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# Game Theory: Homework

Module by: UniqU, LLC. E-mail the author

Summary: This chapter covers principles of game theory. After completing this chapter students should be able to: solve strictly determined games and solve games involving mixed strategies.

## STRICTLY DETERMINED GAMES

### Exercise 1

Determine whether the games are strictly determined. If the games are strictly determined, find the optimal strategies for each player and the value of the game.

1. 12231223 size 12{ left [ matrix { 1 {} # 2 {} ## - 2 {} # 3{} } right ]} {}

2. 63216321 size 12{ left [ matrix { 6 {} # 3 {} ## 2 {} # 1{} } right ]} {}

3. 132031124132031124 size 12{ left [ matrix { - 1 {} # - 3 {} # 2 {} ## 0 {} # 3 {} # - 1 {} ## 1 {} # - 2 {} # 4{} } right ]} {}

4. 204342023204342023 size 12{ left [ matrix { 2 {} # 0 {} # - 4 {} ## 3 {} # 4 {} # 2 {} ## 0 {} # - 2 {} # - 3{} } right ]} {}

5. 0211113202111132 size 12{ left [ matrix { 0 {} # 2 {} ## - 1 {} # - 1 {} ## - 1 {} # 1 {} ## 3 {} # 2{} } right ]} {}

6. 532314532314 size 12{ left [ matrix { 5 {} # - 3 {} # 2 {} ## 3 {} # - 1 {} # 4{} } right ]} {}

#### Solution

1. The game is strictly determined. Optimal strategy for the row player is to always play row 1 and never row 2. In other words, his strategy is 1010 size 12{ left [ matrix { 1 {} # 0{} } right ]} {} . The optimal strategy for the column player is to always to play column 1 and never play column 2. We write it as 1010 size 12{ left [ matrix { 1 {} ## 0 } right ]} {}. When both players play their optimal strategy, the value of the game is 1.
2. The game has no saddle point, therefore, it is not strictly determined.
3. The game is strictly determined. The optimal strategy for the row player is to always play row 4, and never play any other row. We write his strategy as 00010001 size 12{ left [ matrix { 0 {} # 0 {} # 0 {} # 1{} } right ]} {} . The column player’s strategy is 0101 size 12{ left [ matrix { 0 {} ## 1 } right ]} {} . The value of the game is 2.

### Exercise 2

Two players play a game which involves holding out one or two fingers simultaneously. If the sum of the fingers is more than 2, Player II pays Player I the sum of the fingers; otherwise, Player I pays Player II the sum of the fingers.

1. Write a payoff matrix for Player I.
2. Find the optimal strategies for each player and the value of the game.

### Exercise 3

A mayor of a large city is thinking of running for re-election, but does not know who his opponent is going to be. It is now time for him to take a stand for or against abortion. If he comes out against abortion rights and his opponent is for abortion, he will increase his chances of winning by 10%. But if he is against abortion and so is his opponent, he gains only 5%. On the other hand, if he is for abortion and his opponent against, he decreases his chance by 8%, and if he is for abortion and so is his opponent, he decreases his chance by 12%.

1. Write a payoff matrix for the mayor.
2. Find the optimal strategies for the mayor and his opponent.

#### Solution

1. .05.10.08.12.05.10.08.12 size 12{ left [ matrix { "." "05" {} # "." "10" {} ## - "." "08" {} # - "." "12"{} } right ]} {}
2. The optimal strategy for the mayor is 1010 size 12{ left [ matrix { 1 {} # 0{} } right ]} {} and for his opponent is 1010 size 12{ left [ matrix { 1 {} ## 0 } right ]} {} . In other words, both candidates should oppose abortion rights.

### Exercise 4

A man accused of a crime is not sure whether anybody saw him do it. He needs to make a choice of pleading innocent or pleading guilty to a lesser charge. If he pleads innocent and nobody comes forth, he goes free. However, if a witness comes forth, the man will be sentenced to 10 years in prison. On the other hand, if he pleads guilty to a lesser charge and nobody comes forth, he gets a sentence of one year and if a witness comes forth, he gets a sentence of 3 years.

