A Seemingly Impossible Puzzle
After a church service, a priest and a cantor had a conversation.
The priest asked, "Do you know the ages of the three visitors we had today?"
The cantor said, "I do not know."
The priest said, "What if I told you the product of their ages is 2450?"
The cantor did some calculations, and then replied, "I still don't know."
The priest said, "Interestingly the sum of their ages is two times your age."
The cantor thought about it and said, "There is still not enough information."
The priest then said, "As you remember, I did not eat cake at my birthday party, choosing to be healthier. I wonder if the three people who came today will stop eating cake when they reach my age."
The cantor then said, "Now I know their ages!"
How old is the priest?
The answer is 50.
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1 solution
2450 factors as 2 ∗ 5 2 ∗ 7 2 . We need to organize these primes into 3 factors. Either one or two of these numbers will themselves be prime, and we get the possibilities as follows (the last number is the sum):
2 25 49 76
2 35 35 72
5 10 49 64
5 14 35 54
7 14 25 46
7 10 35 52
2 7 175 182
5 5 98 108
5 7 70 82
7 7 50 64
1 50 49 100
1 35 70 136
1 25 98 124
Other possibilities exist, but they would mean the cantor was hubdreds of years old, so we ignore them. Now, if the cantor knows his own age, he should know which possibility is correct. He doesn’t which means the sum of the ages must not be unique. This only occurs for 5,10,49 and 7,7,50 , which both sum 64.
Finally, we examine the last clue. The priest’s wording implies that all the people are younger than he is, and since this allows the cantor to determine the clorrect solution, we know this is not true of the other group. If the priest is 50, then he is older than all three in the 5,10,49 group, but not all in the 7,7,50 group. If he were older, it both would be satisfied; were he younger, neither would. Therefore, the priest is 5 0