Physical Representation
The first method of problem solving is to totally emerge yourself in the problem
where you solve it by taking the actions of the main character. If more characters are
involved, then you solicit the help of your friends, family or classmates to reenact the
problem. By using this form to spatially organize the information, you are creating a
physical representation of the problem. This physical representation allows you to gain
insight into the process of problem-solving. This makes the problem easier to understand
for some people. The various restrictions on identities or movements within the problem
are effortlessly imposed through the acting. Plus, when others are involved, you have
additional minds to help work out the problem as well as keep the restrictions in the
forefront.
Example 1: Handshake
A group of six people meet at a party. If each person shakes hands with each of
the other people at the party exactly one time, how many handshakes take place?
Get into groups of six people and act it out. Keep track of the handshakes and the
solution will present itself.
Example 2: River Crossing
Three adults and three children need to cross a river. There is only one small boat
available to use. The boat is only large enough to accommodate one adult or two children
at one time. How will the adults and children cross the river? How many times does the
boat cross the river?
Use your group of six people to reenact this situation. Use something to represent
the boat and remember the boat can not cross by itself.
Many times involving others is not practical. You may need more people then
there are present or others are too busy to act out the situation. In place of people, you
may use objects to represent the characters. Although the input of others is sacrificed, the
objects can still be manipulated to solve the problem. This may require some ingenuity to
create appropriate objects; but, many are close at hand. Coins, blocks, or paper depending
on the needs of the problem can make clever substitutes. Either way, movement, actions,
or placement can be quickly considered and incorporated into the solving process without
the need for volumes of paper.
Example 3: Post Office
You have gone to the post office to buy four stamps. The postal worker tears the
four stamps from a sheet so that they are still attached to one another. How many
different ways could the four stamps be attached?
Use four objects to represent the stamps. Move the objects around to find as many
combinations as possible. Keep in mind that stamps are the same size and the orientation
of their different sides (top, bottom, left, and right) are distinguishable so choose your
representative objects accordingly.
Example 4: Puzzles
Arrange the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the boxes of the puzzle so that
the statements are true and each number is used only once.
x
4
+
=
x
-4 =
2
Write each of the digits on a separate piece of paper. Now, move the pieces
around until all the statements are true.
Exercises: Act these out.
1.
Friends: Five friends were sitting together on a long bench. Will sat next to Angie.
Jason sat next to Rodney. Kia was the third person away from Angie. Will was the third
person away from Jason. Who sat on the other side of Rodney?
2.
Boy Farmer: A boy with his dog tagging along went into town to purchase some
corn. He also found that store had a sale on hens so bought one to replace a missing one
on the farm. On taking his favorite shortcut home, he forgot he would have to cross over
a tall link fence followed by a ravine. This meant he would have to climb the fence and
cross the ravine with the corn, the hen and his dog each individually. He also knew that if
left alone the hen would eat the corn and the dog would attack the hen. How could he get
across the fence and ravine with everything safely?
3.
Guard Duty: Two daytime guards, David and Derek, are on duty. When the two
evening guards, Ed and Evan, appear to replace them, the four guards begin their change
performance. One guard takes two steps, turns 90, takes one step, and stops when he has
finished the L-shaped maneuver. (Or he takes one step, turns 90, and then takes two
steps.) Then, the next guard does the same and so forth until the four have completed a
set where each has moved. The diagram below represents the initial positions of the four
guards with the blocks representing the steps apart. The guards will not go outside of this
configuration during the maneuvers nor will they occupy the same block at the same time
as another guard. How many sets of maneuvers must the guards repeat for the two
daytime guards to have exchanged positions with the two evening guards?
David
Ed
Derek
Evan
Use manipulatives for the following exercises.
4.
Arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the following triangle so that
the sum of each side equals 23.
5.
Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 in the following boxes so that no two
consecutive numbers are in touching squares. (Touching includes squares which share a
side or simply one point.)
6.
A pentamino is a figure made up of five connected squares. The squares are
connected by sharing a complete side. Note the diagrams below.
a)
How many different pentaminoes can be made?
b)
Join four (different) pentaminoes together to form the capital letter T?
Now try to form the capital letters U, P, W or Z.
7.
Magic Triangle:
a)
Arrange the numbers 1, 2, 3, 4, 5, and 6 in the above triangle so that the
sum of each side equals 11.
b)
Arrange the numbers 1, 2, 3, 4, 5, and 6 in the above triangle so that the
sum of each side equals 9.
8.
Magic Square: Arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the following
square so that the sum of each side and the sum of each diagonal equals 15.
9.
Magic Square II: Rearrange the numbers is this 4 x 4 array so that the sum of each
row, each column, and each diagonal equals 10.
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4