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Survivor It has produced 45 seasons, and continues to grow. If you remember the buzz from Season 1 but can’t believe it’s been 45 years, it hasn’t. They have produced two seasons per year for the most part to feed their insatiable fan base. Each season, the mega-successful reality show pits a charming cast of conflicting personalities on a tropical island to compete for a million dollar prize. Contestants must complete physical, endurance, and mental challenges to earn survival gear or immunity to avoid elimination from the show. The island is rapidly turning into a pressure cooker of scheming and backstabbing, much to the delight of the audience.
This week’s puzzle comes from an actual mental challenge the show used in the season 5 premiere, set in Thailand. Two teams compete in a strategy game, but as you’ll find, one team can always win by making the right moves. The castaways failed to find the optimal strategy in real time, instead exchanging blunders until it was too late for one team. This is understandable under time pressure, but some fans have lamented that the existence of an unbeatable strategy made the game unfair.
If you think you might have seen this on a warm beach with a television camera in your face, let me know. Good luck—your community is counting on you.
Did you forget last week’s puzzle? check it out Here, and find its solution below today’s article. Be careful not to read too far if you haven’t solved last week yet!
Puzzle #30: Survivor flags
Jeff Probst has placed 21 flags on the field. Tribe A and Tribe B will take turns removing one, two or three flags at a time (zeros are not allowed). The tribe that removes the flag last wins. If tribe A goes first, Which team can force a win and what is the winning strategy?
I will be back next Monday with the solution and a new puzzle. Do you know any good puzzles that you think should be featured here? message me on twitter @jackpmurtagh or email me [email protected]
Solution to Puzzle #29: A Game of Chance
Did you win? Last week Competitive puzzles?
I missed the Super Bowl. All I know is that both teams were exactly matched in skill (suspend your disbelief) and the score was No Tied at halftime. I want to know the chances that a team that trailed at halftime comes back to win the game. Given only this information, what should I bet on?
By perfectly matched I mean that teams have equal probabilities of achieving different scores and furthermore these probabilities do not change depending on the state of play (ie which half they are in or who is ahead). Remember that Super Bowl games cannot end in a tie: if the scores are still tied after the second half, they enter overtime.
A team trailing at halftime has a 25% chance of coming back and winning the game. Consider each half of the game separately. We will ignore all ties because we are told the game is not tied at halftime and we know the game is not allowed to end in a tie. Let’s call teams A and B and note who “wins” while ignoring each half. There are four possibilities: AA, AB, BA, BB (for example BA means B scored more marks than A in the first part, but A scored more marks than B in the second part).
Of course, it is important not only who wins each half but also by how much. In BA, B can win by 40 points in the first half and lose by only 2 points in the second half and thus he will still win the entire game. Let’s present all the possibilities in a table:
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<p><figcaption class="sc-7s1ndr-0 fPOdhF no-caption">graphic<!-- -->, <!-- -->jack murtagh</figcaption></p>
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<p class="sc-77igqf-0 fnnahv">A key insight is that, since the teams are evenly matched, all eight possibilities occur with equal probability. Both teams have a 50/50 chance of scoring more points in either half. And the first half has an equal chance of having a larger or smaller spread than the second half. Only two of the eight rows (25%) represent games where the team trailing at halftime came back: Row 4 and Row 6. For example, line 5 means that A trailed at halftime and despite scoring more than B in the second half, it was not enough to overcome A’s deficit.</p>
<p class="sc-77igqf-0 fnnahv"><strong>Dr. A.S. emily lizard </strong>Estimated the correct answer at 25% because football broadcasts often quote that number. He shared a link <span><a class="sc-1out364-0 dPMosf sc-145m8ut-0 jCErAQ js_link" data-ga="[["Embedded Url","External link","https://theathletic.com/713793/2018/12/18/nfl-halftime-scores-winners-losers-houston-texans-bill-obrien-best-winning-percentage-when-leading/",{"metric25":1}]]" href="https://theathletic.com/713793/2018/12/18/nfl-halftime-scores-winners-losers-houston-texans-bill-obrien-best-winning-percentage-when-leading/" target="_blank" rel="noopener noreferrer">this article</a></span> It looks at nearly 40 years of NFL games and finds that the team that leads at halftime wins between 73% and 82% of the time. One <span><a class="sc-1out364-0 dPMosf sc-145m8ut-0 jCErAQ js_link" data-ga="[["Embedded Url","External link","https://www.stat.berkeley.edu/~aldous/157/Papers/stern.pdf",{"metric25":1}]]" href="https://www.stat.berkeley.edu/~aldous/157/Papers/stern.pdf" target="_blank" rel="noopener noreferrer">analysis of nba games</a></span> found that in 74.8% of cases, the team leading at halftime wins the game. So this simplified model may be on to something! </p>
<p class="sc-77igqf-0 fnnahv">Now, onto last week’s bonus puzzle.</p>
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<p class="sc-77igqf-0 fnnahv">A simple round-robin tournament involves several teams participating (ie each team plays every other team once). Call a team a super-winner if all the other teams in the tournament have either lost to them or have lost to someone who has lost to them. <strong>Argue that every tournament has at least one super-winner. </strong></p>
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<p class="sc-77igqf-0 fnnahv">shout out to <strong>Eugene </strong>For a very concise solution to the bonus puzzle.</p>
<p class="sc-77igqf-0 fnnahv">Look at the team that has won the most games. If there are relations, choose one of them. Let’s call them goats. We chose goats because <strong>No team has won more games than them</strong>, I bet the goats will definitely be super-winners. suppose they <em>were not</em> Super winner. This assumption would lead to a contradiction, leading us to conclude that the goats were actually super-winners.</p>
<p class="sc-77igqf-0 fnnahv">If they’re not the super-winners, there must be some team, let’s call them The Underdogs, that beat The Goats <em>And</em> Defeat all those who were defeated by goats. But this would mean that The Underdogs won more games than The Goats! Because The Underdogs Have Beaten Every Team The Goats Have Beaten <em>plus</em> The goats themselves. No one won more games than The Goats, so this is a paradox and our assumption may have been wrong: The Goats were actually super-winners. </p>
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