Friday, 16 September 2011

Word puzzle

Here's a puzzle that's on the Crystal Maze iphone app (I don't have the app, I'm not recommending it, just stole this puzzle which is on the review page):
Mary is 24. Mary is twice as old as Ann was when Mary was the age Ann is now. How old is Ann?
Answer after the break.

When I first read this it was just nonsense. I couldn't make any sense out of the words, I couldn't parse the sentence, and I didn't think there was enough information in there to provide the answer. However, with a bit of linguistics and a bit of algebra, I got there.

OK, first the linguistic knowledge: the bracketed string is a relative clause, modifying the time X when Ann was a certain age.
Mary is twice as old as Ann was [when Mary was the age Ann is now].
So now we have the relatively easily-parsed sentence
Mary is twice as old as Ann was (at some time in the past).
Now for the algebra. Let's call this time in the past X. Let's call Mary's age now M, Ann's age now A, and Mary and Ann's ages at time X m and a respectively. We're looking for the age Ann is now, or A. We're given the value of M, which is 24. We can easily work out a, which is the age of Ann at time X: it's half of M because we're told that Mary is twice as old as Ann was at time X. So a is 12.

We can assume the time is constant for both girls. So the difference between Mary's age now and her age at time X is equal to the difference between Ann's age now and her age at time X. Or, in alegbraic terms:
M - m = A - a
Let's plug in the values that we know:
24 - m = A - 12
We can remove that -12 on the right by adding 12, and then we have to do it to the other side too:
36 - m = A
Now we need some logic again. m is in fact the same as A, because m is Mary's age at X and A is Ann's age now, and we're told that these are the same value in the relative clause bracketed above: at time X, [Mary was the age Ann is now]. In other words,
m = A

36 - A = A
And continue on from there cancelling out the -A:
36 = 2A
And finally:
18 = A
So Ann is 18. What a lovely age to be.
So we can simply replace the m in our equation with A:

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