Super Strategies For Puzzles and Games by Saul X. Levmore and Elizabeth Early Cook Doubleday & Company, Inc, Garden City, New York 1981 ISBN 0-385-17165-X -------------------------------------------------- On the page following the title page, which is the copyright page, we have: Our thanks go to the following authors and publishers for allowing us to use examples from their challenging books. Russell L. Ackoff, The Art of Problem Solving, copyright 1978 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc. E. R. Emmet, Brain Puzzler's Delight copyright 1967, 1970 by E. R. Emmet. Reprinted by permission of Emerson Books, Inc. L. H. Longley-Cook, More Puzzle Fun, copyright 1977 by L. H. Longley-Cook. Reprinted by permission of Fawcett Books. H. E. Dudeney's: books, particularly 536 Puzzles and Curious Problems, published by Charles Scribner's Sons, have also been a source of inspiration. ISBN: 0-385017169-X Library of Congress Catalog Number 81-43275 Copyright 1981 by Saul Levmore All Rights Reserved Printed in the United States of America -------------------------------------------------- << The first two paragraphs of chapter 1, begins on page 3. They are as follows: >> It is now time to begin making you an expert at seeing through and solving puzzles - and enjoying yourself at the same time. Chapter 1 is full of puzzles of all kind and all difficulties. To help you measure your progress, each puzzle is numbered in terms of how hard it is to solve. A "1" is fairly easy and and "5" is very difficult. Be forewarned that these problems are very challenging, so any step in the direction of the correct solution deserves some applause. Now dig in! When someone mentions expansion, what comes to your mind? A river rising to fill its bed? The aftermath of a good meal? Expansion can be unlimited, but it can also take place within certain boundaries or in only one direction at a time. When the idea of expansion is applied to puzzles, it means that the components of the puzzle are exaggerated in some way. It is a technique that helps to solve some puzzles because the enlarged factors highlight the essence of the problem's real identity. Here's an example of how expanding a puzzle can help you see a solution. The Bridge Problem: Difficulty: 3(without paper); 2(with paper and pencil) Four men must cross a bridge over a deep ravine in enemy territory in the middle of the night. The treacherous bridge will hold only two men at once and it is necessary to carry a lantern while crossing. One of the men takes 5 minutes for the trip accross. one takes 10, a third man requires 20, and the last needs 25 minutes. Unfortunately, they have only one lantern among them. How can they make it across if they have only 60 minutes before the bridge is blown up? This is a good problem with a good answer, and to keep you on the right track, here is a list of several possible answers that, although clever, are not correct. The men do not throw the lantern back and forth across the bridge; this would crate a new problem in which the total crossing time would be 35 minutes. Perhaps you should assume that the bridge has a surface that is buckling, because the men also do not roll the lantern across. Nor does one large man juggle the smaller ones across the bridge (although juggling is a cute answer to the problem of the ninety pound man who must, in one crossing carry two ten pound weights across a bridge with a capacity of one hundred pounds)! Think about the Bridge Problem and work on it for a while. Even if you think you have found the answer, do not skip over the explanation that follows (or any others in this chapter), because understanding this answer will help you work out some of the other problems coming up. (The blank spaces throughout the book are meant to keep you from looking at the answer immediately after having read the problem. You might use these spaces for you own work although many of these problems are more enjoyable if tackled only in the mind.) << The page blank to the bottom >> The most obvious way to attack this problem is to begin with trial and error attempts. It is possible to hit on the right solution this way, but it makes for slow work and it's just as likely that you will continue to come up with combinations that require more than sixty minutes. Now if you try expanding the problem you may see the crucial "trick" upon which the solution depends. Think of the same problem, but imagine now that the third man requires 1,000 minutes to cross and the fourth man needs 1,005. (Since the men are still racing against the clock, they must cross the bride in 1040 minutes to stay alive.) Try to find a solution to this version of the problem. Does this change make you see something that you overlooked before? It should be clear that the two slow travelers must cross together because it would be an extravagant (and deadly) waste of time to make both the 5 and 10 minute travelers dawdle across the bridge with a lethargic compatriot. Bringing this insight back to the original problem should result in a solution quickly reached. Since the two slower men should cross the bridge together (and only once), the two quicker men must start by crossing together. This will absorb 10 minutes because they must adopt the pace of the slower of the two so that the lantern serves both. Since the l lantern must return to the departure side of the ravine for the two slow men to use, one of the quicker men brings it back by himself. It can be either one, but assume the quickest man makes the trip, and another 5 minutes have passed. The two slow men leave immediately with the lantern, and their crossing last 25 minutes. The 10 minutes man who was left alone on the shore grabs the lantern and crosses the bridge (in 10 minutes) to pick up his buddy. They travel again across the ravine in 10 minutes, and the four men are united on the far shore just in time to hear the explosion and watch the bridge crumple and crumble. With the solution just reached it would seem appropriate for you to ask how you're supposed to know which elements in a puzzle to expand. In this bridge problem, you might have tried expanding the time the men had to cross the bridge, but that wouldn't show you a new approach to the problem, it would just simplify it. If you were curious during your first reading of the puzzle, you might have wondered why the men's crossing times were 5, 10, 20, and 25 instead of 5, 10, 15 and 20, and so on. The gap is important but can easily be overlooked as it's so small. Expanding the problem by increasing the two slower traveling times then highlights the time gap. With some experience and insight, the potential solver should realize that the four men are really divided into two groups, fast and slow, and that they must cross the bridge accordingly to avoid wasting time. The point of applying the technique of expansion to a puzzles to acquire a new perspective on the problem. Sometimes this technique, or any other like it, will give direct results and yield a solution. Often its virtue lies in merely jogging the mind away from a chain of thought that has been unsuccessful. Since a puzzle frequently will not relinquish its secret to the simple persistent effort, any approach that encourages the mind to bend a little and entertain new options for solutions must be valuable. << The next problem is listed. It happens to be the 9 dots in a 3x3 square that you are suppose to mark through with four straight lines >> << There are no diagrams for this puzzle. The chapter does not end with any notes, and the book does not have an index or bibliography. >> -------------------------------------------------- I have seen the bridge problem other places. The following book sets up a similar problem. Perhaps you will consider it an earlier version. Regards, Conrad Schlundt Dallas, Texas On page 3: The Bridge Problem: Difficulty: 3(without paper); 2(with paper and pencil) Four men must cross a bridge over a deep ravine in enemy territory in the middle of the night. The treacherous bridge will hold only two men at once and it is necessary to carry a lantern while crossing. One of the men takes 5 minutes for the trip accross. one takes 10, a third man requires 20, and the last needs 25 minutes. Unfortunately, they have only one lantern among them. How can they make it across if they have only 60 minutes before the bridge is blown up?