Last one alive 2

Logic Level 3

1000 people stand in a circle. There is a sword in the hand of the 1st person, he kills the 2nd person and passes on the sword to the third,who kills the fourth and gives the sword to the 5th, this goes on until one person is left. At what number was that person standing at the start?

453 875 731 977

Rishabh Sood by Rishabh Sood , 20, India

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2 solutions

This is a version of the Josephus Problem .

If 2 a 2^a is he largest possible power of 2 less than or equal to the number given, n n ,

then l = n − 2 a l=n-2^a

The person left is 2 l + 1 2l+1 .

In this question,

n = 1000 n=1000

l = 1000 − 2 9 = 1000 − 512 = 488 l=1000-2^9=1000-512=488

⟹ 2 l + 1 = 2 × 488 + 1 = 976 + 1 = 977 \implies 2l+1=2\times488+1=976+1=\boxed{977} .

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Kamalpreet Singh
Feb 6, 2016

Use A.P. series.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Absolutely, that is what I have done

Rishabh Sood - 5 years, 4 months ago

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I solved your question in my way. Again I noticed that it was the same as what you told. You just compressed it. In my solution of 5 p a g e s 5 pages I have derived why we are using the power of 2 2 and then found the answer.

Ashish Menon - 5 years, 4 months ago

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