1. Write a payoff matrix for the accused.

2. If you were his attorney, what strategy would you advise?

## NON-STRICTLY DETERMINED GAMES

### Exercise 5

Determine the optimal strategies for both the row player and the column player, and find the value of the game.

1. 11111111 size 12{ left [ matrix { - 1 {} # 1 {} ## 1 {} # - 1{} } right ]} {}

2. 11401140 size 12{ left [ matrix { 1 {} # - 1 {} ## - 4 {} # 0{} } right ]} {}

3. 32243224 size 12{ left [ matrix { 3 {} # - 2 {} ## 2 {} # 4{} } right ]} {}

4. 32143214 size 12{ left [ matrix { - 3 {} # 2 {} ## 1 {} # - 4{} } right ]} {}

#### Solution

1. The optimal strategy for the row player is 1/21/21/21/2 size 12{ left [ matrix { 1/2 {} # 1/2{} } right ]} {} . The optimal strategy for the column player is 1/21/21/21/2 size 12{ left [ matrix { 1/2 {} ## 1/2 } right ]} {} . The value of the game is 0.

2. Optimal strategy for the row player is 2/75/72/75/7 size 12{ left [ matrix { 2/7 {} # 5/7{} } right ]} {}. The optimal strategy for the column player is 6/71/76/71/7 size 12{ left [ matrix { 6/7 {} ## 1/7 } right ]} {} . The value of the game is 16/716/7 size 12{"16"/7} {}.

### Exercise 6

Find the expected payoff for the given game matrix GG size 12{G} {} if the row player plays strategy RR size 12{R} {}, and column player plays strategy CC size 12{C} {}.

1. G=3214R=2/31/3C=1/43/4G=3214R=2/31/3C=1/43/4 size 12{G= left [ matrix { - 3 {} # 2 {} ## 1 {} # - 4{} } right ]R= left [ matrix { 2/3 {} # 1/3{} } right ]C= left [ matrix { 1/4 {} ## 3/4 } right ]} {}

2. G=1111R=1/32/3C=2/31/3G=1111R=1/32/3C=2/31/3 size 12{G= left [ matrix { - 1 {} # 1 {} ## 1 {} # - 1{} } right ]R= left [ matrix { 1/3 {} # 2/3{} } right ]C= left [ matrix { 2/3 {} ## 1/3 } right ]} {}

### Exercise 7

Two players play a game which involves holding out one or two fingers simultaneously. If the sum of the fingers is even, Player II pays Player I the sum of the fingers. If the sum of the fingers is odd, Player I pays Player II the sum of the fingers.

1. Write a payoff matrix for Player I.

2. Find the optimal strategies for both the row player and the column player, and the value of the game.

#### Solution

1. 23342334 size 12{ left [ matrix { 2 {} # - 3 {} ## - 3 {} # 4{} } right ]} {}

2. Optimal strategy for the row player is 7/125/127/125/12 size 12{ left [ matrix { 7/"12" {} # 5/"12"{} } right ]} {} . The optimal strategy for the column player is 7/125/127/125/12 size 12{ left [ matrix { 7/"12" {} ## 5/"12" } right ]} {}. The value of the game is 1/121/12 size 12{ - 1/"12"} {}.

### Exercise 8

In December 1995, President Clinton ordered the first of 20,000 U. S. troops to be sent into Bosnia-Herzegovina as a peace keeping force. Unfortunately, the heavy fog made visibility very poor at the Tuzla airfield, and at the same time increased the threat of sniper attacks from the Serbian forces. U. S. Air Force Col. Neal Patton, and Lt. Col. Sid Kooyman, the advance specialists, had two choices: either to send in the troops by air with the difficulties already described or by road thus exposing the troops to ambush by the Serbian forces. The Serbian army, with its limited resources, had a choice of deploying its forces near the airport or along the road route.

If the U. S. lands its troops on the airfield in the fog while the Serbs are concentrating on the road route, the payoff for U. S. is 20 points. But if the U. S. lands its troops on the airfield, and Serbians are there hiding in the fog, U. S. wins only 5 points. On the other hand, if U. S. transports its troops by road and avoids Serbs its payoff is 35 points, but if U. S. meets Serb resistance on the road route, it loses 50 points.

