Logic Puzzles

Show your work for each problem.

1.
Ana, Ben and Cleo each own one different pet: a cat, a dog or a fish. Ana's pet lives in water. Ben does not own the cat. Who owns the cat?
1. a)Ben
2. b)Cleo
3. c)It cannot be determined
4. d)Ana
Answer: Cleo
Ana owns the fish. Ben does not own the cat, and the fish is taken, so Ben owns the dog. Only the cat is left, so Cleo owns the cat.
2.
Mia, Tom and Leo are $10$ , $12$ and $14$ years old, one age each. Leo is the youngest. Mia is older than Tom. How old is Mia?
Answer: 14
Leo is $10$ . Mia and Tom share $12$ and $14$ , and Mia is older, so Mia is $14$ and Tom is $12$ .
3.
Ana, Ben, Cleo, Dan and Eva line up by height, shortest first (position $1$ is the shortest). Cleo is the shortest. Ben is the tallest. Eva is taller than Ana but shorter than Dan. At which position is Eva?
Answer: 3
1. At which position is Cleo? 1
2. At which position is Ben? 5
3. Ana is shorter than Eva, and Eva is shorter than Dan. At which position is Ana? 2
4. At which position is Eva? 3
4.
On the island of knights (always tell the truth) and knaves (always lie), which sentence can nobody ever say?
1. a)"I am a knight."
2. b) $2 + 2 = 4$
3. c)"I am a knave."
4. d) $2 + 2 = 5$
Answer: "I am a knave."
A knight saying "I am a knave" would be lying, and a knave saying it would be telling the truth. Both are impossible. A knight can say "I am a knight" and $2 + 2 = 4$ ; a knave can say "I am a knight" and $2 + 2 = 5$ .
5.
Two whole numbers have a sum of $15$ and a difference of $3$ . What is the larger number?
Answer: 9
Pairs with sum $15$ : $8$ and $7$ (difference $1$ ), $9$ and $6$ (difference $3$ ). So the numbers are $9$ and $6$ , and the larger one is $9$ . Check: $9 + 6 = 15$ and $9 - 6 = 3$ .
6.
Ana, Ben and Cleo live in houses number $1$ , $2$ and $3$ , one house each. Ana does not live in house $1$ or house $3$ . Ben's house number is greater than Cleo's. In which house does Ben live?
Answer: 3
1. In which house does Ana live? 2
2. Houses $1$ and $3$ are left. Cleo has the smaller number. In which house does Cleo live? 1
3. In which house does Ben live? 3
7.
Three houses stand in a row: left, middle and right. They are painted red, blue and green, one color each. The red house is not in the middle. The green house is directly to the right of the red house. What color is the middle house?
1. a)It cannot be determined
2. b)Red
3. c)Green
4. d)Blue
Answer: Green
If the red house were on the right, no house could stand to its right. So red is on the left, green directly to its right in the middle, and blue on the right. The middle house is green.
8.
Eva, Leo, Mia and Tom stand in a line at positions $1$ (front) to $4$ (back). Tom stands directly behind Eva. Leo is not at either end. Mia stands somewhere behind Leo. At which position is Leo?
Answer: 3
If Leo were at $2$ , Mia would be at $3$ or $4$ , and the two places left ( $1$ and $4$ , or $1$ and $3$ ) are not neighbors, so Eva and Tom would not fit. So Leo is at $3$ , Mia at $4$ , Eva at $1$ and Tom at $2$ .
9.
Lea, Max and Sam ate $2$ , $3$ and $5$ slices of pizza, a different number each. Max ate more slices than Lea. Lea and Sam ate $7$ slices together. How many slices did Sam eat?
Answer: 5
1. Which two of the numbers $2$ , $3$ and $5$ add up to $7$ ? Enter the larger one. 5
2. Lea and Sam have those two numbers. How many slices did Max eat? 3
3. Max ate more than Lea. How many slices did Lea eat? 2
4. How many slices did Sam eat? 5
10.
On the island of knights (always tell the truth) and knaves (always lie), A says: "B is a knave." B says: "A and I are the same type." What are A and B?
1. a)A is a knave, B is a knight
2. b)A is a knight, B is a knight
3. c)A is a knight, B is a knave
4. d)A is a knave, B is a knave
Answer: A is a knight, B is a knave
If A is a knight, B is a knave; then they are different types, so B's sentence is false, which fits a knave. If A is a knave, B is a knight; then they are different types, so B's sentence is false, but a knight cannot say something false. Only the first case works: A is a knight, B is a knave.
