How many quarts of red paint are in the new mixture?

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Goal-oriented, state-to-state search
Most notes from Dr. Michael Wick
Curly used a shovel to dig his own swimming
pool. He figured he needed a pool because
digging it was hard work and he could use it
to cool off after working on it all day. He also
planned to build a rectangular concrete deck
around the pool that would be 6 feet wide at
all points. The pool is rectangular and
measures 14 feet by 40 feet. What is the area
of the deck?
52 feet + 26 feet + 52 feet + 26 feet = 156 feet
Counts each corner twice!
156 feet x 6 feet = 936 square feet
Two lengths: 40 ft x 6 ft x 2 = 480 sq ft
Two widths: 14 ft x 6 ft x 2 = 168 sq ft
Four corners: 6 ft x 6 ft x 4 = 144 sq ft
Total 792 sq ft
52 ft x 6 ft = 312 sq ft
312 sq ft x 2 = 624 sq ft for extended lengths
14 ft x 6 ft = 84 sq ft
84 sq ft x 2 = 168 sq ft for widths
Total = 624 sq ft + 168 sq ft = 792 sq ft
Area of entire figure = 52 ft x 26 ft = 1352 sq ft
Area of pool alone = 40 ft x 14 ft = 560 sq ft
Area of deck = 1352 – 560 = 792 sq ft
The object of the game Frisbin is to throw
three Frisbees at three different-sized bins
that are set up on the ground about 20 feet
away from the player. If a Frisbee lands in the
largest bin, the player scores 1 point. If a
Frisbee lands in the medium-sized bin, the
player scores 5 points. If a Frisbee lands in
the smallest bin, the player scores 10 points.
Kirk McCoy is playing the game. If all three of
his Frisbees land in bins, how many different
total scores can he make?
Tom, John, Fred, and Bill each brought his
favorite food to a dinner. From the clues below,
determine each man’s occupation and favorite food.
1. Tom is neither the nurse nor the teacher.
2. Fred and the pilot play in a jazz band together.
3. The burger lover and the teacher are not musically
inclined.
4. Tom brought hot dogs.
5. Bill sat next to the burger fan and across from the
steak lover.
6. The secretary does not play an instrument or sing.
Tom Pilot Hdog
John Scty Burg
Fred Nurse Steak
Bill Tchr Chkn
A mixture is 25% red paint, 30% yellow paint,
and 45% water. If 4 quarts of red paint are
added to 20 quarts of the mixture, what is
the percentage of red paint in the new
mixture?
1. How many quarts of red paint are in the
new mixture?
2. How many quarts of paint are in the new
mixture?
3. What percentage of the new mixture is red
paint?
1. How many quarts of red paint are in the
new mixture?
How many quarts of red paint are in the
original mixture?
And how many quarts of paint are in the
original mixture?
How many quarts of paint are in the original
mixture?
How many quarts of red paint are in the original
mixture?
How many quarts of red paint are in the new
mixture?
2. How many quarts of paint are in the new
mixture?
3. What percentage of the new mixture is
red paint?
How many quarts of paint are in the
original mixture?
20 (given)
How many quarts of red paint are in the
original mixture?
25% of 20 = 5 quarts
How many quarts of red paint are in the
new mixture?
5 + 4 = 9 quarts
How many quarts of paint are in the new mixture?
20 + 4 = 24 quarts
What percentage of the new mixture is red paint?
9 / 24 = 0.375 = 37.5%
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Use a number instead of a variable
Use smaller or easier numbers
Do a set of specific easier examples and
look for a pattern
Do a specific easier example and figure out
an easier process
Change, fix, or get rid of some conditions
Eliminate unnecessary information
The average of a group of quiz scores is 31.8.
There are k quiz scores in the group. The
average of 10 of these quiz scores is 24.3.
Find the average of the remaining quiz scores
in terms of k.
The average of a group of quiz scores is 31.8
30. There are k quiz scores in the group. The
average of 10 of these quiz scores is 24.3 25.
Find the average of the remaining quiz scores
in terms of k.
The average of a group of quiz scores is 30.
There are k 50 quiz scores in the group. The
average of 10 of these quiz scores is 25. Find
the average of the remaining quiz scores in
terms of k 50.
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Sum of all scores is 30 x 50 = 1500.
Sum of 10 scores is 25 x 10 = 250.
Sum of other scores is 1500 – 250 = 1250.
Average of those 40 scores is
1250
 31.25
40
1250 1500  250 30  50  25  10


