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.

And a Few More Puzzles

 If the prior puzzle was a bit too much for you, a few below that are more manageable...

Been reading "Mathematical Curiosities," new from Alfred Posamentier and Ingmar Lehmann. It is, as the subtitle suggests, "a treasure trove of unexpected entertainments" -- especially entertaining if you have a geometry bias.

In the middle of it come 90 "curious problems with curious solutions." Several of these are classics with which you'll be familiar, and others are a little fresher, all interesting. I'll pass along three to whet your appetite (these are paraphrased from the volume):


#1.  I feel like EVERYone should know this first one, so just passing it along for any readers not already familiar with it:
On a certain pond the water lilies double in number every single day. After the 50th day the pond is completely covered. How many days were required for the pond to be half-covered?

#2. Given the following four numbers:

8,932
12,668
85,423
9,165

What percentage of their sum, is their average?

#3. What time is it now if in 2 hours it will be one-half as long 'til noontime as in 1 hour from now?

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1)  49 days

2)  25%  (if you work this out the 'long' way, you may then see there's an easier, more general solution)
3)   9 am.

Geometer/Author Extraordinaire...

Who doesn't love geometry!!? (well, I know there are some of you, but I'm not sure I trust you ;-) ...I recently mentioned geometrician Alfred Posamentier's "The Secrets of Triangles" on my list of 10 books to consider during the Holidays for your math-enthusiast friends, and I was delighted to wake this morning and find that Sol Lederman's latest 'Inspired by Math' podcast is with Dr. Posamentier. They not only discuss his most recent book, but geometry and math education more generally, as well as some history; and also good to hear he has yet another new book on the way (on mathematical mistakes)!
This is a great interview with a great educator. It's 50 minutes, so you'll need to set aside some time (or just play it in the background while doing other Webby things), but ought not be missed:

http://wildaboutmath.com/2012/12/16/alfred-posamentier-inspired-by-math-13/

Thanks for another inspiring podcast Sol!

There's Something About Triangles

What is it about triangles that make them and their geometry endlessly fascinating?
...I don't know, but I can't think of anyone better to elucidate the topic than Alfred Posamentier. He and co-author Ingmar Lehmann have yet another new book out, a delight for geometry fans, "The Secrets of Triangles," devoted entirely to these simple geometric figures. I think Euclid would be pleased! ;-)

From the Amazon-listed reviews: "an irresistable exploration," "enticing and addictive," "wonders of geometry emerge," "reveals a compelling and natural aesthetic beauty."

"Making Math Fun"

Alfred Posamentier is one of my favorite math writers, but I'd never actually seen him... until now, that is (courtesy of YouTube):



...and here's an Amazon page for his books:

http://tinyurl.com/3d8ef9c