4- Nods

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

J Kenji Higa
Jun 29, 2015

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 .

Since the total number of nods was 1296, n² = 1296.

Simply take the square root of both sides, and you get 36 .

Muhammad Ardivan
Jun 18, 2015

I think , x*x=1296 ,except the king And x^2=1296 , so x=36

I am sorry for asking that, but, if the answer is x^2 = 1296, it means the subjects nodded themselves. Shouldn't it be x.(x-1)= 1296?

Bruno Oggioni Moura - 5 years, 12 months ago

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x is the number of subjects. Every subject present nodded to each to the other once, so there is x(x-1) nods. And every subject present nodded to the king, so the total number of nods is x(x-1)+x=x^2.

Mountassir Farid - 5 years, 12 months ago

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I didn't pay attention to that. My bad. Thanks.

Bruno Oggioni Moura - 5 years, 12 months ago

Instead of themselves they nodded to king which makes it nxn

Gaurav Sinha - 5 years, 11 months ago
Makayla Thomas
Jul 5, 2015

By adding the digits in 1296 (1+2+9+6=18), and deriving at 18 means the total is divisible by 3 which takes out three of the possible answer. Since 30 ends in zero, its power would also end in zero as well. Leaving 36 as the answer

Steve Shifflett
Jul 4, 2015

The other three answers ended in 0 and only one ended in 6 and the total nods were 1296.

Sandeep P
Jun 27, 2015

36 square guys!

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