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The statement “Raphael runs every Sunday” is always true. Which of the following statements is also true?

a. If Raphael does not run, then it is not Sunday.

b. If Raphael runs, then it is Sunday.

c. If it is not Sunday, then Raphael does not run.

d. If it is Sunday, then Raphael does not run.

e. If it is Sunday, it is impossible to determine if Raphael runs.



If the statement “No penguins live at the North Pole” is true, which of the following statements must also be true?

a. All penguins live at the South Pole.

b. If Flipper is not a penguin, then he lives at the North Pole.

c. If Flipper is not a penguin, then he does not live at the North Pole.

d. If Flipper does not live at the North Pole, then he is a penguin.

e. If Flipper lives at the North Pole, then he is not a penguin.


If the statement “All students take the bus to school” is true, then which of the following must be true?

a. If Courtney does not take the bus to school, then she is not a student.

b. If Courtney takes the bus to school, then she is a student.

c. If Courtney is not a student, then she does not take the bus.

d. all of the above

e. none of the above



All of Mark’s former students go to college.

12. If the statement above is true, which of the following must also be true?

(A) If Ethan was not Mark’s student, then he is not going to college.

(B) If Joyelle goes to college, then she was not Mark’s student.

(C) If Ginger goes to college, then she was Mark’s student.

(D) If Stephanie was Mark’s student, then she is not going to college.

(E) If Steve does not go to college, then he was not Mark’s student.

Gregory must inspect 12 working devices, labeled alphabetically from A to L, that are arranged in a linear array. He must start with device A and proceed alphabetically, returning to the beginning and repeating the process after inspecting device L, stopping when he encounters a defective device. If the first defective device he encounters is device D, which of the following could be the total number of devices that Gregory inspects, including the defective one?

(A) 64

(B) 68

(C) 72

(D) 74

(E) 78



14. In the correctly worked addition problem above, A and B represent digits. What is digit A?

(A) 5

(B) 6

(C) 7

(D) 8

(E) 9


15. The operation ⊗ is defined for all nonzero numbers a and b by a ⊗ b = a/b – b/a. If x and y are nonzero numbers, which of the following statements must be true?

I. x ⊗ (xy) = x(1 ⊗ y)

II. x ⊗ y = -(y ⊗ x)

III. 1/x ⊗ 1/y = y ⊗ x



A. I only

B. II only

C. III only

D. I and II

E. II and III



Logic Problems Without Answer-Choices.



16. Imagine a long thin strip of paper stretched out in front of you, left to right. Imagine taking the ends in your hands and placing the right hand end on top of the left. Now press the strip flat so that it is folded in half and has a crease. Repeat the whole operation on the new strip two more times. How many creases are there?


17. A number like 12321 is called a palindrome because it reads the same backwards as forwards. A friend of mine claims that all palindromes with four digits are exactly divisible by 11. Are they?



18. Three slices of bread are to be toasted under a grill. The grill can hold two slices at once but only one side is toasted at a time. It takes 30 seconds to toast one side of a piece of bread, 5 seconds to put a piece in or take a piece out and 3 seconds to turn a piece over. What is the shortest time in which the three slices can be toasted?



19. Ross collects lizards, beetles and worms. He has more worms than lizards and beetles together. Altogether in the collection there are 12 heads and 26 legs. How many lizards does Ross have?



20. Male bees hatch from unfertilized eggs and so have a mother but no father. Female bees hatch from fertilized eggs. How many ancestors does a male bee have in the twelfth generation back?



21. There are five statements written in a list:

"There is only 1 false statement in the list."

"There are only 2 false statements in the list."

"There are only 3 false statements in the list."

"There are only 4 false statements in the list."

"There are only 5 false statements in the list."


Which of the statements is true?

22. A Friday the thirteenth is known as a black Friday. What is the most/least number of black Fridays you can get in one year? In a 12-month period?



23. I wish to share 30 identical individual sausages equally amongst 18 people. What is the minimum number of cuts I need to make?



24. A woman on her way to market, when asked how many eggs she had, replied that, taken in groups of 11, 5 would remain over, and taken in groups of 23, 3 would remain over. What is the least number of eggs that she could have had? On another occasion she replied that taken in groups of 2, 3, 4, 5, 6 and 7 there would remain over 1, 2, 3, 4, 5 and no eggs respectively.



25. How much cardboard do you need to make a carton to hold 1 litre of milk?



26. Amongst nine apparently identical cricket balls, one is lighter than the rest which all have the same weight. How quickly can you guarantee to find the light ball using only a makeshift balance?



27. Which numbers have an odd number of divisors?



28. Take any two numbers that sum to one. Square the larger and add the smaller. Square the smaller and add the larger. Which do you expect will be larger?



29. Write down a sequence of 0s and 1s. Underneath each consecutive pair write a 0 if they are the same and a 1 if not. Repeat this process until you are left with a single digit. Can you predict what the final digit will be?



30. It is rumoured in some countries that the police will not stop you for speeding unless you are going at least 10% over the limit. One such country recently changed from miles to kilometres on all road signs. What is the new rule of thumb?



31. If a population alternately grows and shrinks by 10% each month, what happens in the long run?



32. An old, broken-down car has to travel a two-mile route, up and down a hill. Because it is so old, the car travels the first mile (the ascent) at an average of 15 mph. How fast must it go so as to achieve 30 mph for the whole journey?



33. Winnie-the-Pooh and Piglet went to visit each other. They started at the same time and walked along the same path. However, Pooh was absorbed in a new ‘hum’ and Piglet was counting birds overhead so they walked right past each other without noticing. One minute after they met, Pooh was at Piglet’s and three minutes after that Piglet reached Pooh’s. How long had they each walked?



34, In a list of positive integers, the arithmetic mean is 5, and the number 16 is known to appear. If the 16 is removed, the mean drops to 4. What is the largest possible number that could occur in the original list, and how many numbers were in that list?






Assume that the above subtraction problem with digits P and R is without error. If P and R are not equal to each other, how many distinct digits from 0 to 9 could R symbolize?

A. One

B. Four

C. Five

D. Nine

E. Ten



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