1. Write a payoff matrix for the game.

2. If you were Air Force Col. Neal Patton's advisor, what advice would you give him?

## REDUCTION BY DOMINANCE

Reduce the payoff matrix by dominance. Find the optimal strategy for each player and the value of the game.

### Exercise 9

210320210320 size 12{ left [ matrix { 2 {} # - 1 {} ## 0 {} # 3 {} ## - 2 {} # 0{} } right ]} {}
(1)

#### Solution

2103, R=1/21/20, C=2/31/3, The value=12103 size 12{ left [ matrix { 2 {} # - 1 {} ## 0 {} # 3{} } right ]} {}, R=1/21/20 size 12{R= left [ matrix { 1/2 {} # 1/2 {} # 0{} } right ]} {}, C=2/31/3 size 12{C= left [ matrix { 2/3 {} ## 1/3 } right ]} {}, The value=1 size 12{"The value"=1} {}
(2)

### Exercise 10

234554234554 size 12{ left [ matrix { 2 {} # 3 {} ## 4 {} # 5 {} ## 5 {} # 4{} } right ]} {}
(3)

### Exercise 11

132294501132294501 size 12{ left [ matrix { 1 {} # 3 {} # - 2 {} ## - 2 {} # 9 {} # 4 {} ## - 5 {} # 0 {} # 1{} } right ]} {}
(4)

#### Solution

1224, R=2/31/30, C=2/301/3, The value=01224 size 12{ left [ matrix { 1 {} # - 2 {} ## - 2 {} # 4{} } right ]} {}, R=2/31/30 size 12{R= left [ matrix { 2/3 {} # 1/3 {} # 0{} } right ]} {}, C=2/301/3 size 12{C= left [ matrix { 2/3 {} ## 0 {} ## 1/3 } right ]} {}, The value=0 size 12{"The value"=0} {}
(5)

### Exercise 12

011012124313011012124313 size 12{ left [ matrix { 0 {} # 1 {} # 1 {} ## 0 {} # 1 {} # 2 {} ## 1 {} # 2 {} # 4 {} ## 3 {} # 1 {} # 3{} } right ]} {}
(6)

### Exercise 13

20480662224624802048066222462480 size 12{ left [ matrix { 2 {} # 0 {} # - 4 {} # 8 {} ## 0 {} # - 6 {} # - 6 {} # 2 {} ## 2 {} # - 2 {} # 4 {} # 6 {} ## - 2 {} # - 4 {} # - 8 {} # 0{} } right ]} {}
(7)

#### Solution

0424, R=.60.40, C=0.8.20, The value=0.80424 size 12{ left [ matrix { 0 {} # - 4 {} ## - 2 {} # 4{} } right ]} {}, R=.60.40 size 12{R= left [ matrix { "." 6 {} # 0 {} # "." 4 {} # 0{} } right ]} {}, C=0.8.20 size 12{C= left [ matrix { 0 {} ## "." 8 {} ## "." 2 {} ## 0 } right ]} {}, The value=0.8 size 12{"The value"= - 0 "." 8} {}
(8)

### Exercise 14

1324120513241205 size 12{ left [ matrix { - 1 {} # 3 {} # 2 {} # 4 {} ## 1 {} # 2 {} # 0 {} # 5{} } right ]} {}
(9)

### Exercise 15

5113101284015380551131012840153805 size 12{ left [ matrix { - 5 {} # - 1 {} # - 1 {} # 3 {} ## - "10" {} # 1 {} # 2 {} # - 8 {} ## 4 {} # 0 {} # 1 {} # 5 {} ## 3 {} # - 8 {} # 0 {} # 5{} } right ]} {}
(10)

#### Solution

2815, R=02/75/70, C=0013/141/14, The value=9/72815 size 12{ left [ matrix { 2 {} # - 8 {} ## 1 {} # 5{} } right ]} {}, R=02/75/70 size 12{R= left [ matrix { 0 {} # 2/7 {} # 5/7 {} # 0{} } right ]} {}, C=0013/141/14 size 12{C= left [ matrix { 0 {} ## 0 {} ## "13"/"14" {} ## 1/"14" } right ]} {}, The value=9/7 size 12{"The value"=9/7} {}
(11)

### Exercise 16

13411413111101111341141311110111 size 12{ left [ matrix { 1 {} # - 3 {} # - 4 {} # 1 {} ## 1 {} # - 4 {} # - 1 {} # 3 {} ## 1 {} # 1 {} # - 1 {} # 1 {} ## 0 {} # - 1 {} # 1 {} # 1{} } right ]} {}
(12)

## CHAPTER REVIEW

### Exercise 17

Determine whether the games are strictly determined. If the games are strictly determined, find the optimal strategies for each player and the value of the game.

