25
30
36
40
50

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This is a variant on the classic "handshake problem", in which given a group of

npeople that shake hands with each other only once, the number of handshakes is equal to "n choose 2", or (n × (n-1)) ÷ 2.The difference between the "handshake problem" and the nodding is that in every pair, each subject nods to the other...therefore there are 2 nods per pair.

Therefore the number of nods between subjects is: (n × (n-1)) = n² - n

However, the trick in this problem is the king. If all subjects nod to him, but he doesn't nod back, we have to add

nsubjects to the total number of nods.Therefore we add

nnods from the subjects to the king ton² - nnods between only the subjects, that leaves us with justn².Since the total number of nods was 1296, n² = 1296.

Simply take the square root of both sides, and you get

36.