It’s such fun to melt your brain every week, but today’s solution will be the last part of the solution Gizmodo Monday Puzzle. Thanks to everyone who commented, emailed, or was just silently confused. Since I can’t keep you waiting around idle, check out some of the puzzles I recently created for the Morning Brew newsletter:
I will also write one Math Curiosity Series In Scientific American, I take my favorite exciting ideas and stories from mathematics and present them to a non-mathematical audience. If you liked any of my prologue here, I guarantee you there will be plenty of intrigue in there.
Stay connected with me on X @JackPMurtagh As I continue to try and leave the internet scratching its head.
Thank you for the fun,
Jack
Solution to Puzzle #48: Hat Trick
did you survive last week’s Dystopian nightmare? speak out Beibei Solve the first puzzle and Gary Abramson An impressively concise solution is provided to the second puzzle.
1. In the first puzzle, the team can ensure that all but one person survives. The people in the back don’t know the color of their hats. Therefore, they will use their only guess to convey enough information that the remaining nine people can deduce their own hat color with certainty.
Those in the back will count the number of red hats they see. If it’s an odd number, they shout “red” and if it’s an even number, they shout “blue.” Now, how does the next person in line deduce the color of his or her hat? They saw eight hats. Suppose they count an odd number of reds in front of them; they know that the person behind them sees an even number of reds (because that person yelled “blue”). This information is enough to deduce that their hats must be red, so that the total number of reds is equal. The next person also knows whether the person behind them saw an even or odd number of red hats, and can make the same inference for themselves.
2. For the second puzzle, we will come up with a strategy that guarantees the survival of the entire group, unless all 10 hats happen to be red. Only one person in the group needs to guess right, one wrong guess automatically kills everyone, so once one person guesses a color (rejects a pass), everyone after that gets a pass. The goal is to have the blue hat closest to the front of the line guess “blue” and then let everyone else pass. To achieve this, everyone passes unless they only see the red hat in front of them (or if the person behind them has already guessed it).
To understand how this works, note that people at the back of the queue will pass unless they see nine red hats, in which case they will guess blue. If they say blue, then everyone else passes and that group wins, unless all ten hats are red. If people from behind pass by, it means they see a blue hat in front of them. If the second-to-last person sees eight reds in front of them, they know they must have a blue hat, so the guess is blue. Otherwise, they pass. Everyone passes until someone at the front of the line only sees the red hat in front of them (or no hat in the case of the front of the line). In this case, the first guess is blue.
The chance that all 10 hats are red is 1/1,024, so the chance of this group winning is 1,023/1,024.
