and playing with bounds gives n0>20, so the number is 21 (specifically, 21, 5, and 4; or 21, 4, and 5).

]]>7a: Bob can win by symmetrical play. Define a mapping of suits (e.g. spadeshearts, clubsdiamonds), and always play the card of same rank, and corresponding suit, as Alice’s last move.

9: Two things contribute to the 230 sums: pairs of multiples of three, and pairs of a 3n+1 and a 3n+2. That works out to n0(n0-1)/2 + n1n2 = 230. The problem is symmetric in n1 and n2, and we only have to figure out n0, so let’s start with n1n2 = 230 – n0(n0-1)/2, and n1+n2 = 30-n0. Requiring n1n2 to be positive leads to a quadratic equation implying n020, so the number is 21 (specifically, 21, 5, and 4; or 21, 4, and 5).

]]>21 numbers of the original 30 numbers are divisible by 3.

]]>Minimum possible number of liars is 3. Let the persons having numbers 1 to 17 be all truth tellers. Their sum is 153. The remaining 3 persons with numbers 18, 19, 20 will lie and tell their numbers as 1, 1 and 1. The total sum will be 156.

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