31. 
The following four geometric pieces are arranged two different ways to create two shapes that appear to have the same dimensions which would suggest that they should have the same area. This would be impossible though because the bottom shape has an extra unit. Where did the extra unit come from? 


ANSWER
If you look below the 4 shapes don't truly create a triangle They actually create a quadrilateral. In the top figure point B' is in the interior of the triangle shown and in the bottom figure point B is in the exterior of that triangle. So, there is a little space above and below the hypotenuse that actually creates what appears to be the room for the missing unit of space. As an extension could you prove the sliver of spaces created would actually have an area of exactly 1 square unit?


32. 
Consider the following puzzle pieces below. Cut out each of the 3 rectangles and then try to arrange the puzzle pieces so that it appears that both jockeys are riding horses at the same time. (You shouldn't need to fold or cut any of the 3 rectangles to make this happen. You should only need to 'arrange' the pieces face up.) 



ANSWER
Place the pieces in the way shown in the picture below and you will create a picture of two jockeys riding two galloping horses by using half of each horse to create a new horse.


33. 

The path to enlightenment lies behind one of two doors. In front of each door stands a guard who knows which door leads to enlightenment, but one of the guards always lies and the other one always tells the truth. In your search for enlightenment, you are allowed to ask one guard only one question that can be answered "yes" or "no", but unfortunately, you do not know which guard is the liar. You will be banished to the dungeon of logical illiteracy if you fail in your quest. What question should you ask to find the path to enlightenment? 


ANSWER
You can ask either guard "Would the other guard say that the door you stand in front of is the path to enlightenment?"
Possible Case #1 (Questioning a liar in front of the door to Enlightenment)
If the liar is the one to which you posed the question and the guard is standing in front of the door to enlightenment, the guard must answer "NO" because the guard would lie about the other guard saying yes.
Possible Case #2 (Questioning a truth teller in front of the door to Enlightenment)
If the truth teller is the one to which you posed the question and the guard is standing in front of the door to enlightenment, the guard must answer "NO" because the guard would tell the truth about the other guard saying no.
So regardless of who you ask, if they say "NO" they are standing in front of the door to enlightenment.
Possible Case #3 (Questioning a liar in front of the door to a dungeon)
If the liar is the one to which you posed the question and the guard is standing in front of the door to the dungeon, the guard must answer "YES" because the guard would lie about the other guard saying no.
Possible Case #4 (Questioning a truth teller in front of the door to a dungeon)
If the truth teller is the one to which you posed the question and the guard is standing in front of the door to the dungeon, the guard must answer "YES" because the guard would tell the truth about the other guard saying yes.
So regardless of who you ask, if they say "YES" they are standing in front of the door to the dungeon. 

34. 
The grandson is about as many days old as the son is in weeks. The grandson is approximately as many months old as the father is in years. The ages of the grandson, the son, and the father add up to 120 years. What are their ages? 


ANSWER
Grandson = 6 years old = 313 weeks old= 2191 days old
Son = 42 years old = 2191 weeks old = 15340 days old
Father = 72 years old = 3756 weeks old = 26298 days old 

35. 
A bookworm eats from the first page of an encyclopedia number 1 to the last page of encyclopedia number 5 all on a shelf as shown in the picture. (Book 1 is on the far left and book 5 is on the far right). The bookworm eats in a straight line. Each encyclopedia consists of 200 pages (i.e. 200 physical sheets of paper per book). Each page is 0.01 inches thick and each cover is 0.1 inches thick. What distance does the bookworm eat through? 


ANSWER
Since book 1 is on the left side, page 1 is located on the inner part of the stacking for that book. So, the worm only eats through one page and one cover of the first book and for similar reasons one page and one cover of the last book.
Book 1 (1 page & 1 cover) = 0.01 + 0.1 = 0.11 inches
Book 2 (entire book) = 0.01(200)+0.1(2) = 2.2 inches
Book 3 (entire book) = 0.01(200)+0.1(2) = 2.2 inches
Book 4 (entire book) = 0.01(200)+0.1(2) = 2.2 inches
Book 5 (1 page & 1 cover) = 0.01 + 0.1 = 0.11 inches
Total = 6.82 inches


36. 
Can you think of a reason based on the examples below, why all numbers might lead to 4? 


ANSWER
Each sequence is created by counting the number of letters in the previous numbers spelling and FOUR is the only number that describes the number of letters in it's own spelling.
(e.g. 10 is spelled TEN which has 3 letters in its spelling. So, 10 would lead to 3. 3 is spelled THREE which has 5 letters in its spelling. So, 10 would lead to 3 and 3 would lead to 5. Similarly, 5 would then lead to 4 but after that 4 would just keep repeating because FOUR has 4 letters in it's own spelling)


37. 
Given x^{2} + y^{2} = 36 and xy=32, what is the value of (x+y)? 

ANSWER
x^{2} + 2xy + y^{2} = 36 + 2(32)
x^{2} + 2xy + y^{2} = 100
(x+y)^{2} = 100
x + y = 10


38. 
What can you take from seven other than an odd number to make it even? 

ANSWER
If you remove the letter"s" the word seven becomes "even"


39. 
Two automobiles are approaching each other at a constant velocity of 60 mph. When the autos are two miles apart, a very fast fly leaves the front bumper of one of the autos and travels toward the other at a speed of 120 mph. Upon reaching that auto, the fly immediately reverses direction. This continues until the autos would collide (or, to make it less frightening, let's say narrowly miss colliding). How far did the fly travel? 

ANSWER
If we investigate the first part of the problem and determine just how much time it will take the 2 automobiles to meet, the problem can be simplified. Consider that the rate that the distance between the automobiles is decreasing at a rate of 120 mph since both are traveling at 60 mph towards each other.
D = R*T
2 = 120*T
So, it would take the cars 1/60 of an hour or a minute to reach each other. The fly will be traveling 120 mph for 1/60 of an hour.
D = 120 (1/60)
D = 2 miles


40. 
Can you rearrange exactly 4 coins to change the way the direction the arrow points?


ANSWER


41. 
Can you rearrange exactly 3 coins to change the way the direction the arrow points?


ANSWER


42. 
Using exactly four 4’s and any mathematical operators create an expression equal to 25. 

ANSWER


43. 
Using exactly three 0’s and any mathematical operators create an expression equal to 6. 

ANSWER



44. 
What is the number of the parking space containing the car?


ANSWER
The parking space numbers are just upside down 87.


45. 
Suppose that next week Northside Hospital delivers exactly 40 babies and East Side Medical only delivers 10 babies. If having a boy or girl is equally likely, which hospital is more likely to have exactly a 50/50 split between boys and girls being born? (Hint: The probability is not the same). 

ANSWER
East Side is more likely to have EXACTLY a 50/50 split.
 The probability of exactly a 50/50 split of boys and girls at Northside could be calculated as a binomial probability as
P(20 girls and 20 boys) = (_{40}C_{20})(0.5)^{20}(0.5)^{20} ≈0.125
 The probability of exactly a 50/50 split of boys and girls at East Side could be calculated as a binomial probability as
P(5 girls and 5 boys) = (_{10}C_{5})(0.5)^{5}(0.5)^{5} ≈0.246
The lower the number of total babies the higher the probability of having exactly a 50/50 split. A hospital that delivers 2 babies would have a 50% chance of exactly a 50/50 split. However, the Law of Large Numbers does suggest that the more deliveries a hospital has the more likely they are to be "CLOSE" to 50% but not EXACTLY 50/50.
. 
