MATH MONDAY - Evapoweight
Today’s Puzzle
Suppose you have 100 pounds of cucumbers that are 99% water by weight. You leave the cucumbers out in the sun until they’re 98% water (again, by weight). How much do the cucumbers weigh now?
Thursday’s Puzzle - Let them eat cake
Your puzzle-obsessed friend is having a logic-themed birthday party. You thought that was kind of weird, but you decided to go anyway because she said there’d be cake. As soon as you arrive, she announces the first party game. She says she is going to line you and nine other friends up in height order, such that you can see all of the people in front you, but no one behind. She will then put either a black or white party hat on each person’s head, assigned randomly (there may be ten white hats, or there may be none). Each person, she says, starting from the tallest and going down the line, will have to say either the word “black” or the word “white” to indicate the color of their own hat. Your friend announces that there will be no cake unless at least nine of you can correctly state the color of your own hat. She allows you to confer as much as you want before you line up and get hats, but reminds you that, once you are lined up, the only words you can say are “black” and “white”. How do you get the cake?!
[SPOILER] Answer to Thursday’s Puzzle
The tallest person should say “white” if they see an even number of white hats, and “black” if they see an odd number of white hats (or vice versa). They might get their own hat color wrong, but this strategy gives enough information for everyone else to get their hat color right.
Now go get your cake! Or, if you want to know why this works, read on 🙂
The answer is easiest to explain if we go through an example. Let’s say the order of hats is the following (tallest to shortest):
B W B B B W B W W B
The tallest person sees nine hats ahead of them, four of which are white, so they say “white” to indicate that the number of white hats is even. They get their hat color wrong, but that’s okay (since you only have to get 9/10 correct).
Next, the second person sees eight hats ahead, three of which are white. But since the first person said there were an even number of white hats ahead of them, the second person knows that their own hat must be white (i.e. the fourth hat that the tallest person saw). So, they correctly say “white”.
Now, the third person knows that originally there was an even number of white hats, but because the second person said “white”, there’s now one fewer white hats, making the number odd. The third person sees three white hats ahead, which is already odd, so their own hat must be black.
Similarly, the fourth person knows that still only one white hat has been identified since the first person said there was an even number. The fourth person also see three white hats ahead, so they’re also black.
This continues down the line, each person going through the following thought process:
- Start with the odd/even declaration from the first person (in this case, even).
- Subtract the number of times that color has been stated since the first person.
- If the result is odd, and you see an odd number of hats of that color, your hat is the other color; if the result is odd and you see an even number of hats of that color, your hat is that color.