It can seem like a miracle that some puzzle solutions are so good. Some of the qualities exhibited by the solution to the third question in the most recent Insights puzzle are shown. The question about the best way to use three balance scales, including a broken one that gives random results, to find a counterfeit coin could be solved in four weighings.
A puzzle that improves upon the more obvious solution by at least 60 times is presented today. Beautiful proofs are from the Book, which is a divine volume where the perfect proof of all math problems can be found. If the book has any puzzles, this one must be in it.
Our puzzle imagines a Star Trek adventure in which Lieutenant Uhura faces a life-and-death dilemma.
Lieutenant Uhura was given her sixth surface mission as the enterprise approached a hitherto unknown planet. The landing party was taken to the planet's surface. The landing party was on its own after the Enterprise was diverted to respond to a distress call.
The planet was populated by an advanced civilization with an advanced cloaking technology even though the crew had failed to find any intelligent life. The landing party was tried for being on the wrong side of the tracks.
The judge said that you have a high chance of being found guilty. Our laws know that all doubt can never be eliminated and therefore all punishments are probabilistic. While we prepare the roulette chamber, you will be held in a common room. Each of the eight buttons will respond to one of you. Each button will be labeled with one of your names in order to confuse you. It won't be yours.
We will lead you one at a time in an order of our choosing once the chamber is ready. Each of you will have the opportunity to find your true button. A display will let you know who the button is for. The buttons can be pressed in any order. You need to find your own button. All of you will be executed if anyone attempts to press a fifth button. If all of you succeed, you'll be free.
Once a person is done with the chamber, they will be taken to their own solitary cell with no way to communicate with others. Until the chamber is ready, you have a few hours with each other. The landing party was left to its own devices.
There was a great deal of consternation. At least we have a chance. Things don't look good and I would like to be optimistic. Each of us has a chance of finding our button. There is a small chance that all of us will do it. Nyet! Nyet, huh? The old Russian roulette is worse than this one.
A person chimed in. We have to keep trying. Our chances of survival will be only 0.4% if all of us press the buttons at the same time. We can come up with a strategy to make sure all the buttons are pressed at the same time. Won't that increase our chance?
For a second, Uhura thought about this. I think that the improvement will be small. We might not break 1%.
How much can the crew's survival probability be improved if each button is pressed equally frequently?
Our story is going on.
Uhura frowned at something. She appeared to have an idea. She drew a picture that looked like a Star of David. She remembered a game she used to play as a child. I think it has something to do with it.
The game develops both running and memory skills. The points are labeled A to F on the playground. A is the base of the player. The children place themselves on the corner with the letter they picked. The player who was on the home base ran and tagged the other player. The child runs and displaces the player whose home base he was on until the player who was on the empty base runs to the one he left. The play goes clockwise to the next player. The child doesn't have to run if they are already at home. When play has been passed to all eligible players, the round is done.
In the figure shown above, the player on base C runs and displaces the one on E. He went to the empty base C where the player was since he was on A. The other three players formed a cycle of length 3. The round is complete. Their running paths form a six-pointed star, but a lot of other patterns can be found.
There were lots of rounds with no cycles of more than three legs.
If the children play well, what is the likelihood that there will be a cycle greater than 3?
Click below to find a clue if you get stuck.
You can click for a hint.
Try it for a four-cornered diamond-shaped field. What is the probability that there will be no cycle greater than length 2?The story needs to be back to it's starting point.
Uhura said that was it. We have a good chance of going free if my calculations are correct.
What strategy was suggested by the game and how it improves the landing party's chances of freedom from less than 1% to over 34%?
One of the Catenati has taken a dislike to the crew and is watching them. He thinks that they have come up with a good strategy. He is going to change the order of the buttons before the roulette starts to foil their plan. Is he able to stop the plan? The landing party has to be careful about concealing their identity.
As the size of the landing party grows, what is the maximum percentage of success approach? Is it possible that this method is more efficient than random button pressing?
The crew escaped and continued to explore other strange and dangerous worlds after Uhura executed his plan.
For our space adventure, that's it. Happy thinking and may your brain speed up.
The reader who submits the most interesting, creative or insightful solution in the comments section will be given a T-shirt or one of the two Quanta books, Alice and Bob Meet the Wall of Fire. If you would like to suggest a puzzle for a future Insights column, please submit it as a comment below, marked "New Puzzle Sagittarius." The solution to the puzzle above should be submitted separately.
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