On a recent American Airlines flight, I was looking for a way to pass the time. Knowing that the inflight magazine, AmericanWay, has a Mensa Quiz, I opened to the page and perused the problems. Usually, I can solve the math problems pretty easily, but some of the others take some thought. I looked over the quiz, and saw that four of the ten questions were easy, but I had no idea how to solve the others. One of the questions included a hint: Work Backward. I thought this kind of strange, since I would never consider approaching a problem this way. Usually, when I do this quiz, I can get eight or nine correct, so this particular one seemed harder than most. So I put the magazine away and didn’t try to solve any of them. After I got off the plane, I didn’t think any more about the quiz.
A few days later, I was reading my local Long Island newspaper, Newsday, and came across an opinion piece by a guy named Alfred S. Posamentier. The title of the piece was, “Put drilling out and thinking in.” In the article, Posamentier writes that math students don’t learn enough about ways to approach problems, and if they did it might simplify their work. He also included a couple of math problems in the article. One of them went like this: We have two one-liter bottles. One contains a half-liter of red wine and the other a half-liter of white wine. We take a tablespoonful of red wine and pour it into the white wine. Then, we take a tablespoon of the new mixture (white wine and red wine) and pour it into the bottle of red wine. Is there more red wine in the white wine bottle or more white wine in the red wine bottle? The answer to this did not seem obvious to me, so I read on. The author easily solved this problem through a technique called “extreme situations.”
Later on in the article, he posed another problem: How many games must be played in a single-elimination basketball tournament with 25 teams competing to get a winner? This seemed easy enough, but the author describd a very fast way to solve the problem by using a technique called “considering another point of view.” You can read this article by CLICKING HERE. (Newsday refers to this article as premium content, so you'll have to register. Hint: Enter your information even if you don't get Newsday, and they'll let you read the article.)
Being the problem-solving type of guy that I am, I was naturally intrigued by this article. The author’s bio mentioned a book called “Problem-Solving Strategies for Efficient and Elegant Solutions,” so I looked it up on Amazon (www.amazon.com). I found the book, but it was fairly expensive and seemed to be written for teachers of high school mathematics. Instead of purchasing it, which I originally intended to do, I looked through the information Amazon provides for the book. This led me to a few more problems as well as the Table of Contents. Here the author lists 10 chapters on problem solving strategies with the following headings:
• Working Backwards
• Finding a Pattern
• Adopting a Different Point of View
• Solving a Simpler Analogous Problem
• Considering Extreme Cases
• Making a Drawing (Visual Representation)
• Intelligent Guessing and Testing (Including Approximation)
• Accounting for All Possibilities
• Organizing Data
• Logical Reasoning
The first one, Working Backwards, made me think back to the Mensa Quiz mentioned earlier. This led me to look up AmericanWay on the Internet and re-visit the quiz. You can check it out by CLICKING HERE. Now that I had it in front of me, I knew I would not be able to rest until I figured out all of them (or gave up). As mentioned, problems 5, 7, 8 and 10 were easy enough, even though I missed one of the three answers for problem 7. At least now I understand why the hint was given in problem 8—apparently working backwards is a legitimate problem solving technique. Problems 1, 2, 3, 4, 6 and 9 all seemed insolvable at first glance. So I picked the one I thought would be the easiest of these, problem 6, which is given below.
Problem 6:
A simple substitution code has been used to conceal a “quote.” Work out the code to decipher the original words.
NZ GBWPSJUF QMBDF JO UIF
XIPMF XPSME JT UBIJUJ
EFONBSL JT UPP DPME.
I used my usual technique of checking the frequency of the letters and matching them up to letters frequently used in English. That didn’t help. I tried figuring out words by how they were used within the context of the quote. That didn’t help either. By now, I was in full “got to solve this problem” mode. I entered the quote into my favorite software program, Excel, just to make it easier to do letter substitutions. In other words, substitute E for J and all the Es change to Js. This didn’t help either, although it helped me see which letters could not possibly be substituted for others. As I spent more and more time trying to solve this one, I would occasionally take breaks and try to solve the others. This exercise actually helped me to solve all the other problems that I had thought were so difficult. I found this to be very interesting. So let me add a another problem solving technique to the ten listed above: When trying to solve multiple tough problems at once, tackle the easiest one of all and, while trying to solve it, take a break every so often and go back and try to solve the others. If you still can’t solve any of these others, return to the problem at hand. Repeat the process until you solve the problem at hand.
After spending way more time than I should have on problem 6, I started wondering how I would solve it, if I really had to. (Of course, looking at the solution given in the magazine was not one of the options.) So I searched on Google. I found a few pages of interest. One is titled: The Creation & Solution of Simple Substitution Ciphers. It can be found at www.codasaurus.com/Cipher-1.htm. I had already used most of these techniques, so it wasn’t much help, other than to reaffirm that I was attacking the problem correctly. Another is titled: Solving CryptoQuotes with PVM (Parallel Virtual Machine) by by Ko-Ming Chang. You can find this at: http://home.wlu.edu/~whaleyt/classes/parallel/projects/pvm/crypto/proj3_intro.html. Another is a Java applet written by David Eppstein, which I decided not to use, since it would be akin to looking up the answer in the magazine. After I solved the quote myself, which I eventually did, I checked the applet and found that it did a reasonably good job, though not perfect. The applet is at www.ics.uci.edu/~eppstein/cryptogram/.
Using my own technique of solving problems in between solving the quote, I was able to correctly solve 8-2/3 out of 10. The only one that stumped me was problem 4—I did not quite understand what it wanted me to do. I now have another problem solving technique to offer: Give yourself enough time. Apparently if you do this, it’s possible to solve 80-90% of these Mensa Quiz problems.
)
