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.

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 m...

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?

In Round V, survivors are #9 {(1+8*1)}, #25 {(1+8*3)}, #41, #57 & #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.

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.