Question 1

Find a so that the statement is true.  a = __________

Accepted Answer

The answer of Find a so that the statement is...

Question 2

Find a so that the statement is true.  a = __________

Accepted Answer

The answer of Find a so that the statement is...

Question 3

Find a so that the statement is true.  a = __________

Accepted Answer

The answer of Find a so that the statement is...

Question 4

Find a so that the statement is true.  a = __________

Accepted Answer

The answer of Find a so that the statement is...

Question 5

Suppose I have three bags, one with two peaches, another with two plums, and a third mixed bag with one peach and one plum. Now I give the bags (in mixed-up order) to Alice, Annie, and Nancy. I tell the three to look into their bags and that I want each to make a statement about the contents of her bag, but I want them to lie. Here is what they say: ​
Alice : I have two peaches.
Annie : I have two plums.
Nancy : I have one peach and one plum.
​
Now here is the game. I want you to develop a strategy of asking one of the three to reach into their bag and pull out one fruit and show it to you. The fruit is then returned to the bag and you ask another to do the same thing. Continue until you can deduce which bag is the mixed bag. What is the minimum possible number of necessary moves?
​
A) Only one question is necessary: Ask Nancy (who falsely claimed to have the mixed bag) to pull out one fruit. Suppose she pulls out a peach; this means she has the bag containing two peaches. Then, Alice, who falsely claimed two peaches, must have two plums. This leaves Annie with the mixed bag. Suppose she pulls out a plum; this means she has the bag containing two plums. Then, Annie, who falsely claimed two plums, must have two peaches. This leaves Alice with the mixed bag.
​
B) Only one question is necessary: Ask Annie (who falsely claimed to have the mixed bag) to pull out one fruit. Suppose she pulls out a peach; this means she has the bag containing two peaches. Then, Alice, who falsely claimed two peaches, must have two plums. This leaves Nancy with the mixed bag. Suppose she pulls out a plum; this means she has the bag containing two plums. Then, Nancy, who falsely claimed two plums, must have two peaches. This leaves Alice with the mixed bag.
​
C) Only one question is necessary: Ask Nancy (who falsely claimed to have the mixed bag) to pull out one fruit. Suppose she pulls out a peach; this means she has the bag containing two peaches. Then, Annie, who falsely claimed two peaches, must have two plums. This leaves Alice with the mixed bag. Suppose she pulls out a plum; this means she has the bag containing two plums. Then, Alice, who falsely claimed two plums, must have two peaches. This leaves Annie with the mixed bag.
​

Accepted Answer

By asking Nancy to pull out one fruit, you can determine her actual bag. If she pulls out a peach, she has the two-peach bag, making Alice's statement false, indicating she has two plums, leaving Annie with the mixed bag. If Nancy pulls out a plum, she has the two-plum bag, making Annie's statement false, indicating she has two peaches, leaving Alice with the mixed bag.

Question 6

Find a so that the statement is true. 
A) 7
B) 4
C) 5
D) 6
E) 0

Accepted Answer

The answer of Find a so that the statement is...

Question 7

Find a so that the statement is true.  a = __________

Accepted Answer

The answer of Find a so that the statement is...

Question 8

Suppose you are given 12 coins, one of which is counterfeit and weighs a little more or less than the real coins. Using a two-pan balance scale, what is the minimum number of weighings necessary to find the counterfeit coin? ​
A) 7 weighings
B) 3 weighings
C) 1 weighings
D) 4 weighings
E) 6 weighings

Accepted Answer

The solution involves dividing the 12 coins into three groups of 4 coins each, and weighing two of the groups against each other. If the scales balance, the counterfeit coin must be in the third group. We can then take that third group and divide it into two groups of 2 coins each and weigh them against each other. If the scales balance, the counterfeit coin must be one of the other two coins, and we can weigh them against each other to find the counterfeit coin in a total of 3 weighings. If the scales do not balance the third group, we pick the heavier group, and then divide it into two groups of 2 coins each and weigh them against each other. If the scales balance, the counterfeit coin must be one of the other two coins in the remaining group, and we can weigh them against each other to find the counterfeit coin in a total of 3 weighings. If the scales do not balance, we pick the heavier group, and then weigh any two of the coins against each other. If the scales balance, the remaining coin is the counterfeit coin, and we find it in a total of 4 weighings. If the scales do not balance, we can determine which coin is the counterfeit by weighing any two of the three coins on one pan against any two of the three coins on the other pan. This will leave us with two possibilities for the counterfeit coin, and we can determine which one it is by weighing it against any real coin. This process takes a total of 3 weighings (one to isolate the third group, one to isolate the heavier subgroup, and one to isolate the counterfeit coin). Therefore, the minimum number of weighings necessary is 3.