11.
Six friends want to cross a river in a boat that holds at most $2$ people. The boat cannot cross empty, so someone must row it back each time. What is the smallest number of crossings? (Each one-way trip counts as one crossing.)
Answer: 9
Two go over and one comes back, so each pair of crossings moves one person. After $4$ such pairs ( $8$ crossings) only $2$ people are left on the start bank, and they cross together: $8 + 1 = 9$ . This matches $2n - 3 = 2 \times 6 - 3 = 9$ .
12.
On the island of knights (always tell the truth) and knaves (always lie), A says: "At least one of us two is a knave." How many knights are there among A and B?
Answer: 1
1. Suppose A is a knave. Would A's sentence then be true? Write $1$ for true or $0$ for false. 1
2. A true sentence from a knave is impossible, so A is a knight and the sentence is true. What is B? Write $1$ for knight or $0$ for knave. 0
3. How many knights are there among A and B? 1
13.
Two whole numbers have a sum of $11$ and a product of $24$ . What is the smaller number?
Answer: 3
1. If the smaller number is $2$ , the other one is $9$ . What is $2 \times 9$ ? 18
2. If the smaller number is $3$ , the other one is $8$ . What is $3 \times 8$ ? 24
3. Which pair gives the product $24$ ? Enter the smaller number. 3
14.
Three boxes are labeled "Apples", "Oranges" and "Mixed". Every label is wrong. You may take one fruit out of one box without looking inside. From which box should you take it so that you can then label all three boxes correctly?
1. a)One fruit is never enough
2. b)The box labeled Apples
3. c)The box labeled Mixed
4. d)The box labeled Oranges
Answer: The box labeled Mixed
The box labeled Mixed is wrongly labeled, so it holds only apples or only oranges, and one fruit tells you which. Say it is an apple: that box is Apples. The box labeled Oranges cannot hold only oranges (its label is wrong) or only apples (that box is already found), so it is Mixed, and the box labeled Apples holds the oranges. From any other box one fruit could come from the mixed box and would not be enough.
15.
A two-digit number has a digit sum of $12$ . If you swap its two digits, the new number is $18$ more than the original. What is the original number?
Answer: 57
With tens digit $a$ and units digit $b$ : $(10b + a) - (10a + b) = 9(b - a) = 18$ , so $b - a = 2$ . Together with $a + b = 12$ this gives $a = 5$ and $b = 7$ . The number is $57$ ; check: $75 - 57 = 18$ .
16.
Ana, Ben, Cleo and Dan live in houses $1$ , $2$ , $3$ and $4$ , which stand in this order along a street. Ben's house number is twice Ana's. Cleo lives right next door to Ben. Dan's house number is odd. In which house does Dan live?
Answer: 1
1. Case 1: Ana lives in house $1$ . In which house does Ben live? 2
2. Houses $3$ and $4$ are left. Dan's number is odd. In which house does Dan live? 3
3. Cleo gets house $4$ . Is house $4$ next door to Ben's house $2$ ? Write $1$ for yes or $0$ for no. 0
4. Case 1 fails, so Ana is in $2$ and Ben in $4$ . Houses $1$ and $3$ are left. In which house does Cleo live? 3
5. In which house does Dan live? 1
17.
On the island of knights (always tell the truth) and knaves (always lie), A says: "B is a knave." B says: "C is a knave." C says: "A and B are both knaves." How many of the three are knights? In the steps, write $1$ for knight and $0$ for knave.
Answer: 1
1. Suppose A is a knight. Then what is B? 0
2. Then B lies about C. What is C? 1
3. C would be a knight, but C's sentence is false because A is a knight. So what is A really? 0
4. A lies about B. What is B? 1
5. B tells the truth about C. What is C? 0
6. C's sentence is false, which fits a knave. How many knights are there? 1
18.
Three adults and two children must cross a river. The boat can carry one adult, or one child, or two children, never more, and it cannot cross empty. What is the smallest number of crossings? (Each one-way trip counts as one crossing.)
Answer: 13
1. Both children cross, one child returns, an adult crosses, the other child returns. How many crossings is that? 4
2. Each adult needs this pattern once. How many crossings for all $3$ adults? 12
3. At the end the two children cross together. How many crossings in total? 13