There are 3 people A, B and C. Three of them have positive integers written on their hats. One can only see the
numbers written on other hats and can not see the number written on his own hat. The number on one of the
hats is the sum of the numbers on the other 2 hats. Now the following event occurs
A was asked about the number on his hat. He replies "Don't know".
B was asked about the number on his hat. He also replies "Don't know".
C was asked about the number on his hat. He also replies "Don't know".
A was asked again the number on his hat. He replies "65".
Find the product of the three numbers on the hats.
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Solution :-
Case 1) :-
Let us assume that,
- A = x, B = 2x, C = x .
- x = Any Positive integer .
Conclusion :-
- A = x or 3x (But not confirm) .
- B = The only possibility is that I have 2x .
- B states that his number is 2x .
Case 2 :-
- Let A = x , B = 2x, C = 3x .
Conclusion :-
- A = x or 5x (But not confirm) .
- B = 2x or 4x (But not confirm) .
- C = x or 3x. If it's x, we would have the Case 1 and B would state his number. therefore, it's not x but 3x.
- C states that his number is 3x .
Case 3 :-
- Let A = 5x, B = 2x, C = 3x .
Conclusion :-
In the first round, no one can tell what the numbers are.
But in the second round,
- A = x or 5x.
- If A have x, it's Case 2 (A = x, B = 2x, C =3x) and C would state his number.
- However, he didn't, which means A have not x, but 5x.
- A states his number is 5x .
therefore,
- A : B : C = 5x : 2x : 3x .
→ A = 5x = 65 => x = 13 .
→ B = 2x = 2 * 13 = 26 .
→ C = 3x = 3 * 13 = 39 .
hence,
→ A * B * C = 65 * 26 * 39 = 65910 (Ans.)
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