Question 9

Find a so that the statement is true. 
A) 6
B) 0
C) 5
D) 8
E) 7

Accepted Answer

The answer of Find a so that the statement is...

Question 10

A fox, hunting for a morsel of food, spotted a huge bear about 175 yards due east of him. Before the hunter could become the hunted, the crafty fox ran due north for 175 yards but then realized the bear had not noticed him. Thus he stopped and remained hidden. At this point the bear was due south of the fox. What was the color of the bear? ​
A) Purple
B) White
C) Brown
D) Black
E) Green

Accepted Answer

The bear is white because the scenario described (with the bear due south of the fox after the fox runs north) could only happen at the North Pole, where polar bears, which are white, live.

Question 11

Find a so that the statement is true.  a = __________

Accepted Answer

The answer of Find a so that the statement is...

Question 12

Find a so that the statement is true. 
A) 6
B) 5
C) 0
D) 3
E) 4

Accepted Answer

The answer of Find a so that the statement is...

Question 13

I visited a beautiful flower garden yesterday and counted exactly 50 flowers. Each flower was either white or orange, and the flowers were not all the same color. My friend made the following observation: No matter which two flowers you might have picked, at least one was bound to be white. From this can you determine how many were white and how many were orange? ​
A) 1 orange flower, 49 white flower(s)
B) 1 white flower, 49 orange flower(s)
C) 32 orange flower(s), 18 white flower(s)
D) 24 orange flower(s), 26 white flower(s)
E) 26 orange flower(s), 24 white flower(s)

Accepted Answer

Given the observation that among any two flowers picked, at least one is white, it implies there can be at most one orange flower to satisfy this condition. Thus, there must be 49 white flowers and 1 orange flower.

Question 14

Four boys were playing marbles; their names were Gary, Henry, John, and Alice. One of the boys had 9 marbles, another had 15, and each of the other two had 12. Their ages were 3, 10, 17, and 18, but not respectively. Gary shot before Henry and Alice. Alice was older than the boy with 15 marbles. Henry had fewer than 15 marbles. Alice shot before Henry. John shot after Henry. If John was 10 years old, he did not have 15 marbles. Gary and Alice together had an even number of marbles. The youngest boy was not the one with 15 marbles. If Henry had 12 marbles, he wasn't the youngest. The 10-year-old shot after the 17-year-old. How old was each boy? ​
A) John - 3, Henry - 17, Gary - 10, Alice - 18.
B) John - 10, Henry - 3, Gary - 17, Alice - 18.
C) John - 17, Henry - 18, Gary - 3, Alice - 10.
D) John - 3, Henry - 10, Gary - 17, Alice - 18.
E) John - 17, Henry - 10, Gary - 3, Alice - 18.

Accepted Answer

From the clues given, we can deduce the following:
- One of the boys had 9 marbles, another had 15, and each of the other two had 12.
- Gary shot before Henry and Alice.
- Alice was older than the boy with 15 marbles.
- Henry had fewer than 15 marbles.
- Alice shot before Henry.
- John shot after Henry.
- If John was 10 years old, he did not have 15 marbles.
- Gary and Alice together had an even number of marbles.
- The youngest boy was not the one with 15 marbles.
- If Henry had 12 marbles, he wasn't the youngest.
- The 10-year-old shot after the 17-year-old.

From the first clue, we can determine that the boy with 9 marbles must be one of John or Henry. This eliminates Gary and Alice from being either of the boys with 9 marbles.

From the second and fifth clues, we know that Gary and Alice come before Henry, so neither of them can have 9 marbles. This means that either John or Henry has 9 marbles.

From the third clue, we know that Alice is older than the boy with 15 marbles. This eliminates John from being the boy with 15 marbles, since he is only 10 years old.

From the fourth clue, we know that Henry has fewer than 15 marbles. Since Gary and Alice each have 12 marbles, Henry must have 9 or 12 marbles. From the eleventh clue, we know that if Henry has 12 marbles, he wasn't the youngest, so he must have 9 marbles. This means that John has 12 marbles.

From the sixth clue, we know that John comes after Henry, so John must be either 17 or 18 years old. From the tenth clue, we know that the 10-year-old shot after the 17-year-old, so John must be 18 years old.

From the seventh clue, we know that the 10-year-old did not have 15 marbles, which means that John does not have 15 marbles. This means that either Gary or Alice has 15 marbles.

From the eighth clue, we know that Gary and Alice together have an even number of marbles. Since we know that one of them has 15 marbles, this must mean that the other has 9 marbles (since John and Henry each have 12 and the sum of their marbles is even).

From the ninth clue, we know that the youngest boy did not have 15 marbles, which means that Gary must be the youngest at 3 years old. This means that Alice is 18 years old.