1. 23342334 size 12{ left [ matrix { 2 {} # 3 {} ## 3 {} # 4{} } right ]} {}

2. 031132125031132125 size 12{ left [ matrix { 0 {} # 3 {} # - 1 {} ## 1 {} # 3 {} # - 2 {} ## - 1 {} # 2 {} # - 5{} } right ]} {}

3. 321534321534 size 12{ left [ matrix { 3 {} # 2 {} # - 1 {} ## 5 {} # 3 {} # 4{} } right ]} {}

4. 4213431342134313 size 12{ left [ matrix { 4 {} # 2 {} ## - 1 {} # 3 {} ## 4 {} # 3 {} ## 1 {} # - 3{} } right ]} {}

#### Solution

1. R=01R=01 size 12{R= left [ matrix { 0 {} # 1{} } right ]} {}, C=10C=10 size 12{C= left [ matrix { 1 {} ## 0 } right ]} {}, value=3value=3 size 12{"value"=3} {}

2. R=100R=100 size 12{R= left [ matrix { 1 {} # 0 {} # 0{} } right ]} {}, C=001C=001 size 12{C= left [ matrix { 0 {} ## 0 {} ## 1 } right ]} {}, v=1v=1 size 12{v= - 1} {}

3. R=01R=01 size 12{R= left [ matrix { 0 {} # 1{} } right ]} {}, C=010C=010 size 12{C= left [ matrix { 0 {} ## 1 {} ## 0 } right ]} {}, value=3value=3 size 12{"value"=3} {}

4. R=0010R=0010 size 12{R= left [ matrix { 0 {} # 0 {} # 1 {} # 0{} } right ]} {}, C=01C=01 size 12{C= left [ matrix { 0 {} ## 1 } right ]} {}v=3v=3 size 12{v=3} {}

### Exercise 18

Two players play a game which involves holding out a nickel or a dime simultaneously. If the sum of the coins is more than 10 cents, Player I gets both the coins; otherwise, Player II gets both the coins.

1. Write a payoff matrix for Player I.

2. Find the optimal strategies for each player and the value of the game.

#### Solution

1. 510510510510 size 12{ left [ matrix { - 5 {} # "10" {} ## 5 {} # "10"{} } right ]} {}
2. R=01R=01 size 12{R= left [ matrix { 0 {} # 1{} } right ]} {}, C=10C=10 size 12{C= left [ matrix { 1 {} ## 0 } right ]} {}, value=5 centsvalue=5 cents size 12{"value"=5" cents"} {}

### Exercise 19

Lacy's department store is thinking of having a major sale in the month of February, but does not know if its competitor store Hordstrom's is also planning one. If Lacy's has a sale and Hordstrom's does not, Lacy's sales go up by 30%, but if both stores have a sale simultaneously, Lacy's sales go up by only 5%. On the other hand, if Lacy's does not have a sale and Hordstrom's does, Lacy's loses 5% of its sales to Hordstrom's, and if neither of the stores has a sale, Lacy's experiences no gain in sales.

1. Write a payoff matrix for Lacy's.

2. Find the optimal strategies for both stores.

#### Solution

1. 5305053050 size 12{ left [ matrix { 5 {} # "30" {} ## - 5 {} # 0{} } right ]} {}
2. R=10R=10 size 12{R= left [ matrix { 1 {} # 0{} } right ]} {}C=10C=10 size 12{C= left [ matrix { 1 {} ## 0 } right ]} {}, value=5%value=5% size 12{"value"=5%} {}

### Exercise 20

Mr. Halsey has a choice of three investments: Investment A, Investment B, and Investment C. If the economy booms, then Investment A yields 14% return, Investment B returns 8%, and Investment C 11%. If the economy grows moderately, then Investment A yields 12% return, Investment B returns 11%, and Investment C 11%. If the economy experiences a recession, then Investment A yields a 6% return, Investment B returns 9%, and Investment C 10%.

