Chapter 9 Solve an easier related problem

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Chapter 9Solve an easier related problem
Some problems are just too complex and too
challenging. It is necessary to solve an easier
version first to gain experience or to test a
method. Some times, you may find out that it is
better (or necessary) to use a totally different
approach.
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We choose to go to the moon, not because it is
easy, but because it is hard.
JFK, 1962
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Painting a swimming pool is a pretty challenging
task, so it is better to solve an easier but
similar task first, such as painting the spa.
This will give you the necessary experience.
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Example 0 Jack and Jill are both in the same
line for the Soarin over California ride in
Disney World. Jack is in the 25th place and Jill
is in the 125th. How many people are between them?
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Example 1 How many squares (of all
possible sizes) are there in the following 8 8
checker board?
We need to do some easier and related problems.
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1. How many squares are in this baby checker
board?
Answer 4 1 5
2. How many squares are in this toddler checker
board?
How many squares of size 1 1
9
How many squares of size 2 2
4
How many squares of size 3 3
1
Therefore there are totally 1 4 9 squares.
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3. How many squares are in this junior checker
board?
How many squares of size 1 1
16
How many squares of size 2 2
9
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3. How many squares are in this junior checker
board?
How many squares of size 1 1
16
How many squares of size 2 2
9
How many squares of size 3 3
4
How many squares of size 4 4
1
Therefore there are totally 1 4 9 16
squares.
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Now return to the original question How many
squares (of all possible sizes) are there in the
following 8 8 checker board?
Answer 1 4 9 16 25 36 49 64
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Example 2 How many paths are there from corner A
to corner B if you have to stay on the black
lines, and you can only go up or right?
B
A
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Example 3 There are 4 different
containers, and 10 identical balls. If the
containers are all big enough to hold 10 balls,
how many ways can you put these 10 balls into
some or all of these containers?
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Solve some smaller problems first and then find a
pattern.
1 container 2 containers 3 containers 4 containers
1 ball
2 balls
3 balls
4 balls
5 balls
6 balls
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Solve an easier and related problem
Example 4. Following recess, the 1000
students of a school lined up and enter the
school as follows The 1st student opened
up all of the 1000 lockers in the school. The
2nd student closed all lockers with even
numbers. The 3rd student changed all lockers
that were numbered with multiples of 3 (by
closing those that were open and opening
those that are closed). The 4th student
changed all lockers that were numbered with
multiples of 4, and so on. After all 1000
students had entered the building in this
fashion, which lockers were left open?
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An easier related problem is to consider the same
situation with only 26 lockers. The following are
26 locker doors. Click to open and click to close.
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(Use Control A to show the cursor )
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Example 5 What is the sum of
?
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• Example 6. The Tower of Hanoi Puzzle
• In the beginning there are 64 discs on the left
  pole. The goal is to move all the discs from the
  left pole to the right pole while following the
  rules
• Only one disc can be moved at each step,
• Smaller discs must be on top of bigger discs,
• Each disc must be on any one pole at the end of
  each step.

How many steps are required to move all the discs
to the right pole?
(we only show 8 discs for simplicity.)
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Number of discs Min number of steps
1 1
2
3
4
5
6
3
7
15
31
63
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• The Tower of Hanoi Puzzle
• The goal is to move all the discs from the left
  pole to the right pole while following the rules
• Only one disc can be moved at each step,
• Smaller discs must be on top of bigger discs,
• Each disc must be on any one pole at the end of
  each step.

If there are 64 discs on the left pole, how many
steps are required to move the discs over?
(we only show 3 discs for simplicity.)
(Please click to see the moves)
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Example 7. Counterfeit Coin
In front of you are 12 identical looking gold
coins, but one is counterfeit. The only
difference is its weight - it may be heavier or
lighter than a real one. If you are given a pan
balance, and are allowed to use it only 3 times,
how can you determine which one is fake, and
whether it is heavier or lighter?
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Method 2 Fix some variables or
parameters Using a batting tee to practice your
swing is example of fixing some parameters.
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Method 2 Fix some variables or parameters Putting
a pair of training wheels on a bike is an
example of solving an easier related problem.
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Method 3 Use a totally different
approach.
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The Invention of Dynamite
The explosive chemical in dynamite is
Nitroglycerin, which was first invented by
Italian chemist Ascanio Sobrero in 1846. In its
natural liquid state, nitroglycerin is very
volatile. It is very dangerous to manufacture and
to be transported.
In 1860, Nobel started experimenting with
chemical additives to tame the explosive power of
nitroglycerine. None of the additives works.
Alfred Nobel
Then, one day in 1866, at age 33, Nobel had
produced a full test tube of the
substance--enough to easily blow up his
laboratory. He was just ready to pour a drop into
another test tube. He was very nervous, when
suddenly the test tube full of nitroglycerin
slipped out of his hands and fell to the floor!
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Luckily, the tube fell into a packing box
filled with sawdust. If it would have hit the
floor, there probably would have been a great
explosion, killing Nobel and others around him.
The nitroglycerin ran out of the test tube
and was absorbed in the sawdust. Not letting the
expensive material go to waste, Nobel started to
test the mixture and found that it still had
great explosive power but could be handled
easier.
He then realized that the approach to the problem
is not adding chemical substances, but mixing
nitroglycerin with absorbent and chemically inert
substances! His final choice was a porous
siliceous earth called kieselguhr .
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Example 8. What is the sum of 1 2 3 4 ?
? ? 100 ?
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Example 8 Johann Carl Friedrich Gauss (1777
1855) was a German mathematician and scientist
who contributed significantly to many fields. He
is known as the Prince of Mathematicians. There
are several stories of his early genius.
According to one, his gifts became very apparent
at the age of three when he corrected, mentally
and without fault in his calculations, an error
his father had made on paper while calculating
finances.
Another famous story, has it that at the age of
8, his primary school teacher, J.G. Büttner,
tried to occupy his pupils by making them add a
list of whole numbers 1 2 3 4 5
95 96 97 98 99 100 The young
Gauss reputedly astonished his teacher and
assistant Martin Bartels by producing the correct
answer within seconds. How did he solve the
problem so quickly?
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Here is Gausss secret
1
2
3
4
5 95
96
97
98
99
100




