Skeptophilia (skep-to-fil-i-a) (n.) - the love of logical thought, skepticism, and thinking critically. Being an exploration of the applications of skeptical thinking to the world at large, with periodic excursions into linguistics, music, politics, cryptozoology, and why people keep seeing the face of Jesus on grilled cheese sandwiches.

Monday, September 14, 2020

Solution to the Census Taker Puzzle

A few days ago, I posted a puzzle, and challenged my readers to try to solve it.  (If you haven't seen it yet, it's in my post "Pieces of the Puzzle.")  I promised I'd post a solution, so here it is.  (If you're still working on it, read no further!  It's always more fun to work something out yourself than to have someone simply tell you the answer.)

Here's the puzzle:
A census taker goes to a man's house, and asks for the ages of the man's three daughters.  
The man says, "The product of their ages is 36."  
The census taker says, "That's not enough information to figure it out." 
The man says, "Okay. The sum of their ages is equal to the house number across the street."  
The census taker looks out of the window at the house across the street, and says, "That's still not enough information to figure it out."  
The man says, "Okay.  My oldest daughter has red hair."  
The census taker says thank you and writes down the ages of the three daughters.  
How old are they?
Clue #1 -- that the product of the three girls' ages is equal to 36 -- gives us eight possible combinations of ages:
1, 1, 36
1, 2, 18
1, 3, 12
1, 4, 9
1, 6, 6
2, 3, 6
2, 2, 9
3, 3, 4
So the census taker is quite right that this is insufficient information.

The second clue is that the sum of their ages is equal to the house number across the street. So let's see what the house number could be:
1 + 1 + 36 = 38
1 + 2 + 18 = 21
1 + 3 + 12 = 16
1 + 4 + 9 = 14
1 + 6 + 6 = 13
2 + 3 + 6 = 11
2 + 2 + 9 = 13
3 + 3 + 4 = 10
The census taker looks at the house number through the window, and still can't figure it out.  This is the key to the puzzle. 

Suppose the house number had been 21.  Then looking at the house number would have been sufficient information for solving it; the children would be 1, 2, and 18.  The only way that looking at the house number would be insufficient is if there were two sets of ages that added to the same thing -- which is only true for 1, 6, and 6, and 2, 2, and 9, both which add to 13.

The third clue is that the oldest daughter has red hair.  In the first of our remaining possibilities, 1, 6, and 6, there is no oldest daughter -- the eldest children are twins.  Therefore the daughters are 2, 2, and 9.

I hope you enjoyed this puzzle -- I think it's one of the cleverest ones I've ever seen!

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