1. Write a payoff matrix for Mr. Halsey.

2. What would you advise him?

#### Solution

1. .14.08.11.12.11.11.06.09.10.14.08.11.12.11.11.06.09.10 size 12{ left [ matrix { "." "14" {} # "." "08" {} # "." "11" {} ## "." "12" {} # "." "11" {} # "." "11" {} ## "." "06" {} # "." "09" {} # "." "10"{} } right ]} {}
2. 010010 size 12{ left [ matrix { 0 {} # 1 {} # 0{} } right ]} {}, 010010 size 12{ left [ matrix { 0 {} ## 1 {} ## 0 } right ]} {} or 010010 size 12{ left [ matrix { 0 {} # 1 {} # 0{} } right ]} {}, 001001 size 12{ left [ matrix { 0 {} ## 0 {} ## 1 } right ]} {}, value=.11value=.11 size 12{"value"= "." "11"} {}

### Exercise 21

Mr. Thaggert is trying to decide whether to invest in stocks or in CD's(Certificate of deposit). If he invests in stocks and the interest rates go up, his stock investments go down by 2%, but he gains 1% in his CD's. On the other hand if the interest rates go down, he gains 3% in his stock investments, but he loses 1% in his CD's.

1. Write a payoff matrix for Mr. Thaggert.

2. If you were his investment advisor, what strategy would you advise?

#### Solution

1. .02.03.01.01.02.03.01.01 size 12{ left [ matrix { - "." "02" {} # "." "03" {} ## "." "01" {} # - "." "01"{} } right ]} {}
2. stocks=2/7stocks=2/7 size 12{"stocks"=2/7} {}, CD's=5/7CD's=5/7 size 12{"CD's"=5/7} {}

### Exercise 22

Determine the optimal strategies for both the row player and the column player, and find the value of the game.

1. 22222222 size 12{ left [ matrix { 2 {} # - 2 {} ## - 2 {} # 2{} } right ]} {}

2. 22502250 size 12{ left [ matrix { - 2 {} # 2 {} ## 5 {} # 0{} } right ]} {}

3. 35413541 size 12{ left [ matrix { 3 {} # 5 {} ## 4 {} # - 1{} } right ]} {}

4. 25432543 size 12{ left [ matrix { - 2 {} # 5 {} ## 4 {} # - 3{} } right ]} {}

#### Solution

1. 1/21/21/21/2 size 12{ left [ matrix { 1/2 {} # 1/2{} } right ]} {}, 1/21/21/21/2 size 12{ left [ matrix { 1/2 {} ## 1/2 } right ]} {}, value=0value=0 size 12{"value"=0} {}
2. 5/94/95/94/9 size 12{ left [ matrix { 5/9 {} # 4/9{} } right ]} {}, 2/97/92/97/9 size 12{ left [ matrix { 2/9 {} ## 7/9 } right ]} {}, value=10/9value=10/9 size 12{"value"="10"/9} {}
3. 5/72/75/72/7 size 12{ left [ matrix { 5/7 {} # 2/7{} } right ]} {}, 6/71/76/71/7 size 12{ left [ matrix { 6/7 {} ## 1/7 } right ]} {}, value=23/7value=23/7 size 12{"value"="23"/7} {}
4. 1/21/21/21/2 size 12{ left [ matrix { 1/2 {} # 1/2{} } right ]} {}, 4/73/74/73/7 size 12{ left [ matrix { 3/7 {} ## 4/7 } right ]} {}, value=1value=1 size 12{value=1} {}

### Exercise 23

Find the expected payoff for the given game matrix G if the row player plays strategy R, and the column player plays strategy C.

