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| Even percents (Posted on 2022-05-10) |
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Two players decided to play many rounds of the same two-player game. One of the players checked their winning percentages after every round and found they had winning percentages of exactly 30%, 40%, 50%, 60%, & 70% at some point, at least once and not necessarily in this order. What's the minimum number of rounds they played?
Extension: What if the set of percentages included all of 10% through 90%?
Adapted from an article in The Guardian.
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No Solution Yet
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Submitted by Jer
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Rating: 5.0000 (2 votes)
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soln (entirely proved)
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Comment 3 of 3 | 
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I was able to verify Paul's conjectures with a thorough search of solutions. For the 30%-70% problem, I examined all games less than 60 rounds and for 10%-90% I looked at games up to 300 rounds. I found the games that met the requirements, sorted by the least rounds, and listed the top 10 below.
30 rounds and 150 rounds were indeed the minimal numbers. All solutions come in complementary pairs (with the players switching roles) as was pointed out. There are at lease five distinct routes to a 30-round solution and two distinct routes in the 150-round solution, along with their paired complements.
The programs are here and here. The programs search for the decadal percentages (10%, 20%, ...) being reached in every possible order. The search is made efficient by considering solutions for wining percentages from the highest "total rounds" games to the lowest, because then the search is quickly cut-off by the number of rounds already reached.
lord@rabbit 12848 % o4
====== total rounds: 150 ====== ( a, b, tot % -------------------- ( 1, 4) 5 20% ( 1, 9) 10 10% ( 6, 9) 15 40% ( 6, 14) 20 30% ( 14, 14) 28 50% ( 21, 14) 35 60% ( 35, 15) 50 70% ( 60, 15) 75 80% (135, 15) 150 90%
====== total rounds: 150 ====== ( a, b, tot % -------------------- ( 1, 1) 2 50% ( 1, 4) 5 20% ( 1, 9) 10 10% ( 6, 9) 15 40% ( 6, 14) 20 30% ( 21, 14) 35 60% ( 35, 15) 50 70% ( 60, 15) 75 80% (135, 15) 150 90%
====== total rounds: 150 ====== ( a, b, tot % -------------------- ( 1, 1) 2 50% ( 4, 1) 5 80% ( 9, 1) 10 90% ( 9, 6) 15 60% ( 14, 6) 20 70% ( 14, 21) 35 40% ( 15, 35) 50 30% ( 15, 60) 75 20% ( 15,135) 150 10%
====== total rounds: 150 ====== ( a, b, tot % -------------------- ( 4, 1) 5 80% ( 9, 1) 10 90% ( 9, 6) 15 60% ( 14, 6) 20 70% ( 14, 14) 28 50% ( 14, 21) 35 40% ( 15, 35) 50 30% ( 15, 60) 75 20% ( 15,135) 150 10%
====== total rounds: 160 ====== ( a, b, tot % -------------------- ( 1, 1) 2 50% ( 4, 1) 5 80% ( 9, 1) 10 90% ( 9, 6) 15 60% ( 14, 6) 20 70% ( 14, 21) 35 40% ( 15, 35) 50 30% ( 16, 64) 80 20% ( 16,144) 160 10% lord@rabbit 12848 % o5
====== total rounds: 30 ====== ( a, b, tot % -------------------- ( 2, 3) 5 40% ( 3, 7) 10 30% ( 8, 8) 16 50% ( 12, 8) 20 60% ( 21, 9) 30 70%
====== total rounds: 30 ====== ( a, b, tot % -------------------- ( 2, 3) 5 40% ( 3, 7) 10 30% ( 7, 7) 14 50% ( 12, 8) 20 60% ( 21, 9) 30 70%
====== total rounds: 30 ====== ( a, b, tot % -------------------- ( 2, 3) 5 40% ( 3, 3) 6 50% ( 3, 7) 10 30% ( 12, 8) 20 60% ( 21, 9) 30 70%
====== total rounds: 30 ====== ( a, b, tot % -------------------- ( 2, 3) 5 40% ( 3, 3) 6 50% ( 7, 3) 10 70% ( 9, 6) 15 60% ( 9, 21) 30 30%
====== total rounds: 30 ====== ( a, b, tot % -------------------- ( 2, 3) 5 40% ( 7, 3) 10 70% ( 9, 6) 15 60% ( 9, 9) 18 50% ( 9, 21) 30 30%
====== total rounds: 30 ====== ( a, b, tot % -------------------- ( 2, 2) 4 50% ( 2, 3) 5 40% ( 3, 7) 10 30% ( 12, 8) 20 60% ( 21, 9) 30 70%
====== total rounds: 30 ====== ( a, b, tot % -------------------- ( 2, 2) 4 50% ( 2, 3) 5 40% ( 7, 3) 10 70% ( 9, 6) 15 60% ( 9, 21) 30 30%
====== total rounds: 30 ====== ( a, b, tot % -------------------- ( 2, 2) 4 50% ( 3, 2) 5 60% ( 3, 7) 10 30% ( 6, 9) 15 40% ( 21, 9) 30 70%
====== total rounds: 30 ====== ( a, b, tot % -------------------- ( 2, 2) 4 50% ( 3, 2) 5 60% ( 7, 3) 10 70% ( 8, 12) 20 40% ( 9, 21) 30 30%
====== total rounds: 30 ====== ( a, b, tot % -------------------- ( 1, 1) 2 50% ( 2, 3) 5 40% ( 3, 7) 10 30% ( 12, 8) 20 60% ( 21, 9) 30 70% lord@rabbit 12848 %
Edited on May 18, 2022, 11:04 pm
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