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A287150 Fairest turn sequence for 3 players where the probability of a win for a player on his turn approaches 0. 0
0, 1, 2, 2, 1, 0, 2, 1, 0, 0, 1, 2, 1, 2, 0, 0, 2, 1, 0, 2, 1, 1, 2, 0, 2, 1, 0, 0, 1, 2, 1, 0, 2, 2, 0, 1, 0, 2, 1, 1, 2, 0, 2, 1, 0, 0, 1, 2, 1, 0, 2, 2, 0, 1, 0, 2, 1, 1, 2, 0, 1, 2, 0, 0, 2, 1, 0, 2, 1, 1, 2, 0, 2, 1, 0, 0, 1, 2, 0, 1, 2, 2, 1, 0, 2, 1, 0, 0, 1, 2, 1, 0, 2, 2, 0, 1, 2, 0, 1, 1, 0, 2, 1, 0, 2, 2, 0, 1, 0, 2, 1, 1, 2, 0, 1, 2, 0, 0, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

0, 1, and 2 are three archers of equal skill, but which have infinitesimally low chances of hitting a target. Still, they try: taking turns until one of them hits: 0 goes first, if 0 misses then 1 will try (these are the first two terms). Subsequent terms are found by giving a turn to the player who has the lowest probability of winning so far.

Joshua Cooper and Aaron Dutle showed that for two players this is the Thue-Morse sequence (A010060).

The following observations are made and verified to be true for the first 5000 terms:

- Every 3 terms completes a new round with each player going once per round.

- Every other round is the same as the last but in reverse order.

- The 48th-101st terms form a block of 54 terms which are repeated from then on.

LINKS

Table of n, a(n) for n=0..119.

Joshua Cooper and Aaron Dutle, Greedy Galois Games, Amer. Math. Monthly, 120 (2013), 441-451, arXiv:1110.1137 [math.CO].

Daniel Hug, Generate fairest turn sequence for n players (web app written in JavaScript)

MATHEMATICA

n=3; a=Range[n]; pw=(1-p)^(Range[n]-1)*p;

Do[

next = First[Select[Range[n], And@@NonNegative/@Limit[Sign[pw-pw[[#]]], p->0]&, 1]];

AppendTo[a, next]; pw[[next]]+=p*(1-p)^k

, {k, n, 50}];

a-1

(* Andrey Zabolotskiy, Jun 05 2017 *)

CROSSREFS

Cf. A010060.

Sequence in context: A153241 A257400 A194853 * A257109 A096830 A141647

Adjacent sequences:  A287147 A287148 A287149 * A287151 A287152 A287153

KEYWORD

nonn

AUTHOR

Daniel Hug, May 20 2017

STATUS

approved

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Last modified October 21 15:56 EDT 2018. Contains 316424 sequences. (Running on oeis4.)