Puzzles and dystopian fiction have a recurring theme in common: those in power forcing their subjects to play twisted games for entertainment. Crack the spine of any brainteaser book, and it reads like The Hunger Games–prisoners vying for their lives and monarchs sacrificing their servants in the name of logic.
One of the most popular puzzle genres involves a group of people wearing colored hats. It has hundreds of possible setups. Typically, people can see others’ hats but not their own, and the goal is to maximize the number of people who guess their own hat color. Sounds like a wholesome party game, yet the penalty for failure always involves slitting throats or eternal imprisonment.
I’ve given two hat puzzles below. The first is a classic and an absolute gem of puzzledom. Since some of you will have heard it before, I’ve also posed a fascinating variant that you won’t have encountered. Start the rebel uprising, and may the odds be ever in your favor.
Did you miss last week’s puzzle? Check it out here, and find its solution at the bottom of today’s article. Be careful not to read too far ahead if you haven’t solved last week’s yet!
Puzzle #48: Hat Trick
1. Ten friends, er, I mean terrified prisoners, are lined up. Everybody receives a hat that is either red or blue. Everybody can see all of the hats of the people in front of them in line but cannot see their own hat, nor the hats of those behind them in line. (For example, the back of the line can see nine hats and the front can’t see any). One at a time, starting at the back of the line, they’re each going to shout either “red” or “blue.” Everybody can hear other people’s shouts. If you shout your own hat color, you succeed, and if you shout the wrong color, you fail.
The people can strategize in advance, but once the game begins absolutely no information may be communicated beyond each shouting red or blue in turn. (No poking the person in front of you, no shouting in an accent, nothing.) You may assume that everyone cooperates with the group’s plan. What is the maximum number of people that the group can guarantee succeeds? (Not probabilistically). Oh, and anyone who fails will be drawn and quartered.
2. The same exact setup as puzzle 1, but with two wrinkles. First, the hats are assigned at random (in puzzle 1, the person assigning hats could know the people’s strategy, try their best to thwart it, and still the solution strategy will work!) Second wrinkle: people can say “pass.” If anybody guesses an incorrect color, then they all lose. If everybody passes, then they all lose. They all win if at least one person guesses a color and all guessed colors are correct. What strategy maximizes their chances of survival?
I’ll be back Monday with the answers and a new puzzle. Do you know a cool puzzle that you think should be featured here? Message me on X @JackPMurtagh or email me at gizmodopuzzle@gmail.com
Solution to Puzzle #47: All in the Family
Did you make your parents proud on last week’s kinship puzzles? Shout-out to Veronica, who emailed three correct answers complete with a family tree spreadsheet.
A man was looking at a photograph and said, “Brothers and sisters I have none, but that man’s father is my father’s son.” Who is in the photograph?
The man is looking at a photo of his son. “My father’s son” either refers to the man or the man’s brother, but the man says that he has no brothers, so we can rewrite his claim as, “That man’s father is me,” which clearly means he’s looking at his son.
A girl has an equal number of brothers and sisters. But each of her brothers has only half as many brothers as sisters. How many kids are in the family?
The family has seven kids: four girls and three boys. Each girl has three brothers and three sisters–an equal number. But the boys in the family each have four sisters and only two brothers–half as many.
A family picnic had the following people in attendance:
1 grandfather, 1 grandmother, 2 fathers, 2 mothers, 4 children, 1 brother, 2 sisters, 2 sons, 2 daughters, 3 grandchildren, 1 father-in-law, 1 mother-in-law, and 1 daughter-in-law, but there were only 7 people present. How is this possible?
The picnic consists of two sisters and their brother, the three kids’ mom and dad, and their grandfather and grandmother on their dad’s side. If you’re not sure who’s who, the family tree below should help you trace through the relationships. For example, 4 children refer to the two sisters, the brother, and the dad, because the dad’s parents are present too. The two mothers refer to the grandmother and the mom, etc.
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