Earlier today, I had the chance to delve into a fascinating puzzle that has sparked significant interest within the academic community due to its surprising outcomes. Let’s revisit the puzzle and explore its intriguing solution.
In this scenario, we have two players, Andrew and Barbara, who are faced with a grid containing fifteen boxes arranged as follows:
[Insert Image]
Prizes are hidden in two randomly selected boxes. Andrew searches the boxes row by row in the order A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, while Barbara takes a column-by-column approach, checking in the order A, F, K, B, G, L, C, H, M, D, I, N, J, O.
This raises a compelling question: if Andrew and Barbara open their boxes simultaneously on every turn—starting with A for both in the first round, then B for Andrew and F for Barbara in the second, and so on—who has a better chance of finding a prize first?
a) Andrew
b) Barbara
c) Both equally likely
The answer is a) Andrew.
At first glance, one might think both players have an equal chance of finding a prize, especially given that the boxes are selected at random. Indeed, if there were only one prize in a single box, they would have equal probabilities. However, the situation becomes more complex when two prizes are involved, as the game ends upon the discovery of the first prize.
Let’s analyze the scenario with a single prize. An illustration clarifies who would triumph based on the prize’s location. If the prize is in box A, H, or O, both players have an equal shot because they check those boxes simultaneously. In six boxes (turns 2, 3, 4, 5, 9, and 10), Andrew has the upper hand, being the first to check, while Barbara secures six boxes as well (turns 2, 3, 5, 6, 9, and 12).
[Insert Image]
Now, considering the case with two prizes, if both prizes happen to be in boxes Andrew checks first, he wins. Conversely, if both are in boxes Barbara reaches first, she wins. While both scenarios are equally likely, the critical factor arises when one prize is in a box Andrew checks first and the other in a box Barbara checks. In this instance, Andrew has a slight advantage, as he tends to discover the prize earlier on average. For example, if a prize is hidden in box N—checked by Barbara on turn 12—she is guaranteed a loss since Andrew will have already claimed victory by turn 10.
This puzzle was initially posed by Timothy Chow in 2010 and gained popularity a few years later on StackExchange, with mathematician Gil Kalai recently discussing it on his blog. The intricacies of this puzzle capture the attention of professional mathematicians, many of whom find it challenging to intuitively understand Andrew’s advantage.
If anyone has a clear and intuitive explanation, I encourage you to share your insights in the comments.
I hope you enjoyed this brain teaser! I’ll return in two weeks with another challenge.
By the way, had I known about Chow’s puzzle sooner, I might have included it in my recent book, “Think Twice,” which features various counter-intuitive puzzles. The aim of the book is for readers to enjoy it individually and in groups, as these puzzles spark fantastic discussions and debates.
“Think Twice: Solve the simple puzzles (almost) everyone gets wrong” (Square Peg, £12.99). To support the Guardian and Observer, you can order through guardianbookshop.com, though delivery charges may apply.
Since 2015, I’ve been sharing puzzles here every other Monday and am always on the lookout for new and interesting ones. If you have suggestions, feel free to email me!
