Click the link below the picture
.
Sun Bin, allegedly descended from The Art of War author Sun Tzu, was a legendary Chinese military strategist who lived more than 2,300 years ago. After one king imprisoned and tortured him, forcing him to use a wheelchair for the rest of his life, he rose to fame as a military advisor for a rival king and lived to record two great victories against his former captor.
One of Sun Bin’s greatest exploits came not on the battlefield but on a racetrack. Like his supposed ancestor, who wrote, “Warfare is a way of deception,” Sun Bin discovered a “way of deception” in horse races that can be boiled down to a simple mathematical puzzle. For reasons that I cannot explain, the most general version of this puzzle remained unsolved and unexamined until 1980, when a college classmate of mine rediscovered it. Thirty years later he told me about the puzzle, and I was hooked. Besides transporting me back to ancient China, solving this problem took me back to my own roots as a mathematician.
I like to approach Master Sun’s problem through several simpler puzzles:
Puzzle One: Suppose you toss a coin six times. What is the probability of getting exactly three heads and three tails? What if you toss the coin 10, 100 times or any even number of times? How likely are you to get exactly the same number of heads and tails?
Puzzle Two: Now pick an even number, which we’ll call 2N, and a target number, N – k, which is less than N. If you flip a coin 2N times, do you think it is more likely that you will get exactly N heads or that you will get (N – k) or fewer heads? For example, are you more likely to get exactly 10 heads in 20 flips or to get eight or fewer heads in 20 flips? Experiment with different values of k and N and see if you can find a general rule.
The next two puzzles seem, on the surface, to have nothing to do with the first two.
Puzzle Three: Two teams, each with three players, are planning to compete in a match. Each team is supposed to rank its players from best to worst. The number one seed on one team is supposed to play the number one seed on the other, and so on. One coach is honest and ranks their players by true order of ability. The other coach is dishonest and decides to strategically rank their players in a different order. Can the latter coach tilt the odds in their favor? If so, what ranking strategy gives their team the greatest chance of winning the match? What if each team has 10, 100 or N players?
Puzzle Four: The dishonest coach decides to place bets on the individual games rather than the overall team match. For this purpose, they want to know what ranking strategy will give their team the greatest expected number of games won. If there are three players per team, what strategy is optimal, and how many individual games can the coach expect to win? What if there are 10, 100 or N players on each team?The last two puzzles harken back to a popular legend featuring Sun Bin. According to the story, General Tian Ji and King Wei were planning a friendly series of horse races, and the general asked Master Sun, his trusted retainer, for advice. The wily strategist told Tian Ji to race his best horse against the king’s second best, his second best against the king’s third best and his worst against the king’s best. I call this Master Sun’s strategy. “Just bet heavily, my lord, and I can make you the winner,” Sun Bin said. And indeed, according to the legend, Tian Ji won two out of three races and took home 1,000 gold coins.
That is how the legend goes, but any modern mathematician would be skeptical. Certainly Sun Bin could not guarantee a win. But did the strategy improve Tian Ji’s chances of winning the match? And if so, by how much did it do so? Those questions lead us straight to Puzzles Three and Four.
Before I reveal the answers, we must go back and consider the first two puzzles.
.
.
.
Click the link below for the complete article:
.
__________________________________________
Leave a Reply