When the ship is as old as the captain is now, then the captain will be exactly 32 years older than the ship was when the captain was half as old as the ship is now. How old can the captain be?
Let $h$ denote the difference between the captain's and the ship's current age, which is positive according to the text. Let $a$ be the captain's age in the past, $b$ the current age, and $c$ the age in the future as described in the problem. Then $c-h=b$, $c-32=a-h$, and $2a=b-h$.

From the first two equations, we get $2h+b=a+32$. Eliminating $a$, we have $5h+b=64$. Since $h>0$, it follows that $b<64$. Eliminating $b$, we get $2a+3h=a+32$, or $a+3h=32$. Given $a \geq h$, we have $h \leq 8$, hence $b \geq 24$.

Since $c$ is always at least as large as $b$, we don't need to consider it further. $a$ will be between $b$ and $h$ under these conditions. Therefore, the captain is at least 24 and less than 64 years old.

Remarks. Several contestants assumed that the ages involved are integers, although this was not stated. In real life, we often round ages to smaller units (e.g., a child can be 4 and a half years old, 13 weeks, or just 2 minutes), but the information that "the captain is exactly 32 years older than..." is only valid without rounding. If we round, conditions such as one person being exactly three times as old as another at a certain point in time may never be true. For example, if the ship is 20 years old and the captain is 15 years old, how many years ago was the ship three times as old as the captain? In the original problem, the ages in the Sam Lloyd problem are also not integers.

"The combined ages of Mary and Ann are 44 years. Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann. How old is Mary?"

The answer: $M=27.5, A=16.5$. It is worth checking the intermediate ages as well; not just half numbers appear. To solve the Lloyd problem, a linear equation or system of equations must be solved, but to ensure the complete solution, it is also necessary to check that the different tenses are appropriate. In this KőMaL problem, there are intentionally not enough equations, but the tenses provide one inequality each. Likely due to the underdetermination, many assumed - incorrectly - that the numbers involved could only be integers.