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Witnesses Who Like Dogs and Cats (illustration)
FILE No.43Unsolved

Witnesses Who Like Dogs and Cats

Difficulty ★★☆about 4 minLogic PuzzleSetsSize of overlap
We asked 25 witnesses about their tastes. 18 like dogs and 15 like cats. Everyone likes at least one. How many like both?

Case Record

There were 25 witnesses to an incident in a park. When questioning them, the police also asked which animals they like.

  • People who like dogs: 18
  • People who like cats: 15

All 25 said they like "at least one of dogs or cats." No one likes neither.

There is information that the culprit is among "people who like both dogs and cats."

Your question

How many people like both dogs and cats?

Case Files

Evidence

Diagram

Witnesses’ tastesItemPeopleTotal witnesses25Like dogs18Like cats15Like neither0Everyone likes at least one of dogs or cats.
Diagram

Timeline

  1. An incident takes place in a park.
  2. The 25 witnesses are asked about their tastes.

Hints

If you get stuck, open the hints one at a time, in order.

Open hint 1

What do you get by adding the dog lovers and the cat lovers?

Open hint 2

How many people were there in all?

Open hint 3

Who are the people counted too many times in the sum?

Solve It

Your question

How many people like both dogs and cats?

Choose the answer closest to the truth.

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The Truth

8 people

The answer is 8 people.

Adding 18 dog lovers and 15 cat lovers gives 33. But there are only 25 people in all. The excess, 33 - 25 = 8, is the number of people counted in both groups.

So 8 people like both. Since the condition is that "everyone likes at least one," this calculation always settles it.

Add, then subtract the total18 dog lovers + 15 cat lovers = 33The total is 25, so 33 - 25 = 8 were counted twiceThose who like both: 8 people
Diagram of the solution

Key Points

  • Adding two groups counts the overlapping people twice
  • The amount over the total is the number in the overlap
  • "No one likes neither" is the condition for the calculation

Why people get fooled

The dog and cat lovers look like two separate groups, and you forget that they overlap.

Similar tricks

  • The type that counts set overlaps
  • The type that notices double counting in a sum

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