Therefore, the ages of the boys are:
John - 18
Henry - 9
Gary - 3
Alice - 18

The answer is D.

Question 15

I have a habit of getting up before the sun rises. My socks are all mixed up in the drawer, which contains 20 blue and 18 black socks. I reach into the drawer and grab some socks in the dark. How many socks minimum do I need to take from the drawer to be sure that I have a matched pair? ​
A) 18 socks
B) 21 socks
C) 2 socks
D) 3 socks
E) 19 socks

Accepted Answer

The worst case scenario is that you keep picking up socks of different colors until you have 17 blue socks and 17 black socks. At this point, the next sock you pick up will complete a pair, so you only need to take one more sock for a total of 3 socks. Therefore, the minimum number of socks you need to take from the drawer to be sure you have a matched pair is 3.

Question 16

Find a so that the statement is true. 
A) 8
B) 0
C) 7
D) 5
E) 6

Accepted Answer

The answer of Find a so that the statement is...

Question 17

Find a so that the statement is true. 
A) 0
B) 4
C) 7
D) 5
E) 6

Accepted Answer

The answer of Find a so that the statement is...

Question 18

I have a habit of getting up before the sun rises. My socks are all mixed up in the drawer, which contains 8 blue and 18 grey socks. I reach into the drawer and grab some socks in the dark. How many socks do I need to take from the drawer to be sure that I have a matched pair?
​
__________ socks

Accepted Answer

The answer of I have a habit of getting up...

Question 19

Find a so that the statement is true.  a = __________

Accepted Answer

The answer of Find a so that the statement is...

Question 20

Find a so that the statement is true.  a = __________

Accepted Answer

The answer of Find a so that the statement is...

Quiz 3: The Nature of Logic

Find a so that the statement is true. a = __________

Find a so that the statement is true. a = __________

Find a so that the statement is true. a = __________

Find a so that the statement is true. a = __________

Find a so that the statement is true.

Find a so that the statement is true. a = __________

Suppose you are given 12 coins, one of which is counterfeit and weighs a little more or less than the real coins. Using a two-pan balance scale, what is the minimum number of weighings necessary to find the counterfeit coin?

Find a so that the statement is true.

Find a so that the statement is true. a = __________

Find a so that the statement is true.

I have a habit of getting up before the sun rises. My socks are all mixed up in the drawer, which contains 20 blue and 18 black socks. I reach into the drawer and grab some socks in the dark. How many socks minimum do I need to take from the drawer to be sure that I have a matched pair?

Find a so that the statement is true.

Find a so that the statement is true.

I have a habit of getting up before the sun rises. My socks are all mixed up in the drawer, which contains 8 blue and 18 grey socks. I reach into the drawer and grab some socks in the dark. How many socks do I need to take from the drawer to be sure that I have a matched pair? __________ socks

Find a so that the statement is true. a = __________

Find a so that the statement is true. a = __________

Quiz 3: The Nature of Logic

Find a so that the statement is true. a = __________

Find a so that the statement is true. a = __________

Find a so that the statement is true. a = __________

Find a so that the statement is true. a = __________

Find a so that the statement is true.

Find a so that the statement is true. a = __________

Suppose you are given 12 coins, one of which is counterfeit and weighs a little more or less than the real coins. Using a two-pan balance scale, what is the minimum number of weighings necessary to find the counterfeit coin? ​

Find a so that the statement is true.

Find a so that the statement is true. a = __________

Find a so that the statement is true.

I have a habit of getting up before the sun rises. My socks are all mixed up in the drawer, which contains 20 blue and 18 black socks. I reach into the drawer and grab some socks in the dark. How many socks minimum do I need to take from the drawer to be sure that I have a matched pair? ​

Find a so that the statement is true.

Find a so that the statement is true.

I have a habit of getting up before the sun rises. My socks are all mixed up in the drawer, which contains 8 blue and 18 grey socks. I reach into the drawer and grab some socks in the dark. How many socks do I need to take from the drawer to be sure that I have a matched pair? ​ __________ socks

Find a so that the statement is true. a = __________

Find a so that the statement is true. a = __________

Suppose you are given 12 coins, one of which is counterfeit and weighs a little more or less than the real coins. Using a two-pan balance scale, what is the minimum number of weighings necessary to find the counterfeit coin?

I have a habit of getting up before the sun rises. My socks are all mixed up in the drawer, which contains 20 blue and 18 black socks. I reach into the drawer and grab some socks in the dark. How many socks minimum do I need to take from the drawer to be sure that I have a matched pair?

I have a habit of getting up before the sun rises. My socks are all mixed up in the drawer, which contains 8 blue and 18 grey socks. I reach into the drawer and grab some socks in the dark. How many socks do I need to take from the drawer to be sure that I have a matched pair? __________ socks