Goal-oriented, state-to-state search Most notes from Dr. Michael Wick

How many different outfits can you make if you have two shirts and four pairs of pants? Red shirt, blue pantsRed shirt, khaki pants Yellow shirt, blue pantsYellow shirt, khaki pants Red shirt, green pantsRed shirt, black pants Yellow shirt, green pantsYellow shirt, black pants

The sum of two numbers is 27, their product is 180. What are the two numbers? Guess: = * 14 = 182 Guess: = * 15 = 180

What are the possible combinations for families with two children? BG BGBG

Lisa is drawing a pyramid. She puts one block in the top row, two in the second, four in the third, eight in the fourth. If she continues this pattern, how many blocks will be in the tenth row?

Four students sit at each lunch table. Sue is left-handed and doesn’t want to bump elbows with anyone, but she likes to sit next to her best friend Kate. Kate is to the left of Nancy. Allison likes to be on the end. Where do each of the girls sit during lunch?

The night before their debut in Carnegie Hall, the dancers stayed at a hotel on 57 th Street. Fourteen members of the ballet company went to an all-night card room to play poker. Half of the remaining dancers went to Madison Square Garden for a special midnight professional wrestling show featuring Jedidiah Jiggsaw. After about an hour, six of the dancers who had gone to play poker came back to the hotel broke. The eleven dancers now at the hotel went to bed and got enough sleep, but the rest of the dancers were tired for their debut the next day. How many dancers were in the ballet company?

There are 100 fifth graders in the school. One fifth of them like pizza, one half like spaghetti, one fifth like cheeseburgers, one tenth like tacos. How many students like each type of food?

The yard is 2,400cm long and 1,700cm wide. How many meters of fencing is needed to surround the yard? 2,400cm = 24m 1,700cm = 17m P = (24 x 2) + (17 x 2) = 82 meters of fencing

A number is composite, and a multiple of 6. The first digit is prime, but not 2. The number is less than 50 but greater than 20, and is not divisible by 5. What is the number? 1. Composite numbers have factors other than one and themselves. 2. Prime numbers have only one and themselves as factors. 3. Multiples of 6 >20, < 50: 24, 30, 36, 42, First digit prime, not 2: 30, Not divisible by 5: …

Jim tells lies on Fridays, Saturdays, and Sundays. He tells the truth on all other days. Freda tells lies on Tuesdays, Wednesdays, and Thursdays. She tells the truth on all other days. If they both say “Yesterday I lied,” then what day is it today?

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 feet Counts 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 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 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? 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 = 9 quarts

How many quarts of paint are in the new mixture? What percentage of the new mixture is red paint? = 24 quarts 9 / 24 = = 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 There are k quiz scores in the group. The average of 10 of these quiz scores is Find the average of the remaining quiz scores in terms of k.

The average of a group of quiz scores is There are k quiz scores in the group. The average of 10 of these quiz scores is 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 =  Sum of 10 scores is 25 x 10 = 250.  Sum of other scores is 1500 – 250 =  Average of those 40 scores is

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)

≈ 2,022.2 voters per polling place 28,311 14

The divisors of 360 add up to 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

The divisors of 10 are 1, 2, 5, and 10. Their sum is 18. The sum of reciprocals is 0

How many squares are there on a checkerboard?

18x8square 47x7squares 96x6squares 165x5squares 254x4squares 363x3squares 492x2squares 641x1squares 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.

 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. 300 B. 625 C. 275 D. 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. 528 B. 504 C. 625 D. 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. Dragon B. Sheep C. Boar D. 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. 4 B. 6 C. 8 D. 10 Strategy: Change, fix, or get rid of some conditions.

A square has an area of S 2. 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/4 B. h + p = 4T + 6S C. h + p = T/6 + 4S D. 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?  html 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)