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Ziyuan Liu

05/02/2013

MATH 07

My Favorite Puzzle-Solving Strategy: Drawing Pictures

Drawing pictures or diagrams is what I believe to be one of the best ways of

solving puzzles. The following methods and examples will show how drawing

pictures can benefit puzzle solving. I will talk about how visual interpretations, in

general, can be applied not only geometry-related puzzles, but games as well.

As a visual person, I think that drawing pictures is the best way to

comprehend a puzzle. It adds another dimension, which is sometimes not given to a

reader by the books. Visual interpretations work best for geometry-related puzzles;

however, there are many instances in which drawing diagrams of charts or trees can

be best for comprehension.

For example, a geometry-related puzzle, such as the seven cities of gold,

would be easier to solve if it were realized on paper. Given that there are seven

cities of gold and for any three cities, any two are ten leagues apart, is there a

possible configuration that satisfies the conditions? For this specific problem, based

on the second condition, we can be sure that the smaller bits to this puzzle are

composed of three points and three lines where a equilateral triangle is needed. If

we took two of these triangles and combined them with a common line, we will

come across a general rhombus shape. At the same time one can also visualize a

pentagonal shape with edges as the cities. With the pentagonal shape (Figure 1), we

have used five of the seven cities.

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Figure 1. Five points making a pentagonal shape.

However with a pentagonal shape, dividing the innards into equilateral

triangles is impossible (one can do it with a hexagon, but not a pentagon). By this

point, one should realize that equilateral triangles are needed in order to complete

this puzzle. In addition, when drawing different possible solutions, we realize that

the triangles must be drawn inside the pentagon, not outside. For a point inside the

pentagonal bounds can be used to create two separate triangles sharing an edge.

With the sixth point, we can connect three lines from it to three edges and do the

same with the seventh point. As we can see from the answer (Figure 2), the difficulty

when thinking about this puzzle is the part where the two rhombi have to overlap.

Figure 2. Each gold bar/city is ten leagues apart from two other cities of gold.

 

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It is obvious that the seven cities of gold puzzle needed to be drawn out;

however, it is clear that just trying to solve the puzzle through various random tries

is not efficient. We had to follow the conditions of the puzzle in order to generate a

simple shape that allows to progress from one step to the next. Some non-geometry

puzzles or games can also be solved or won with a simple drawing or diagram.

Charts and diagrams can be best used for computable puzzles such as tic-tac-toe.

Figure 3. A tic-tac-toe tree drawn out for the next best move by looking ahead. 1

For games like Tic-Tac-Toe, a tree can be drawn out to examine moves that

would benefit either player the most. Each level of the tree displays the unmade

possibilities of the game. However, it should be noted that such a method is time

consuming to solve by hand and usually a computer is need to do so efficiently.

Nevertheless, for the purpose of generalization, looking at the game tree can

develop winning strategies. For example, even though we do not need a game tree to

generalize this, it becomes clear why it becomes advantageous for the first player to

1 Alpha Beta Tree. Digital image. N.p., n.d. Web. 1 May 2013. <http://www.ocf.berkeley.edu/~yosenl/extras/alphabeta/alphabeta.jpg>.

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place a piece in the middle of the board. The middle piece allows the possibility of

four different ways to win. Every other position on the Tic-Tac-Toe board will only

grant two or three ways of winning. However, counting all the possible outcomes of

Tic-Tac-Toe, one can come to the conclusion that the middle position and the edge

positions carry more weight than the remaining four other positions.

Perhaps the simplest example of exemplifying the power of a drawing is to

draw out a game tree for a game called Nim. Nim is stick-picking game with a

Chinese origin. Different versions of the game can be played with other objects such

as pebbles or matches. The basic game begins with a bowl of n objects where each of

the two players will take turns taking either one or two object(s), where n is greater

than two.

Figure 4. A game tree of Nim. 2

2 Digital image. Think Quest. N.p., n.d. Web. 8 May 2013. <http://library.thinkquest.org/06aug/01132/Gametreepicture.jpg>.

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The winning condition of the game varies. For our example, the player to take

the last piece wins the game. To keep the game interesting we will play a game of

seven objects or pieces. In Figure 4, the edges of tree include either “F” or “S,”

representing either the first player’s turn or the second player’s turn. The letters in

the edges are followed by a number representing the number of objects in the bowl

before the current player’s move. The letters “W” and “L” indicates the win or loss

for the second player.

Ignoring all previous intuition, we can see that in all of the cases where the

letter “L” appears in an edge the parent edge (the move before the Losing move)

was the first player’s move followed by either the numbers “1” or “2.” The winning

case is the exact opposite: the parent edge must contain the results of the second

player’s move, leaving either 1 or 2 pieces. Therefore, the root winning strategy for

player one is leave three pieces left for player two to choose from. We can see from

the game tree that it is a sure loss for either player if he is left with three pieces to

choose from. If the rule of this puzzle was changed to allow players to take from one

to three pieces, we can already visualize from the previous condition and its game

tree that the only strategy to obtain a sure win is to leave your opponent with four

pieces. We are now able to generalize that in a game where one is allowed to take

from one to m pieces, the winning is strategy is leave for your opponent with m+1

pieces. Generalizing further, it is largely in a player’s best interest to leave exactly x

pieces for his opponent, where n is a multiple of x – following the root strategy, x

must be m+1. So in this case, even if we assume that both players have perfect

knowledge of the game tree, the person to make the first move will always win.

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In conclusion, for geometry-related puzzles, drawing diagrams and

pictures is the most helpful method to finding the solution. For other puzzles such as

games, drawing a diagram can vastly help puzzle solver to come to a realization or a

pattern to a game. In cases such as Nim, with perfect knowledge, the first player to

move will win indefinitely. Obviously, the perfect player that has an absolute chance

of winning varies as the rules of the game.

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