Here at PopMech, we love mind-bending math and logic puzzles, which is why we regularly recruit the sharpest minds in the world to concoct riddles that will test your critical thinking, mathematics, and logic skills. The following riddle is one particularly punishing problem—so go grab a pencil and a piece of scratch paper and prepare to rip your hair out (in the best way). No cheating!
➡ The Problem
Three sisters combine their money to buy a mystery box full of an unknown amount of comic books. The box arrives at the sisters’ house one night, and they agree to store it in their younger brother’s room and not open it until the next morning.
But that night, one sister can’t sleep, so she decides to sneak in and open the box. Looking inside, she finds that she can split the comic books into three equal shares, plus one extra comic. She sees her brother awake and spying, so she hands him the extra comic, takes her own share of one-third of the remaining comic books, and goes back to sleep.
An hour later, the second sister does the exact same thing. She opens the box and sees that the comics can be split into three equal shares, plus one extra comic. She bribes the spying little brother with a comic, takes one-third of the remaining books, and goes back to sleep.
Then, an hour after that, the third sister does the same thing yet again, bribing the spying little brother with a comic, taking one-third of the remaining books, and returning to bed.
In the morning, none of the sisters want to reveal that they’ve already taken any comics. So they each pretend to be surprised when they look in the box. They dump out the comics and realize that the number of books can easily be split into three equal shares along with one extra comic, which they gave to their happy little brother.
What’s the smallest possible number of comic books that could have been in that mystery box when it first arrived? And how many comics does each sibling have?
➡ The Solution
It pays to take this one slowly and think about each step headed backward toward the initial Mystery Box. This is a great recursion problem, which means there’s plenty of ways to get confused about what step you’re on. There are four moments to consider: Final Share, 3rd Share, 2nd Share, and 1st Share.
The smallest Final Share is 1 per sister. That means the Final Pile is three times that share, plus 1 extra, for a total of 4 comic books.
For the 3rd Share, those 4 comic books must represent the two-thirds that the third sister left in the box. Each sister would have gotten a share of 2. Which means the 3rd Pile must have had three times the share, plus 1 extra, for a total of 7.
By the time we get to the 2nd Share, we see the problem. If there were 7 left in the box, did the second sister take 3.5 comic books? We need a solution where each share is a whole number.
Hopefully the recursive process is clear, which means we might as well set up a table and work until we find the right answer. Notice that this table includes the Share for each sister at any moment, as well as the Pile just before anyone took their share.
It turns out most of our attempts are doomed to decimal oblivion. The smallest working solution comes when each of the sisters gets 7 comic books as their final share. Working backward, that means the mystery box itself had 79 comic books. But who ended up with those books?
- The first sister snagged 26 comic books during the initial split, plus 7 more the next morning, for a total of 33 comic books.
- The second sister managed to walk away with 24 comic books.
- The third sister managed to gain 18.
- And, of course, the youngest brother got 4 comic books for free.
This puzzle is based on a fantastic old riddle collected by Maurice Kraitchik in Mathematical Recreations.
➡ Pro Perk: Chat With the Puzzle Coach!
Curious to know more about this particular problem and solution? Stumped by another riddle and want the secret to solving it? I’d love to hear from you. Email me with your questions and we can nerd out over all kinds of puzzles, math problems, and brain teasers together!
Richard Malena is a puzzle and game designer, creating codebreaking challenges for the launches of the Tomb Raider movie and Call of Duty: WWII, as well as for board games, film festivals, immersive theater, and all kinds of strange and esoteric endeavors. When he isn’t building puzzles, Rich is likely rewatching National Treasure 2: Book of Secrets and searching for hidden mysteries in the world.