4- Nods
Probability
Level
1
In a palace they were present only the king and some of his subjects. Everyone present nodded to each to the other once, except the king, who did not nod to anyone. In total has 1296 nods. How many subjects (except the king) were present in the palace?
25
30
36
40
50
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5 solutions
This is a variant on the classic "handshake problem", in which given a group of n people 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 n subjects to the total number of nods.
Therefore we add n nods from the subjects to the king to n² - n nods between only the subjects, that leaves us with just n² .
Since the total number of nods was 1296, n² = 1296.
Simply take the square root of both sides, and you get 36 .