1. G=3541G=3541 size 12{G= left [ matrix { 3 {} # 5 {} ## 4 {} # - 1{} } right ]} {}R=1/21/2R=1/21/2 size 12{R= left [ matrix { 1/2 {} # 1/2{} } right ]} {}C=1/43/4C=1/43/4 size 12{C= left [ matrix { 1/4 {} ## 3/4 } right ]} {}

2. G=2543G=2543 size 12{G= left [ matrix { - 2 {} # 5 {} ## 4 {} # - 3{} } right ]} {}R=2/31/3R=2/31/3 size 12{R= left [ matrix { 2/3 {} # 1/3{} } right ]} {}C=1/32/3C=1/32/3 size 12{C= left [ matrix { 1/3 {} ## 2/3 } right ]} {}

#### Solution

1. 19/819/8 size 12{"19"/8} {}

2. 14/914/9 size 12{"14"/9} {}

### Exercise 24

A group of thieves are planning to burglarize either Warehouse A or Warehouse B. The owner of the warehouses has the manpower to secure only one of them. If Warehouse A is burglarized the owner will lose $20,000, and if Warehouse B is burglarized the owner will lose$30,000. There is a 40% chance that the thieves will burglarize Warehouse A and 60% chance they will burglarize Warehouse B. There is a 30% chance that the owner will secure Warehouse A and 70% chance he will secure Warehouse B. What is the owner's expected loss?

\$11,000

### Exercise 25

Two players play a game which involves holding out a nickel or a dime. If the sum of the coins is odd, Player I gets both the coins, and if the sum of the coins is even, Player II gets both the coins. Determine the optimal strategies for both the row player and the column player, and find the expected payoff.

#### Solution

1/21/2, 2/31/3, v=01/21/2 size 12{ left [ matrix { 1/2 {} # 1/2{} } right ]} {}, 2/31/3 size 12{ left [ matrix { 2/3 {} ## 1/3 } right ]} {}, v=0 size 12{v=0} {}
(13)

### Exercise 26

A football quarterback has to choose between a pass play or a run play depending on how the defending team is going to react. If he chooses a pass play and the defending team is expecting a pass, he expects to gain 4 yards, but if the defending team is expecting a run, he gains 20 yards. On the other hand, if he calls a run play and the defending team expects a pass, he gains 7 yards, and if he calls a run play and the defending team expects a run, he loses 2 yards. If you were the quarterback, what would your strategy be?

#### Solution

Pass=9/25, Run=16/25Pass=9/25 size 12{"Pass"=9/"25"} {}, Run=16/25 size 12{"Run"="16"/"25"} {}
(14)

### Exercise 27

The Watermans go fishing every weekend either at Eel River or at Snake River. Unfortunately, so do the Nelsons. If both families show up at Eel River, the Watermans can hope to catch only 3 fish, but if the Watermans fish at Eel River and the Nelsons at Snake River, the Watermans can catch as many as 12 fish. On the other hand, if both families fish at Snake river, the Watermans can catch about 5 fish, and if Watermans fish at Snake river while the Nelsons fish at Eel river, the Watermans can catch up to 15 fish. Determine a mixed strategy for the Watermans, and the expected payoff.

#### Solution

10/199/19, payoff=9.58   fish10/199/19 size 12{ left [ matrix { "10"/"19" {} # 9/"19"{} } right ]} {}, payoff=9.58   fish size 12{"payoff"=9 "." "58"" fish"} {}
(15)

### Exercise 28

Terry knows there is a quiz tomorrow, but does not remember whether it is in his math class or in his biology class. He has time to study for only one subject. If he studies math and there is a quiz in it, he gains 10 points and even if there is no quiz he gains two points for acquiring the extra knowledge which he will apply towards the final exam. If he studies biology and there is a quiz in it, he gains ten points but there is no gain if there is no quiz. Determine a mixed strategy for Terry, and the expected payoff.

#### Solution

10, payoff=2   points10 size 12{ left [ matrix { 1 {} # 0{} } right ]} {}, payoff=2   points size 12{"payoff"=2" points"} {}
(16)

### Exercise 29

Reduce the payoff matrix by dominance. Find the optimal strategy for each player and the value of the game.

