Ruppert 1
Nick Ruppert
Prof. Winkler
Math 7
5/9/13
See It to Solve It
The human tendency to believe in something or understand how something works
often comes in the form of visualization. One can see how something works by simply
drawing a picture of what is happening. A drawing, diagram, or map can clearly show an
individual how or why something works the way it does. Drawings don’t need long
explanation, as the reasoning appears cut and dry on paper. When solving puzzles, if the
solver has the opportunity to draw the puzzle, then the solver ought to do it because
visually finding the solution to a puzzle may allow the solution to be found more
efficiently and accurately.
Some puzzles essentially require that the solver use a drawing. In one instance,
there are seven cities. Someone makes the claim that for any 3 cities, at least 2 of them
will be 10 miles apart. The solver must prove or disprove this claim. On first thought, the
problem seems quite easy. In the brain, this scenario can make perfect sense. When
proving it, the problem becomes another beast.
The solver can begin by making plenty of scratch graphs. As most solvers have
experienced, these scratch graphs won’t work. The solver must start small and work their
way up. Plenty of solvers may find that solving this problem with 6 cities is easy. A
pentagon of cities with one city in the middle, 10 miles from each other city, is an easy
solution. Seven cities, as any solver may find out, won’t work like a pentagon. A hexagon
around the middle city wouldn’t prove the problem, as not any set of three cities has at
Ruppert 2
least three cities 10 miles from each other. In some form or another a solver may come
down to these sets of drawings:
As one can see, the solver has a solution for every set of cities except for the two
on the end. If the solver is clever enough, they may figure the possibility of taking their
first drawing, and squeezing the two end cities together, until they’re 10 miles apart. With
those two cities 10 miles apart, the solver has officially proven the claim.
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This problem isn’t easy, but visualization, and some clever thinking, can help the
solver understand how the puzzle works. This can be useful for many puzzle solvers, as
many solutions seem hard to believe until seen.
In Gardner’s problem 6.4, geometry presents itself again in the form of a 6X6X6
cube. The puzzle asks if that cube can be formed, using only 27 4X2X1 bricks:
Since part of the brick is only one unit long, the solver can infer that if multiple
4X2 shapes can fit an entire side of the cube, the cube can possibly be comprised of 27
4X2X1 bricks. When taking the 4X2 side and filling in the 6X6 side, the solver can
quickly see that this won’t fit. Therefore, the rest of the cube can’t be filled with the
bricks.
Geometric puzzles almost always require a visual argument to prove whether or
not the puzzle is solved. A good rule when solving puzzles is that if drawing the problem
out can be a possibility, then do it. Drawings can make the solver’s understanding of the
problem better, and assist them in presenting a clear solution.
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In the case of The Seven Bridges of Konigsberg, the solver is presented with a
map of the city of Kongisberg with its seven bridges connecting the land:
The puzzle asks the solver to find a route on foot where he or she can cross each
bridge once, and only once, as he or she moves about the city.. For some solvers, the map
suffices. To thoroughly prove the solution to this puzzle, a new drawing can show good
evidence. Since there are 4 land masses, a dot can represent each land mass. The seven
bridges can connect the dots through representation of lines, as such:
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The solver can try every which way possible to find a solution by tracing along
the lines and dots. After never finding a solution, the solver can confirm that what the
puzzle asks them to do is impossible. The non-existent solution has been verified, but
letting the solver see this through drawing can solidify their conclusion.
Combinatorial problems may also offer opportunities to use drawings. In the case
of Gardner’s problem 1.18, the puzzle presents the solver with a scenario where 5
married couples go to a party and shake hands. A man asks his wife and the four other
couples how many hands they shook. Everyone gives him a different answer. The solver
must figure out exactly how many hands the man’s wife shook. The solver can make a
drawing of the people involved in the event, and put them in the circle. Connections
between dots present the solver the chance to visualize who shook whose hand, and how
many times they shook hands, without having to go back and think about it again.
Through visual evidence and various solving techniques the solver can then determine
that his wife shook exactly four hands, just as he did.
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Drawings may even apply to probability problems if they are helpful to the solver
in doing the math. In Gardner’s problem 2.13, a boy can divide ten $10 bills and ten $1
bills into 2 hats. He gets to choose once out of one hat, but he doesn’t know which hat
has the certain bills he put in it earlier. What’s the maximum probability the boy can
create to pull out a $10? The boy may use a drawing to help him figure out the math in
this situation as such:
The boy can draw probabilities next to the lines to assist him in doing the math.
When he figures out his maximum probability of drawing a $10, 14/19, he can use the
drawing to sort out the bills into the hats.
Drawings may be the most underrated technique in problem solving. People are
often too prideful to make use of drawings as they think their brains are sharp enough as
it is. This can be true for some people, but drawings always give the solver clear,
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straightforward evidence. One doesn’t even need to explain solutions in words. The
solution may just be sitting right in front of them, in the most concise way possible. Many
people must see something to believe it. Explanations may not be enough, but visual
proof is, every time.