Jeff Bass
Peter Winkler
Math 007
5/9/13
KISS
Although knowledge often appears to be the key to puzzle solving, knowledge can also be the greatest roadblock to finding a solution. Having certain knowledge can lead to overanalyzing problems, especially those that sound complicated but actually have extremely simple solutions. Sometimes I get so wrapped up in searching for the answer that I fail to see it right in front of me. Although I cannot speak for everyone, I believe that most people have experienced times when they have made a problem much more complicated than it needed to be. When I took the SAT for example, I had been learning calculus all year. This may sound like it would be a great help on the math section of the test, but it was exactly the opposite. Since there is no calculus on the SAT, my mind was prepared for a level of math that was unnecessary, and it actually took me a while to adjust to the lower difficulty. This is not to say that knowledge is a bad thing, but it is easy to overlook its blinding effects. Because of situations like this, the puzzle solving technique of “simplification” is a powerful one to know how to use.
It seems as though the more educated I get, the more accustomed I become to complexity, even to the point of inability to recognize simplicity. This inability to recognize simplicity is the core of many interesting puzzles and riddles. On the internet, I have seen several puzzles that start with a line such as: “97% of Dartmouth grads are unable to solve this puzzle, but almost all kindergarteners can answer it immediately,” in order to taunt the reader’s inability to think simply. (The sites usually use a different university, but I think Dartmouth illustrates the contrast better.) Whether or not the claim that kindergarteners can answer the problem easily holds any validity, it succeeds in making the reader feel silly for overthinking the puzzle when they hear the solution. One such puzzle looks like the one in Figure 1 below.
The puzzle at first appears rather difficult. If you approach the puzzle in the way that I did at first, you might start looking for mathematical relationships and patterns between the numbers that would result in the given output. For example, the answer might have something to do with multiplying some of the digits and subtracting others. Or maybe it has to do with how many times a digit is repeated? Eventually you might run into the solution by treating each digit from 0-9 as a variable and adding up the digits in each number, but this still makes the solution more complicated than necessary. The answer to this puzzle is to simply count how many closed loops are in each number. In the first one, for example, there are two closed loops in each 8, one closed loop in the 0, and one closed loop in the 9, for a total of 6 closed loops. I must admit that it took me a long time to figure out this problem, because even after the puzzle hinted to me to think simply, I continued to think that my knowledge of math would bring me to the answer. However, I was immediately able to see the solution when I stopped looking at the problem through a mathematical lens, and instead saw it through the eyes of a child who would only see the numbers as shapes.
In a book that I love, The Puzzler’s Dilemma by Derrick Niederman, there is an entire chapter devoted to this puzzle solving technique. One point that Niederman makes that I find very interesting is that puzzles can often have an easy way and a hard way to be solved. Although the more difficult way to solve the puzzle is not necessarily wrong, it would save a lot of time to think more simply about the puzzle and solve it the easy way. One example that he gives has to do with tennis matches. He asks:
“In the Wimbledon singles championships, the draw consists of 128 players. How many matches are required to produce a champion?” (Niederman 67)
One way to solve this puzzle, which is probably the intuitive method for most people, would be to work backwards. That is, the finals match is one match, the semifinals consists of two matches, the quarter-finals has four different matches, and so on all the way back to the first round with sixty-four matches. Adding up all these matches you would end up with 127 matches as your answer, which is correct. However, the much simpler way to solve this problem requires hardly any arithmetic. You merely have to realize that each match eliminates one person, and there is one person who never gets eliminated, the champion. Therefore, there must be 127 matches in total. This again demonstrates the power of breaking down a problem into common sense. However, it also shows that common sense can be surprisingly difficult to recognize. You may say that these simpler ways to solve puzzles are useless, since they almost seem more difficult to discover than the answers themselves. However, I believe that by studying these simplified solutions, the mind can be trained to find these solutions more intuitively. Simple thinking does not always come naturally to us, but learning how to work against the desire to think complexly can often work to our advantage. We must learn how problems can be simplified so that we can learn how to simplify future problems ourselves.
Learning how to simplify a problem can also take the form of completely changing the format of the puzzle. Niederman offers a brilliant example of this in his book. The puzzle says that there are three monsters, each holding a globe. One monster is short, one medium, and one tall. They are all initially holding small globes, but a globe can be medium or large as well. The goal is to get all the monsters to hold large globes by following these rules:
“1. Only one globe may be changed at a time,
2. If two globes are the same size, only the globe carried by the bigger monster may change.
3. A monster may not change the size of his globe to a size carried by a bigger monster.” (Niederman 108)
Although this puzzle sounds very complicated, it can be simplified into a puzzle that is much more familiar, intuitive, and simple. The monsters and globes problem is in fact the exact same problem as the Tower of Hanoi problem. In case you are not familiar with this puzzle, it consists of discs and poles organized like in figure 2.
The goal is to move the three discs onto the right pole by moving one disc at a time and without ever stacking a bigger disc on top of a smaller disc. Although this problem is identical to the puzzle with the monsters, it immediately seems easier to solve.
Another way in which knowledge can work against us is through bias. No one is born with bias. One must learn bias through experience. One specific type of bias is the idea that a puzzle must have a solution. On the contrary, the answer to a puzzle might be that there is no solution at all. Consider the following puzzle from Gardner’s book.
The puzzle is to cover the modified chessboard on the left with dominoes, which each cover two adjacent squares. Although you may spend hours trying different arrangements of dominoes, it would be much easier to show that the puzzle is impossible. Since the two removed squares are both white, there are now fewer white squares than black. Since each domino covers two different colors, it is impossible to cover the board with dominoes. I would argue that it is actually very simple to prove the impossibility of this puzzle. However, the idea that a puzzle must have a solution is so strongly rooted in our minds that we fail to see the simple solution.
My favorite example of the advantage of simple thinking comes from The Princess Bride. In one scene, Westley challenges the “genius” Vizzini to a “battle of the wits.” He places two cups of wine in front of them, one of which he says is poisoned. Each of the men is to drink from one of the cups, but Vizzini is allowed to pick which cup each man will drink from. Vizzini spends a long time trying to work out which cup has the poison, using such hilarious logic as:
“You've beaten my giant, which means you're exceptionally strong, so you could've put the poison in your own goblet, trusting on your strength to save you, so I can clearly not choose the wine in front of you! But, you've also bested my Spaniard, which means you must have studied, and in studying you must have learned that man is mortal, so you would have put the poison as far from yourself as possible, so I can clearly not choose the wine in front of me!”
Of course, all of his logic is completely useless, since there is no possible way to know which cup contains the poison. In the end, both cups were poisoned anyway, and Westley was immune to the poison. The puzzle that Vizzini was faced with is also manifested as the “two-envelope paradox,” although the two-envelope paradox gives the puzzle a more mathematical slant. In this version, you are a game show contestant where the host gives you a choice between two envelopes, one of which contains twice as much money as the other. After choosing one envelope, the host gives you the option to switch envelopes. Although many have proposed probabilistic solutions to why you should switch or not switch, it really doesn’t make any difference. Thinking about this problem on the simplest level, you don’t have any information to help you make your decision. Therefore, the puzzle is completely meaningless and it doesn’t matter which envelope you take.
From these examples, it becomes evident that thinking in terms of complicated mathematics is not always the best idea. Taking a step back and thinking in more simple terms, or simplifying the problem, can often times lead to the solution. While math knowledge is useful for many problems, if you feel like you have tried everything to no avail, see if simplicity can do the trick.