1. 312353241312353241 size 12{ left [ matrix { - 3 {} # 1 {} # 2 {} ## - 3 {} # 5 {} # 3 {} ## 2 {} # 4 {} # - 1{} } right ]} {}
2. 123414234122123414234122 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ## 4 {} # 1 {} # 4 {} ## 2 {} # 3 {} # 4 {} ## 1 {} # 2 {} # 2{} } right ]} {}
3. 43977535145835114397753514583511 size 12{ left [ matrix { 4 {} # 3 {} # 9 {} # 7 {} ## - 7 {} # - 5 {} # - 3 {} # 5 {} ## - 1 {} # 4 {} # 5 {} # 8 {} ## - 3 {} # - 5 {} # 1 {} # - 1{} } right ]} {}
4. 2315221323152213 size 12{ left [ matrix { 2 {} # 3 {} # 1 {} # 5 {} ## - 2 {} # 2 {} # 1 {} # 3{} } right ]} {}
5. 03210217495447660321021749544766 size 12{ left [ matrix { 0 {} # 3 {} # 2 {} # 1 {} ## 0 {} # 2 {} # 1 {} # - 7 {} ## - 4 {} # - 9 {} # 5 {} # 4 {} ## 4 {} # - 7 {} # 6 {} # 6{} } right ]} {}
6. 10222202230422321022220223042232 size 12{ left [ matrix { 1 {} # 0 {} # 2 {} # 2 {} ## 2 {} # 2 {} # 0 {} # 2 {} ## 2 {} # - 3 {} # 0 {} # 4 {} ## 2 {} # - 2 {} # - 3 {} # 2{} } right ]} {}

#### Solution

1. 33213321 size 12{ left [ matrix { - 3 {} # 3 {} ## 2 {} # - 1{} } right ]} {}, 01/32/301/32/3 size 12{ left [ matrix { 0 {} # 1/3 {} # 2/3{} } right ]} {}, 4/905/94/905/9 size 12{ left [ matrix { 4/9 {} ## 0 {} ## 5/9 } right ]} {}, value=1/3value=1/3 size 12{"value"=1/3} {}
2. 41234123 size 12{ left [ matrix { 4 {} # 1 {} ## 2 {} # 3{} } right ]} {}, 01/43/4001/43/40 size 12{ left [ matrix { 0 {} # 1/4 {} # 3/4 {} # 0{} } right ]} {}, 1/21/201/21/20 size 12{ left [ matrix { 1/2 {} ## 1/2 {} ## 0 } right ]} {}, value=5/2value=5/2 size 12{"value"=5/2} {}
3. 43–1443–14 size 12{ left [ matrix { 4 {} # 3 {} ## –1 {} # 4{} } right ]} {}, 5/601/605/601/60 size 12{ left [ matrix { 5/6 {} # 0 {} # 1/6 {} # 0{} } right ]} {}, 1/65/6001/65/600 size 12{ left [ matrix { 1/6 {} ## 5/6 {} ## 0 {} ## 0 } right ]} {}, value=19/6value=19/6 size 12{"value"="19"/6} {}
4. 21212121 size 12{ left [ matrix { 2 {} # 1 {} ## - 2 {} # 1{} } right ]} {}, 3/41/43/41/4 size 12{ left [ matrix { 3/4 {} # 1/4{} } right ]} {}, 00100010 size 12{ left [ matrix { 0 {} ## 0 {} ## 1 {} ## 0 } right ]} {}, value=1value=1 size 12{"value"=1} {}
5. 03470347 size 12{ left [ matrix { 0 {} # 3 {} ## 4 {} # - 7{} } right ]} {}, 11/14003/1411/14003/14 size 12{ left [ matrix { "11"/"14" {} # 0 {} # 0 {} # 3/"14"{} } right ]} {}, 10/144/140010/144/1400 size 12{ left [ matrix { "10"/"14" {} ## 4/"14" {} ## 0 {} ## 0 } right ]} {}, value=6/7value=6/7 size 12{"value"=6/7} {}
6. 02200220 size 12{ left [ matrix { 0 {} # 2 {} ## 2 {} # 0{} } right ]} {}, 1/21/2001/21/200 size 12{ left [ matrix { 1/2 {} # 1/2 {} # 0 {} # 0{} } right ]} {}, 01/21/2001/21/20 size 12{ left [ matrix { 0 {} ## 1/2 {} ## 1/2 {} ## 0 } right ]} {}, value=1value=1 size 12{"value"=1} {}

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks