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Red Caps is a puzzle in Professor Layton and the Diabolical Box.

Puzzle

US Version

A preschool teacher had everyone in her class close their eyes. While none of the students were looking, she slipped caps onto their heads and then said:

"OK, everyone, open your eyes and look at the hats all your friends are wearing. Whoever sees four or more people wearing red hats gets a red balloon, and whoever doesn't gets a blue balloon."

In a class of 10 children, only some of the kids got a red balloon. Knowing this, how many kids went home with a red balloon?

UK Version

A teacher had everyone in her class close their eyes. While none of the students were looking, she slipped caps onto their heads, and said:

"OK, everyone, open your eyes and look at your friends' caps. Whoever sees four or more people wearing red caps gets a red balloon, and whoever doesn't gets a blue balloon."

In a class of 10 children, only some got a red balloon. But how many?

Hints




Click a Tab to reveal the Hint.

US Version

The puzzle says that some of the children received red balloons.

From that, you can infer that there were no fewer than four children in red hats.

UK Version

The puzzle says that some of the children received red balloons. This means some children saw four or more red caps.

If there were, say, three children wearing red caps, then no one would get a red balloon.

So there must be four or more children wearing red caps.

US Version

The puzzle says some of the kids received a red balloon.

By definition, therefore, not all the kids received a red balloon.

UK Version

The puzzle says that some of the children received a red balloon.

By definition, therefore, not all the children received a red balloon. What would be the number of red hats that would allow everyone to receive a red balloon?

The answer is less than that number.

US Version

If every preschooler had on a red hat, you'd see every single child go home with a red balloon.

Remembering that the children aren't allowed to add their own hat to their total, think about how many children would recieve a red balloon if nine kids wore red hats. What about if eight did? Seven? Just keep going down the line.

UK Version

If all 10 children were wearing red caps, then everyone would see at least four and would get a red balloon. But no child can see the cap on their own head. With this in mind, ask yourself how many children would receive a red balloon if nine of them wore red hats. What about if eight did? Seven?

Just keep going down the line, and remember that at least four must be wearing red caps.


Solution

Incorrect

Too bad!

US Version

Think hard about the clues you've been given and try again.

UK Version

Think hard about the puzzle and try again.

Correct

That's right!

US Version

Since you know only some of the children received red balloons, you also know that some children didn't receive them. If five or more children were wearing a red hat, every child in the class would get a red balloon. Therefore, four children must have had red hats, meaning that six received red balloons.

UK Version

You know that only some children got red balloons. If five or more children were wearing a red cap, every child in the class would see at least four red caps and get a red balloon, so it must be fewer than five. But if three or fewer children were wearing red caps, no one would see four red caps.

Therefore, four children must have had red caps, meaning that the other six children received red balloons.

DB014S

A big thanks to http://professorlayton2walkthrough.blogspot.com

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