Below are a
series of logic problems. (You may
recognize some of them from Wednesday’s class.) Try as many as you want and submit written answers to those
you are able to do. I will award marks to
you in the following manner:
65% for one solved puzzle
75% for two or three
85% for four through six
95% for more than six
100% for whoever gets the most correct solutions
110% for whoever finds a puzzle undoable by anyone else in the class but which they themselves can do
You must provide
a clear and complete explication of the puzzle(s) to receive credit. If you want to find more logic puzzles than
those below just look at this site
and/or do a search on the internet…
1. Take two large
beakers A and B of equal size. Pour wine into A and water into B until they are
both half-full. Now pour some water from B into A, thoroughly stir the contents
of A so as to obtain a homogeneous mixture, and pour the same amount of mixture
from A into B as you did water from B into A. Which is greater now: the
quantity of water in A or the quantity of wine in B?
2. You are given
nine marbles of identical appearance. All, except one, have the same weight.
You are also given a pair of scales with which to weigh. In three weighings or
fewer, pick out the marble with the different weight. (Now try to solve the
same problem with 12 marbles.)
3. Generalize the
previous problem to the case of n marbles. What is the minimum number of
weighings that will always pick out the distinguished marble?
4. Can you
subdivide a cube into a finite number of smaller cubes, no two of which have
the same size?
5. Take any
collection of 101 natural numbers between 1 and 200 (inclusive). Show that
there must be at least one number in the collection that divides another one in
the collection.
6. You are given
three buckets. One has a capacity of three litres, another a capacity of five
litres, while the third has a capacity of eight litres. The eight-litre bucket
is filled with water. By doing nothing other than transferring water between
the buckets, can you have at least one of the buckets contain exactly four
litres of water?
7. On a certain
TV game show, a prize is hidden behind one of three closed doors. The
contestant must guess the correct door to win the prize. He starts by selecting
a door. As soon as he has done this, the host of the show, who knows behind
which door the prize is hidden, opens one of the remaining two doors to show
the contestant that the prize is not there and gives the contestant the option
of changing his choice. Should the contestant change his initial selection?
8. Can the
product of four consecutive positive integers ever be a perfect square?
9. How is this
possible?
10. A gold brick
in a row boat on a lake is thrown overboard. Does the level of water in the
lake rise or fall as a result?
11. Prove that
given any group of six people it is always possible to find either a subgroup
of three people such that they all know each other, or a subgroup of three
people such that each member of the subgroup is a complete stranger to the
others.
12. (The famous
prisoner puzzle.) A prisoner is held in jail. There are two doors to his prison
cell; one of them leads to freedom, while the other leads to death. Each door
is guarded by a watchman. One of the watchmen always tells the truth, while the
other always lies. However, the prisoner does not know which watchman is the
honest one. If he is allowed to ask only one question, whose answer must be
either "yes" or "no," in order to determine the door to
freedom, what should be the question?
13. Same problem
as the previous one, except that there is only one watchman, and he sometimes
lies and at other times tells the truth.
14. Can you find
1000 consecutive positive integers such that none of them is prime? What about
1,000,000?
15. Consider the
sequence {an} = {9, 98, 987, 9876, ..., 987654321, 9876543219, 98765432198,
987654321987, ...}. How many prime numbers does {an} contain?
16. Four bugs are
placed at the corners of a unit square. Each bug always walks directly towards
the next bug in the clockwise direction. What is the total distance travelled
by the bugs before they meet?
17. On a sheet of
paper there are 100 statements. The nth statement reads: "The are exactly
n false statements on this sheet." (So, the first statement says,
"There is exactly one false statement on this sheet,"; the second
statement says, "There are exactly two false statements on this
sheet,"; and so on.) Which statements on the sheet are true and which are
false? What if we replace "exactly" by "at least" or
"at most"?
18. There are 4 men on one side of a narrow bridge. At most 2 men can
cross at a time, and since it is night and the bridge is narrow and
rickety, they must cross with a flashlight. Unfortunately there is
only 1 flashlight, and all 4 must get across the bridge in at most
17 minutes.
However, the men take the following lengths of time to cross the
bridge:
Man No. 1 takes 1 minute
Man No. 2 takes 2 minutes
Man No. 3 takes 5 minutes
Man No. 4 takes 10 minutes
When two men walk together, they walk at the pace of the slower one.
So if No. 1 and No. 4 cross together, it will take them 10 minutes.
How can they do it and all 4 get across in no more than 17 minutes?
19. What
six-digit number, with six different digits, when multiplied by one through
six, circulates its digits as follows:
ABCDEF x 1 =
ABCDEF
ABCDEF x 3 =
BCDEFA
ABCDEF x 2 =
CDEFAB
ABCDEF x 6 =
DEFABC
ABCDEF x 4 =
EFABCD
ABCDEF x 5 =
FABCDE
20. (The Allais
Paradox) The Allais paradox involves making a choice between the following
alternatives:
Alternative A:
89% chance of
winning an unknown amount of money;
10% chance of
winning $1,000,000;
1% chance of
winning $1,000,000.
Alternative B:
89% chance of
winning an unknown amount (the same amount as in A);
10% chance of
winning $2,500,000;
1% chance of
winning nothing.
Which is the
rational choice? Does this choice remain the same if the unknown amount is now
known to be $1,000,000? What if the unknown amount is zero?
21. How many
times per day does a digital clock display a palindromic number?
22. A clock has
only the hour and minute hands, but no second hand. How many times a day can
the two hands be switched so that they still signify a valid time?
23. How many
times a day do the hour and minute hands of a clock form a right-angle?
24. Find all
series of consecutive positive integers whose sum is exactly 10,000.
25. If a teacher
puts 33 students into each bus, then one student is left unassigned to any bus.
If she puts more students in each bus, but always the same number of students
in each bus, then she has one bus left unused. How many buses and students are
there?
26. You have a
set of 11 integers with the following peculiar property: one can choose any one
of them, and divide the remaining ten into two groups of five, such that the
sum of the numbers of the first group equals that of the second. Show that the
11 integers must be identical.
27. The same as
the previous problem, except that the numbers are not necesarily integers.
28. A farmer, a
goat, a wolf, and a cabbage have to cross a river. A boat nearby only has
enough room for the farmer and one other thing. What is the fewest number of
trips he must take so that the goat does not eat the cabbage, and so the wolf
does not eat the goat?
29. A farmer is
taking his prize lettuce, lion, llama, and pet leviathan (a unique creature
that eats lions unless a lettuce is present) to market. A boat
awaits him to cross a river, but he can only take one item at a time. How can
the farmer cross the river, without the llama eating the lettuce, the lion
eating the llama, and the leviathan eating the lion?
30. There are
three people in front of you. You know that:
One of them is
God. He knows everything, and always tells the truth. One of them is the Devil.
He also knows everything, but lies. The
third person knows nothing, but answers questions as if he knows the answers.
His answers, however, are completely useless and could be right or wrong.
You can ask a
total of three questions that can correctly be answered with yes or no, each to
one of the persons. You may choose whom to ask.
Determine who is
who...