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Gautam Soundararajan
Math 7
Professor Winkler
5/2/13
Drawing a Picture
Puzzles are created in order to perplex the solver. Good ones are not easy, but they do
not usually require advanced knowledge in a specific field. In the case of mathematical puzzles,
understanding how to solve line integrals or determine eigenvectors is of little value. The only
benefit that may be gained from knowledge of these topics is that one would have strong analysis
skills. But that does not necessarily have to be true. At first glance, puzzles may seem simple.
That is rarely the case. There are intricacies that make solving puzzles more complicated. For
instance, in one puzzle we have to determine the shortest time it takes for four people to cross a
bridge. Each person takes a different period of time to cross. Given this information, the puzzle
is relatively simple. But additional conditions make the puzzle more difficult to solve. There is
the condition that only two people can cross the bridge at a time. This adds another level of
complexity to the problem. These small pieces of information turn simple puzzles into diabolical
challenges.
However, there are various strategies that can be taken in order to aid in the problem
solving process. These different methods include trying smaller numbers, examining
assumptions, exploiting symmetries, and counting possibilities. Each strategy can only be
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applied to certain questions. In many cases, multiple strategies can be applied to a puzzle. One
difficulty is determining which strategy is the most useful for solving the puzzle at hand. Simply
knowing and using a strategy, however, does not automatically produce the solution to a puzzle.
A strategy helps guide a solver to the solution.
Drawing a picture is a strategy that proves particularly useful not only for mathematical
puzzles, but also for a wide range of other things. When we want to convey significant data, we
draw pictures, which can manifest themselves in the form of graphs. We could explain the
change in prices over time by stating the prices in each year over a certain span of years, but
doing so is far too complicated. It is hard for listeners to remember what number was just said.
Instead, we use a graphical representation to convey the information more clearly. In this way,
the trend in prices can be seen clearly and takes the task of remembering numbers out of the
listeners head.
The technique of drawing a picture is of large benefit in the world of mathematical
puzzles as well. Pictures can be used in the explanation of the puzzle. It can make the task of
explaining the puzzle much easier. The hackneyed saying “a picture is worth a thousand words”
is very true. Take the following puzzle as an example.
The digits from 1 to 8 are to be placed in the eight circles shown in the figure below, with
this proviso: no two digits directly adjacent to each other in serial order may go in
circles that are directly connected by a line. For example, if 5 is placed in the top circle,
neither 4 nor 6 may be placed in any of the three circles that form a horizontal row
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beneath it, because each of these circles is joined directly to the top circle by a straight
line.
The figure above is vital to solving the puzzle. Explaining the positions of all the circles and
connecting lines could be done using an extensive number of words, but it would be complicated
and the reader may lose interest and move on.
In some cases, a picture will not be given as part of the puzzle. But drawing one can be
helpful in determining the solution to the puzzle. This involves transforming the words in the
puzzle into a picture in order to visualize the problem. Being able to visualize something
provides a more concrete tool than imagining abstract concepts. It is not always clear when it
will be useful to draw pictures. One way is to look for key words that indicate figures would
help. An example will help clarify the benefit of forming pictures. One puzzle asks:
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You have six identical toothpicks. How many topologically distinct shapes can you form
by arranging the six toothpicks, without crossing one toothpick over another and joining
the toothpicks at the ends only?
The puzzle is almost blatantly advising the solver to draw a picture. The hint comes from the
word “shapes” in the question. We automatically think to draw different pictures in order to
determine the solution. Specific pictures are drawn based on the conditions given in the
question. Solvers will immediately start drawing all distinct shapes that they can come up with.
The following are some shapes that he/she may determine.
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After a short while, the solver will probably run dry of ideas. There are still more shapes to be
drawn, but the solver does not know how. This brings us to another technique: searching for the
solution systematically. Drawing different shapes in an organized manner ensures that no figures
have been glossed over. This concept highlights an interesting point. Combining multiple
problem-solving techniques can be very useful when trying to solve a puzzle. Using this
technique, there is a higher chance of determining all of the possible shapes. In the case of this
specific puzzle, a systematic approach could be to create different shapes based on the number of
sticks used to make a loop or the number of loops in the shape. The full list of distinct shapes
that can be created using six toothpicks is 19, as shown below.
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There is a great benefit in using pictures to solve mathematical puzzles. They make it
easier to visualize the situation in the problem and aid in the solving process. There is no
standard procedure for drawing pictures. Each problem can benefit from a different figure.
Some problems would be easier to solve without one since the task of drawing a picture is too
complicated. In the end, it comes down to the contents of the specific puzzle. No two puzzles
are alike so each must be tackled differently.