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Wednesday, May 6, 2015

An "Exemplary" Solution


Another quick problem today adapted from Alfred Posamentier's simple volume, "Problem-solving Strategies in Mathematics," that I'm enjoying (...the volume is very similar to an older Posamentier/Krulik work you might already have). This is actually a classic puzzle most readers are likely familiar with. The key is not so much getting the answer... but HOW you get the answer:

25 basketball teams compete in a single-elimination tournament (as soon as you lose a game you're out of the tournament, and with 25 teams one team will get a bye in the first round).
How many games total will be played by the end in order to crown the champion?
Answer down below... but first a li'l more about the book.

This is a great volume for middle and high school teachers to have on-hand to draw problems from, and many of the problems could be suitably presented to younger math-inclined pupils as well (I'm talking to you Mike Lawler! ;-))

The approach of the book is to offer a problem, and then give a somewhat brute-force (or what they call "common") strategy for solving it, followed by a simpler, quicker, more elegant (or what the authors call "exemplary") solution. Many of the problems are classics, or offshoots of classics, generally requiring nothing more advanced than logical, arithmetic, algebraic, or geometric solutions.

The authors illustrate a total of 10 "strategies" for solving math problems, starting off with "logical reasoning," "pattern recognition," and "working backwards." Students will often experience facepalms when they see the "exemplary" solutions given. The book is supposed to be the first of a series (edited by Posamentier) on mathematical problem-solving from World Scientific Publishers.
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Answer:   24  ...the answer can be tediously found using arithmetic, but all you need realize for the "exemplary" solution is that to arrive at 1 champion in a single-elimination tournament of 25 teams means there MUST be 24 losers... and at one loser per game, there must be 24 games played by the end; no pencil and paper required.


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