The Monkey's Mother

My National Review Online "Diary" column for August 2002 included a brainteaser:  "The Monkey's Mother."  A lot of readers tackled it.  About half of those reporting in got it right.  Some tackled it and gave up, and these people were not always polite in their e-mails (which had an odd tendency to refer to parts of the monkey's anatomy that were not mentioned at all in the puzzle).

The Monkey's Mother

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A rope hangs over a pulley.  On one end is a weight.  Balanced on the other end is a monkey of equal weight.  The rope weighs 4oz. per foot.  The age of the monkey and the age of its mother together equal 4 years.  The weight of the monkey is as many pounds as its mother is years old.  The mother is twice as old as the monkey was when the mother was half as old as the monkey will be when the monkey is three times as old as the mother was when the mother was three times as old as the monkey.  The weight of the weight plus the weight of the rope is half as much again as the difference between twice the weight of the weight and the weight of the monkey.  How long is the rope?

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Here is a worked solution.  First I have written out the problem again, with the sentences numbered for reference.

<1> A rope hangs over a pulley. 

<2> On one end is a weight. 

<3> Balanced on the other end is a monkey of equal weight. 

<4> The rope weighs 4oz. per foot. 

<5> The age of the monkey and the age of its mother together equal 4 years. 

<6> The weight of the monkey is as many pounds as its mother is years old. 

<7> The mother is twice as old as the monkey was when the mother was half as old as the monkey will be when the monkey is three times as old as the mother was when the mother was three times as old as the monkey. 

<8> The weight of the weight plus the weight of the rope is half as much again as the difference between twice the weight of the weight and the weight of the monkey. 

<9> How long is the rope?

SOLUTION

Suppose the rope is L feet long.  There go 16 oz. to the pound, so by <4> the rope weighs 0.25L lbs.

Sentence <8> is almost pure bluff.  You don't notice this because your attention was scrambled from reading <7>.  From <3> you know that the weight of the weight IS EQUAL TO the weight of the monkey, so you can freely substitute the one for the other.  Calling the weight of the weight W, sentence <8> boils down to:  W + 0.25L = 1.5W.  Rearranging:  L=2W.

Now tackle <7>, using D for the difference in ages.  (Which, of course, never changes.  You will NEVER catch up with your older sister!)  The trick is to work backwards from the end of the sentence.

— "When the mother was three times as old as the monkey" can only mean: "When Mom was 1.5D and Junior was 0.5D."

— "When the monkey is three times as old as the mother was..." therefore means: "When Junior is 4.5D."

— "When the mother was half as old as the monkey will be..." therefore means:  "When Mom was 2.25D."

— At that point in time, Junior was of course 1.25D.

— "The mother is twice as old as the monkey was..." therefore means:  "Mom is currently 2.5D."

— It follows that Junior is currently 1.5D.

— Since, from <5>, these have to add up to 4, D must be 1.

— And from <6>, W is 2.5 lbs.

— In my analysis of <8> up above, I showed that L=2W.  Therefore L=5.  The rope is 5 feet long.