MA 490-505 Spring 2014
Logical Problems
Problem 1 Pete’s cat always sneezes before it rains. She sneezed today. “This means it will
be raining,” Pete thinks. Is he right?
Problem 2 A sum of fifteen cents was paid by two coins, and one of these coins was not a
nickel. Find the values of the coins.
Problem 3 A man whose clock had stopped running wound it up, but did not have access
to the correct time to reset it. Leaving the clock at home, he walked to the home of a friend
whose clock was correct, stayed for some time and then walked home (in the same time as he
took earlier). Upon arriving at home, he set his clock to the correct time even though he did
not know how long he had walked! Explain.
Problem 4 During a period of days, it was observed that when it rained in the afternoon, it
had been clear in the morning, and when it rained in the morning, it was clear in the afternoon.
It rained on 9 days, and was clear on 6 afternoons and 7 mornings. How long was this period?
Problem 5 During a trial in Wonderland the March Hare claimed that the cookies were stolen
by the Mad Hatter. Then the Mad Hatter and the Dormouse gave testimonies which, for some
reason, were not recorded. Later on in the trial it was found out that the cookies were stolen
by only one of these three defendants, and, moreover, only the guilty one gave true testimony.
Who stole the cookies?
Problem 6 Assume that a following statements are true:
a) among people having TV sets there are some who are not mathematicians;
b) non-mathematicians who swim in swimming pools every day do not have TV sets.
Can we claim that not all people having TV sets swim every day?
Problem 7 In a box, there are pencils of two different colors, and of two different sizes. Prove
that there are two pencils that differ both in color and in size.
Problem 8 There are three urns containing balls: the first one contains two white balls, the
second–two black balls, and the third–a white ball and a black ball. The labels WW, BB, and
WB were glued to the urns so that the contents of no urn correspond to its label. Is it possible
to choose one urn so that after drawing a ball from it one can always determine the contents
of each urn?
Problem 9 Three people–A, B, and C–are sitting in a row in such a way that A sees B and
C, B sees only C, and C sees nobody. They were shown 5 caps–3 red and 2 white. They were
blindfolded, and three caps were put on their heads. Then the blindfolds were taken away and
each of the people was asked if they could determine the color of their caps. After A, and then
B, answered negatively, C replied affirmatively. How was that possible?
Problem 10 Three friends–sculptor White, violinist Black, and artist Redhead– met in a
cafeteria. “It is remarkable that one of us has white hair, another one has black hair, and the
third has red hair, though no one’s name gives the color of their hair”, said the black-haired
person. “You are right”, answered White. What color is the artist’s hair?
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Problem 11 Three friends are sitting around a table having lunch and discussing their common acquaintance.
Bob said: “Tom has more then thousand books”.
Ted said: “Tom has less then thousand books”.
Jim said: “Tom has at least one book”.
If only one of these statement is true, how many books has Tom?
Problem 12 Four suspects of a crime made the following statements to the police:
Andy: Carl did it.
Bob: I did not do it.
Carl: Dave did it.
Dave: Carl lied when he said I did it.
(a) Given that exactly one of the four statements is true, determine who did it.
(b) Given that exactly one of the four statement is false, determine who did it.
Problem 13 One of five brothers had broken a window. John said, “It was Henry or Thomas.”
Henry said, “Neither Ernest nor I did it.” Thomas said, “You are both lying.” David said, “No,
one of them is speaking the truth, but not the other.” Ernest said, “No, David, that is not
true.” When the father of the boys (a truthful man), was consulted, he said, “Three of my sons
always speak the truth, but I am afraid you cannot always rely on what the other two say.”
Who broke the window?
Problem 14 Four drivers–Bob, Tom, Peter, and Scott–decided to have a race. After the race
they said:
Bob: I did not win the race and I did not finish 4th.
Tom: I did not finish 4th.
Peter: I won the race.
Scott: I finished 4th.
Given that exactly one of the four statements is false, determine who won the race and who
lied.
Topics are taken from Problem-Solving Strategies by Arthur Engel, 2007; and Mathematical
Circles: Russian Experience by Dmitri Fomin, Sergey Genkin, Ilia V. Itenberg, 1996
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