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Ten in a Circle (illustration)
FILE No.149Unsolved

Ten in a Circle

Difficulty ★★☆about 5 minLogic PuzzleCounting outWriting it out
Ten people sit in a circle, and starting from number 1, every second person leaves in turn. Who is left at the end?

Case Record

Ten people sit in a circle in the order 1 to 10. Starting the count at number 1, the 2nd person leaves. Then count again from the person after the one who left, and the 2nd person leaves again. Continue until one person remains.

Your question

Which number is the last person left?

Case Files

Diagram

Ten people seated in a circle12345678910Start counting at No. 1; everysecond person leaves.Then start counting again fromthe person after the one who left,and again remove the second.Continue until one is left.Who is the last one left?
Diagram

Hints

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

Open hint 1

Who leaves first?

Open hint 2

After a full round, where do you start counting?

Open hint 3

Try writing out the order of leaving.

Solve It

Your question

Which number is the last person left?

Choose the answer closest to the truth.

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The Truth

Number 5

The answer is "Number 5."

The order of leaving is 2, 4, 6, 8, 10, 3, 7, 1, 9. The last remaining is number 5.

Numbers 10, 7, and 3 come up partway through the order of leaving.

Order of leavingOrder123456789Person2468103719The last one left is No. 5.
Diagram of the solution

Key Points

  • Actually write out the order
  • After a full round, check where counting starts
  • Recount the people who remain

Why people get fooled

Because it is easy to get the starting point of the count wrong.

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