Jack's Number Conundrum
One day, Jack decides to give his friends Alice and Bob an "impossible" challenge:
"I am thinking of an integer between and I will tell Alice its prime factors (not raised to their exponents), and Bob the digit sum of the integer. Then, I would like you to figure out my number. However, you are forbidden to tell each other the information I've given you."
After handing out the promised information, Jack witnesses this exchange:
Alice: "I do not know the number."
Bob: "How many prime factors does your number have?"
Alice: "Three."
Bob: "Then I know the number."
Alice: "And so do I."
What is Jack's number?
Assume that Alice and Bob are perfect logicians, and made no computational errors.
The answer is 30.
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Possible numbers are: 30, 42, 60, 66, 70, 78.
At first Alice cannot figure it out which means that the number has it's multiple among the possible numbers.
Hence, the number will be either 30 or 60.
After knowing the number of prime factors, Bob figured out the number which means that only 1 number in the list have the sum which is told to him.
Hence, 60 is excluded.
From the given scenario, Bob figured out that the number is 30.
Since Bob could figure out the number, Alice understood the situation and figured out the number.