This might take number of reads to understand, but concentrate on it, you'll get!﻿

100 people standing in a circle in an order 1 to 100.No. 1 has a sword. He kills the next person (i.e. No. 2) and gives the sword to the next (i.e. No. 3). All people do the same until only 1 survives. Which number survives at the last?

The Question (Josephus Problem)

In Round IV, survivors are {1+8*(2n+1)} - starting with #9 who bumps off #17 (1+8*2) and hands the dagger to #25 (1+8*3). The last survivor of this round is (1+8*11) #89 who kills #97 and hands the dagger to #9.

Round #4

In Round VI, survivors are #9, #41 and #73. So, once again, #9 bumps off #41 and hands the dagger over to #73. But, what goes around, comes around.#73 bumps off #9 and emerges the winner of this round and the whole series.The lone survivor in friend kills friend, man kills man sordid puzzle is #73 and of course, the ubiquitous dagger.

Round #6

In Round III, #1 bumps off #5 and hands the dagger to #9. Survivors follow the pattern (1+8n) with #97 being the last one in the round (1+8*12). The person next to him is #1 and it is his turn to get bumped off. Next round starts with #9

Round #3

In Round II, #1 starts by bumping off #3. Thus, the numbers that survive correspond to (1+4n) and therefore, the last one surviving the round is (1+4*24) ie 97 who proceeds to bump off #99 and hands the dagger to #1.So far so good?

Round #2

In Round V, survivors are #9 {(1+8*1)}, #25 {(1+8*3)}, #41, #57 &amp; #73. Alternate knocks and #9 kills #25, dagger to #41 who kills #57 who gives dagger to #73 who kills #89 and hands the dagger over to #9.

Round #5

In round one, number 1 picks up the dagger/sword and kills the person next to him - #2. He proceeds to hand the dagger over to #3 who kills #4 and so on.Thus all even numbers die with #100 being the last one to die and the dagger is with #1 for round II.

Round #1

The Josephus Problem, featuring Daniel Erman from University of Wisconsin-Madison. Winning at Dots and Boxes: https://youtu.be/KboGyIilP6k More links & stuff in full description below ↓↓↓ Correction: Around 9:40 that should be L less than 2^a NOT 2a --- Sorry, typo in editing! But you got the point hopefully. Support us on Patreon: http://www.patreon.com/numberphile NUMBERPHILE Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Subscribe: http://bit.ly/Numberphile_Sub Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile Videos by Brady Haran Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/ Brady's latest videos across all channels: http://www.bradyharanblog.com/ Sign up for (occasional) emails: http://eepurl.com/YdjL9 Numberphile T-Shirts: https://teespring.com/stores/numberphile Other merchandise: https://store.dftba.com/collections/numberphile

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The Josephus Problem, featuring Daniel Erman from University of Wisconsin-Madison.
Winning at Dots and Boxes: https://youtu.be/KboGyIilP6k
More links & stuff in full description below ↓↓↓

Correction: Around 9:40 that should be L less than 2^a NOT 2a --- Sorry, typo in editing! But you got the point hopefully.

Support us on Patreon: http://www.patreon.com/numberphile

NUMBERPHILE
Website: http://www.numberphile.com/
Numberphile on Facebook: http://www.facebook.com/numberphile
Numberphile tweets: https://twitter.com/numberphile
Subscribe: http://bit.ly/Numberphile_Sub

Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile

Videos by Brady Haran

Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/

Brady's latest videos across all channels: http://www.bradyharanblog.com/

Sign up for (occasional) emails: http://eepurl.com/YdjL9

Numberphile T-Shirts: https://teespring.com/stores/numberphile
Other merchandise: https://store.dftba.com/collections/numberphile

Videos about numbers and mathematics. Videos by Brady Haran since 2011.

Round #1

Similar ideas to Round #1

Practice

Getting Technical

#1-#3