Answer Puzzle 1

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Each scientist can discuss with nineteen others. If one scientist has communicated with nineteen, no one could have communicated with none. The possible range is therefore 1 to 19 or 0 to 18, so 0 and 19 cannot be declared at the same time.
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As there are only ever 19 numbers and 20 people, at least two of the twenty must declare the same number.