In 1654 two brilliant mathematicians, Blaise Pascal and Pierre de Fermat, conducted a correspondence in which a variety of seemingly intractable puzzles were presented and solutions proposed. Some of the problems posed in their letters (Fermat's last theorem among them) were only recently solved, after hundreds of years and thousands of subtle solutions worked over by the best and brightest. Some of the problems posed (like "the unfinished game") appeared recalcitrant at the time but can now be solved in minutes by my fourth graders after a two week unit on probability. This leads one to wonder how two puzzles which today seem diametrically opposed in difficulty (one can be solved by only a handful of people in the world and the other can be solved by more than a handful of my students in a simple elementary school) appeared equally mystifying then."The Unfinished Game" puzzle can be summarized thusly: Suppose two gamblers decided to wager some money on a dice game. Each put into the pot $50 and decided the game would consist of five rounds in which each player would roll one die. The player with the higher roll would win each round and the player with the most wins out of the five rounds would take the pot of $100. Now suppose, due to some emergency, the game was forced to come to halt after player A had won 2 rounds, and player B had won 1 round. The question Pascal posed to Fermat was: How should the pot be divvied up at this point in the unfinished game?
Here's how Keith Devlin, author of the highly recommended book pictured above, explains the dilemma: "If the game were tied, there wouldn't be a problem. They could simply split the pot in half. But in the case being examined, the game is not tied. To be fair, they need to divide the pot to reflect the two-to-one advantage that one player has over the other. They somehow have to figure out what would most likely have happened had the game been allowed to continue. In other words, they have to be able to look into the future -- or in this case, a hypothetical future that never came to pass."
Considering that before the mid-seventeenth century scholars generally agreed that it was impossible to predict something by calculating mathematical outcomes, it appeared finding a solution based on "hypothetical future" outcomes was impossible. Fermat thought this puzzle was easily solved: in the case of the best-of-five dice game that is stopped after the third round with one player in the lead by two to one, there are four possible ways the game can be completed (B wins round 4 and A wins round 5, or A wins round 4 and B wins round 5, or A wins rounds 4 and 5, or B wins rounds 4 and 5). Of those four, three are won by player A after the third round. So the two players should split the pot with 3/4 ($75) going to the player A and 1/4 ($25) going to player B. Simple, yes? Well Pascal (that is Pascal of the brilliant Pascal's Triangle) couldn't understand this solution. He proposed a much more detailed analysis (pages long and still inconclusive) and then asked Fermat to explain his solution again (and again) and still didn't comprehend it. How is it that this certified genius couldn't understand such a simple and elegant solution?
Pascal's incomprehension appears to hinge on the wide spread belief (of his contemporaries) that humans could simply not speculate on the future. The future, along with any predictions, belonged in God's realm and no amount of human ingenuity could scale to the heights of such wisdom. But here we live in a time when speculation on the future, based on mathematical models, is seminal to a functioning democracy and economy: insurance tables, weather forecasting, election polling, software design, drug design and testing.
It is ludicrous today to contend that the future cannot be predicted, that probability and statistical analysis are insufficient to build models of future event outcomes. Or should I say it "was" ludicrous to argue against probability, for look at our economy now limping sickly between bouts of influenza and rickets. How best to split the pot of an unfinished game was solved over one hundred years ago, so why are we still muddling our way through a bailout package the sharpest minds of our time have yet to predict the likely success of?
Here's how Keith Devlin, author of the highly recommended book pictured above, explains the dilemma: "If the game were tied, there wouldn't be a problem. They could simply split the pot in half. But in the case being examined, the game is not tied. To be fair, they need to divide the pot to reflect the two-to-one advantage that one player has over the other. They somehow have to figure out what would most likely have happened had the game been allowed to continue. In other words, they have to be able to look into the future -- or in this case, a hypothetical future that never came to pass."
Considering that before the mid-seventeenth century scholars generally agreed that it was impossible to predict something by calculating mathematical outcomes, it appeared finding a solution based on "hypothetical future" outcomes was impossible. Fermat thought this puzzle was easily solved: in the case of the best-of-five dice game that is stopped after the third round with one player in the lead by two to one, there are four possible ways the game can be completed (B wins round 4 and A wins round 5, or A wins round 4 and B wins round 5, or A wins rounds 4 and 5, or B wins rounds 4 and 5). Of those four, three are won by player A after the third round. So the two players should split the pot with 3/4 ($75) going to the player A and 1/4 ($25) going to player B. Simple, yes? Well Pascal (that is Pascal of the brilliant Pascal's Triangle) couldn't understand this solution. He proposed a much more detailed analysis (pages long and still inconclusive) and then asked Fermat to explain his solution again (and again) and still didn't comprehend it. How is it that this certified genius couldn't understand such a simple and elegant solution?
Pascal's incomprehension appears to hinge on the wide spread belief (of his contemporaries) that humans could simply not speculate on the future. The future, along with any predictions, belonged in God's realm and no amount of human ingenuity could scale to the heights of such wisdom. But here we live in a time when speculation on the future, based on mathematical models, is seminal to a functioning democracy and economy: insurance tables, weather forecasting, election polling, software design, drug design and testing.
It is ludicrous today to contend that the future cannot be predicted, that probability and statistical analysis are insufficient to build models of future event outcomes. Or should I say it "was" ludicrous to argue against probability, for look at our economy now limping sickly between bouts of influenza and rickets. How best to split the pot of an unfinished game was solved over one hundred years ago, so why are we still muddling our way through a bailout package the sharpest minds of our time have yet to predict the likely success of?
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