40
50  10
50  10
30k  25  10

k  10
31.8k  243

k  10
In this election, there are 29 issues and
candidates. In the last election, there were
28,311 voters, representing 18,954
households, and they voted at 14 polling
places. This time there will be 34,892 voters.
How many polling places will be needed?
•
•
•
•
•
•
Polling places (last election): 15
Voters (last election): 30,000
Households: 20,000
Issues: 30
Voters (this election): 35,000
Polling places (this election): ?
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Polling places (last election): 15
Voters (last election): 30,000
Households: 20,000
Issues: 30
Voters (this election): 35,000
Polling places (this election): ?
30,000
 2,000 voters/polling place
15
35,000
 17.5 polling places (this election)
2,000
28,311
≈ 2,022.2 voters per polling place
14
34,892
 17.25 polling places (this election)
2,022.2
The divisors of 360 add up to 1170. What is
the sum of the reciprocals of the divisors of
360?
Use divisors of 24, which are 1, 2, 3, 4, 6, 8,
12, and 24. Their sum is 60.
The sum of reciprocals is
1 1 1 1 1 1 1
1
     

1 2 3 4 6 8 12 24

24 12 8
6
4
3
2
1







24 24 24 24 24 24 24 24
24  12  8  6  4  3  2  1
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24
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60 sum of divisors

24
number
The divisors of 10 are 1, 2, 5, and
10. Their sum is 18.
The sum of reciprocals is
1 1 1 1 10 5
2
1
  
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


1 2 5 10 10 10 10 10
10  5  2  1 18 sum of divisors

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
10
10
number
sum of divisors 1170

number
360
How many squares are there on a
checkerboard?
1 8x8 square
4 7x7 squares
9 6x6 squares
16 5x5 squares
25 4x4 squares
36 3x3 squares
49 2x2 squares
64 1x1 squares
total: 204 squares
A train leaves Roseville heading east at
6:00 a.m. at 40 miles per hour. Another
eastbound train leaves on a parallel
track at 7:00 a.m. at 50 miles per hour.
What time will it be when the two trains
are the same distance away from
Roseville?
One possibility: Change the condition
so that the first train travels for an hour
and then stops, and the second travels
at 10 miles per hour.
Second train will “make up” 10 miles per hour.
Let’s reduce the problem to finding when the second train,
running at 10 miles per hour will reach the spot that the first
train reaches at the one hour point.
1 hour  40 mi/hr  40 mi
40 mi
 4 hr
10 mi/hr
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Subproblem
◦ Results from breaking a problem down into its
parts
◦ Use answers to construct solution
◦ CS Terminology
 Divide and Conquer
 Wishful Thinking

Easier Problem
◦ Doesn’t involve solving the original problem
◦ Use method to construct solution
◦ CS Terminology
 Hardcode
 Releases
A diagonal of a polygon is a line segment
that connects two nonadjacent vertices of
the polygon. A certain polygon has 25
sides. How many diagonals can be
drawn?
A.
B.
C.
D.
300
625
275
500
Strategy:
Do a set of specific easier examples and
look for a pattern.
Ted has to load a truck with television
sets. The cargo area of the truck is a
rectangular prism 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.
How many sets can be loaded into the
truck?
A.
B.
C.
D.
528
504
625
508
Strategy:
Use a small or easier number.
In China each calendar year is given one of 12 names,
which rotate year after year. The year 2000 was the
year of the Dragon. The year 2001 was the year of the
Snake. The subsequent ten years are, in order, the
years of the Horse, Sheep, Monkey, Rooster, Dog, Boar,
Rat, Ox, Tiger, and Rabbit. After the year of the Rabbit,
the year of the Dragon will occur again, and then the
whole cycle will repeat. What will the year 3000 be?
A.
B.
C.
D.
Dragon
Sheep
Boar
Monkey
Strategy:
Do a set of specific easier examples and
look for a pattern.
In the land of Kantanu, it was considered good luck to own
goats. Barsanta owned some goats at the time of her death and
willed them to her children. To her first born, she willed onehalf 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.
Assuming that Barsanta had 17 goats and barring a Solomonic
approach, how many goats did the second child receive?
A.
B.
C.
D.
4
6
8
10
Strategy:
Change, fix, or get rid of some
conditions.
A square has an area of S2. A regular
hexagon has a perimeter of T. If p is
the perimeter of the square and h is a
side of the hexagon, then find h + p
in terms of S and T.
Strategy:
Use values instead of variables; use logic
A.
B.
C.
D.
h + p = 6T + S/4
h + p = 4T + 6S
h + p = T/6 + 4S
h + p = T/4 + 6S
Knights always tell the truth.
 Knaves always lie.

Ima says “Dewey is a knave.”
 Dewey says “Neither Ima nor I are
knaves.”
 Who, if any, is a knight and who, if
any, is a knave?
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Strategy:
Use logic
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Summing numbers – what’s the sum of all
numbers between 1 and 100?
Summing digits of numbers – what’s the sum
of all of the digits of all of the numbers
between 1 and 100?
http://mathforum.org/library/drmath/view/5
7919.html

Error reads: “missing semicolon on line XY”

You go to line XY, and it looks like this:
◦ System.out.println(“A perfectly valid statement”);
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What’s the problem?
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G. Polya, “How To Solve It” (2004)

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