101
101
101
101
. . . 50 51 101
Therefore, there will be 50 copies of 101,
hence the sum is 50 101 5050.

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Exercise Find the sum of 7 10 13 16 ?
? ? 103
Formula
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Example 9 Find the area of the following triangle
5 inches
6 inches
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Example 10
A 5 km long straight tunnel is pointing roughly
east-west direction. A bicycle enters the west
end exactly when another bicycle enters the east
end. A fly is flying back and forth between the
two bicycles at 16km per hour, leaving the
eastbound bicycle as it enters the tunnel. If
the bicycles are both traveling at 10 km per
hour, how far has the fly traveled in the tunnel
when the bicycles meet?
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Example 11 Good Luck Goats In the mythical
land of Kantanu, it was considered good luck to
own goats. Basanta owned some goats at the time
of her death and willed them to her children. To
her first born, she willed one-half of her goats.
To her second born, she willed one-third of her
goats. And last she gave one-ninth of her goats
to her third born. As it turned out, when Basanta
died, she had 17 goats. Barring a Solomonic
approach, how should the goats be divided?
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How many triangles (of all possible sizes) are in
the following diagram?
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Look for a pattern
1 triangle
1 3 1 triangles (red means
upside-down)
1 3 6 3 triangles
(red means upside-down)
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1 3 6 10 1 6 triangles
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There are 1 3 6 10 15 21
1 6 15 78 Triangles.
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• Diagonals
• A certain convex polygon has 25 sides. How many
  diagonals can be drawn?

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2. Sum of Odds   Find the sum of the first 5000
odd numbers.  
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3. TV Truck   Theotis has to load a truck with
television sets. The cargo area of the truck is
a rectangular block that measures 8 ft by 21 ft
by 11 ft. Each television set measures 1 1/2 ft
by 1 2/3 ft by 1 1/3 ft. What is the maximum
number of TV sets that can be loaded into the
truck?
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  In order to prepare dinner in the mess
hall, Jamie, who is a member of the 4th battalion
of the 23rd regiment, generally used about 85
pounds of potatoes to feed the 358 people in his
unit. He usually assigned three soldiers to
scrub the potatoes, and it takes them just under
2 hours to complete the job.
4. Potatoes
However, this next week he will need to feed
about 817 people beginning at 1730 hours, due to
a special army event. When he arrived at the
mess hall tent for the field exercises, he
discovered 131 pounds of potatoes had been sent.
He needed to send for the rest right away. How
many pounds should he request, and how many
soldiers does he need if he is going to have them
spend about 2 hours each on scrubbing?
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6. Twenty-five Man Roster   Roger Craig, during
his term as manager of the San Francisco Giants,
received a strange communication from the team
general manager, Al Rosen. Mr. Rosen told him to
select 25 players for his roster according to
this formula   1/2 of the team had to be
outfielders and infielders, 1/4 of the team had
to be starting pitchers, 1/6 of the team had to
be relief pitchers, and 1/8 of the team had to
be catchers.   Roger was a bit confused by Al's
request, yet complied anyway. How did he do it?
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9. Fifty-two Card Pickup   A deck of cards was
dropped on the floor. While Amel was out of the
room, Naoko picked up at least one card. She may
have picked up one, all fifty-two, or any number
in between. How many possible combinations are
there for what she picked up?
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1. Covering the Grid   A grid has lines at
90-degree angles. There are 12 lines in one
direction and 9 lines in the other direction.
Lines that are parallel are 11 inches apart.
What is the least number of 12-inch by 12-inch
floor tiles needed to cover all of the line
intersections on the grid? The tiles do not have
to touch each other. You must keep the tiles
intact-do not break or cut them.
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2. Tupperware Parties   A Tupperware party host
was frustrated at her attempts to get more people
to host sales parties for her. She finally
offered to pay any host 25 who (a) hosted a
party for her, and (b) arranged for two other
friends to host parties (during the next month).
To her surprise, it worked!
She started with five hosts in the first month.
If this continues for the entire year, how many
parties will there be during the year? (Note
each person will host at most one party.)
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4. Jogging Around a Track   Dionne can run around
a circular track in 120 seconds. Basha, running
in the opposite direction as Dionne, meets Dionne
every 48 seconds. Sandra, running in the same
direction as Basha, passes Basha every 240
seconds. How often does Sandra meet Dionne?