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At the time the ship was as old as the captain is now, the captain was twice as old as the ship.

Together they are 56 years old.

How old is the captain and ship?

I've figured out the answer, but I did this through trial and error basically. An equation or way of thinking on how to solve would be much appreciated!

zoli
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Kalle B
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    This does not seem clear to me. Did you mean: At the time the ship was at old as the captain is now, the captain was twice as old as the ship? – Element118 Oct 16 '15 at 10:25
  • Yes! I'll edit it, sorry for the misunderstanding, vital part of the problem... Translating from another language can get confusing. – Kalle B Oct 16 '15 at 10:31
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    Is the captain older than the ship now? Assuming they have positive ages, something seems wrong. – Element118 Oct 16 '15 at 10:36
  • Let's ignore that and see if it falls out of the equations ;) – mathreadler Oct 16 '15 at 10:37
  • Yes the captain is obviously older than the ship. – Kalle B Oct 16 '15 at 10:40
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    I also think something is wrong. "At the time the ship was as old as the caption is now" implies that the Ship is older than the captain, but "the captain was twice as old as the ship" implies (assuming positive ages) that the captain is older than the ship. – Ward Beullens Oct 16 '15 at 10:43
  • I think the correct wording should be "At the time the ship was as old as the captain is now, the captain was twice as old as the $\text {ship is now}$.

    Together they are 56 years old. "

     – Vikram Oct 16 '15 at 10:54
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    Something is very wrong. When the shiw was as old as the captain is now means and the captain was twice the age then - means that he was twice as old as he's now, in the past? – skyking Oct 16 '15 at 10:54
  • I think the original puzzle is : "A ship is twice as old as the captain was when the ship was as old as captain is; and together they are now 56 years old." – mrprottolo Oct 16 '15 at 10:55
  • Assuming that the "was" is misleading tempus a solution would be that the boat is brand new and the captain is 56. In 56 years the boat will be 56 years and the captain will be 104 years - twice as old as the boat that. – skyking Oct 16 '15 at 11:05

3 Answers3

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Putting the facts into equations shows that the question is a paradox. Let's assume the captains age is $C$ and the age of the ship is $S$ now. Then we have $C+S = 56$.

At some other time $t$ from now the captain is $C+t$ old and the ship $S+t$ old. When the ship "was" as old as the captain is now we have $S+t = C$ and then the captain was $C+t$ old and twice as old as the ship that is $C+t$ = $2(S+t)$.

That is the three equations become:

$$C+S = 56$$ $$S+t = C$$ $$C+t = 2(S+t)$$

Substitution in of $C=S+t$ in the last gives:

$$S+t+t = S+2t = 2(S+t) = 2S+2t$$

which gives that $S=0$, substituting this back into the second gives that $C=t$ and that into the first gives $C+S=t+S=t+0=56$.

That is that the time mentioned as "was" is indeed in the future. The captain is now 56 and he "was" 104 when the boat "was" as old as he is now (56).

If on the other hand the question is as mrprottolo pointed out:

"A ship is twice as old as the captain was when the ship was as old as captain is; and together they are now 56 years old."

the equations become:

$$C+S=56$$ $$S-t = C$$ $$S = 2(C-t)$$

which has the solution $C=24$ and $S=32$ and $t=8$: a ship is twice as old (32y) as the captain was (16y, 8yrs ago) when the ship was as old as the captain is(24y, 8yrs ago).

skyking
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  • There seem to be many ways to try and interpret this question :) – mathreadler Oct 16 '15 at 11:58
  • @mathreadler Now it's just the question which of my two "interpretation" that were correct (that made my the accepted answer). The question asked or that mrprottolo formulated? – skyking Oct 16 '15 at 15:08
  • Maybe it is a popular puzzle somewhere which is learnt by heart and taught to be interpret in one specific way. – mathreadler Oct 16 '15 at 15:37
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So let us call their current ages $x$,$y$ and the age difference since they were "double aged": $k$.

$x+y = 56$ : sum of current ages is 56.

$y = x-k = 2(y-k)$ which can be rewritten as 2 separate equations:

$y = x-k$

$y = 2(y-k)$

Now rewriting this as a linear equation system (x,y,k) $$\bf Ax = b$$ where $${\bf A} = \left[\begin{array}{rrr}1&1&0\\1&-1&-1\\0&-1&2\end{array}\right], {\bf b} = \left[\begin{array}{r}56\\0\\0\end{array}\right] $$

we find the solution:

$${\bf x} = \left[\begin{array}{c}33.6\\22.4\\11.2\end{array}\right]$$

$$33.6+22.4 = 56$$ and $$22.4 = 2 \cdot 11.2$$ which was their ages $11.2$ years ago.

So either I got some equation wrong or it is actually solvable despite the confusing grammar.

mathreadler
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The ship is $56 - x$ years old,

The captain was the age of the ship $x - (56 - x)$ years ago.

So, $2x - 56$ years ago was when the captain was the age of the ship.

This indicates the ship was $(56 - x) - (2x - 56)$ years of age.

So $2(112 - 3x) = x$

So, $224 = 7x$

So $x = 32$

It was $32$ years ago.

Orland
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