problem solving

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

156 feet x 6 feet = 936 square feetCounts each corner twice!

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?

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, John, Fred, and Bill each brought his favorite food to a dinner. From the clues below, determine each man’s occupation and favorite food.

Tom Pilot HdogJohn Scty BurgFred Nurse SteakBill 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?

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

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

20 (given)

25% of 20 = 5 quarts

5 + 4 = 9 quarts

How many quarts of paint are in the new mixture?

What percentage of the new mixture is red paint?

20 + 4 = 24 quarts

9 / 24 = 0.375 = 37.5%

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.

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

4031.25

1250

401500 250

50 103050 2510

50 10

30k 2510

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): ?

Polling places (last election): 15 Voters (last election): 30,000 Households: 20,000 Issues: 30 Voters (this election): 35,000 Polling places (this election): ?

voters/polling place

polling places (this election)

30,000

152,000

35,000

2,00017.5

34,892

2,022.217.25 polling places (this election)

≈ 2,022.2 voters per polling place28,311 14

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

2

1

3

1

4

1

6

1

8

1

12

1

24 24

24 12

24 8

24 6

24 4

24 3

24 2

24 1

24

24 1286 4 321

24

60

24sum of divisors

number

The divisors of 10 are 1, 2, 5, and 10. Their sum is 18.

The sum of reciprocals is

1

1

1

2

1

5

1

10

10

10

5

10

2

10

1

10

10521

1018

1sum of divisors

number0

sum of divisors

number

1170

360

How many squares are there on a checkerboard?

1 8x8square

4 7x7squares

9 6x6squares

16 5x5squares

25 4x4squares

36 3x3squares

49 2x2squares

64 1x1squares

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.

1 hour 40 mi/hr 40 mi

40 mi

10 mi/hr4 hr

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.

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. 300B. 625C. 275D. 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. 528B. 504C. 625D. 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. DragonB. SheepC. BoarD. 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 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.

Assuming that Barsanta had 17 goats and barring a Solomonic approach, how many goats did the second child receive?

A. 4B. 6C. 8D. 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.

A. h + p = 6T + S/4B. h + p = 4T + 6SC. h + p = T/6 + 4SD. h + p = T/4 + 6S

Strategy: Use values instead of variables; use logic

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?

Strategy: Use logic

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/57919.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”);

What’s the problem?

G. Polya, “How To Solve It” (2004)