Math Explorer's Club.
Puzzles - Antonio
Montalbán and Yannet
(January 13, 20, February 3, 10, 17 and March
Ithaca High School.
In this module I'll be
giving you many math puzzles of various kinds.
Some of them are easier than others. Some are very
hard. The idea is to try to solve the puzzles and
discuss the solution, or the possible solutions,
with other students. The first day we'll do
different kind of puzzles. In the following days
I'll choose a topic and discuss puzzles related to
Possible topics are:
Liars and Truth-tellers
The setting for
these problems is the following. There is this island in
the middle of the ocean where there are two kinds of
people: the liars and the truth-tellers. The liars always
lie. Any question you ask them will be answered with a
lie. The truth-tellers always answer the truth.
Here are a few examples.
There are more liars and truth-tellers
- You're walking in this island and you meet a native, who
could be either a liar or a truth-teller, and you want to
know which kind of person he is. So you ask him
- are you a truth-teller?
When he is answering, the volcano makes a loud noise and you
cannot hear the answer. So you ask him again
- Excuse me, I couldn't hear what you said, did
you say you were a truth-teller? - and he answers
- no, I didn't say that, I said I was a liar.
Is the native a liar or a truth-teller?
(Hint: Think about what the native's could have
answered to the first question, first assuming he's a
truth-teller and then assuming he's a liar.)
- Suppose now that you visit a second island close by. In
this second island the people is even more weird. They lie
on Tuesdays, Thursdays and Saturdays and they tell the truth
the other days. After spending so many days in these islands
you forgot which day it is. So you ask one of the islanders:
- Excuse me sir, what day is today?
- Saturday - he answers.
- And, what day is it going to be
Can you deduce what day is today?
- Now suppose you meet three people, A, B and C form the first island
(so they are either liars or truth-tellers). You ask A:
- How many truth-tellers are among
Again, the volcano makes a loud noise when A answers and you
can't hear the answer. So you ask B:
- What did A say? - and B answers
- A said that one of us is a
truth-teller and two of us are liars - and then C adds
- Don't believe what B said, he's lying.
What class of people are B
Knots and graphs.
I'll add more stuff here later.
- Draw three houses in a horizontal row. Draw three
utilities suppliers beneath them: Gas, Water, Electricity.
You should now have six points or boxes on your sheet of
paper. The challenge is to connect each house to each
utility supplier without any of the nine connection lines
crossing. Is it possible? (This problem is very hard!)
Arithmetic and geometry.
Here are a few examples (I'll add more later):
- What's wrong
with this picture? (Click to enlarge and translate.)
- In what
direction is the bike going? (click to enlarge)
- I have three clocks. One is a watch that I wear every day.
Then I have an alarm clock that goes back to 12:00am
(midnight) when there is a blackout, and I have an electric
clock in the kitchen which is not digital. The latter clock
stops working if there is a blackout and continues working
when the electricity comes back.
One day I went to sleep after midnight and when
I woke up the following morning , the three watches were
working but they all had different times: 7:45am, 8:15am and
9:30am. Probably it was because there was a blackout during
the night. Assuming that this is what happened, at what time
did the blackout started and when did it end?
- What is the last digit of the product of the numbers 1, 3,
5, 7, 9, 11,...., 2001, 2003 and 2005?
Solving the 3 x 3
x 3 Rubik's cube is very very hard. The smaller 2 x 2 x
2 is not easy either. I really don't expect anybody to
come up with a solution. I just expect you'll understand
the method I'll show you, which is not simple at all.
Note that to solve the 2 x 2 x 2 cube is
equivalent to solve the corners of the bigger cube. So
by learning this method, you're learning the beginning
of the method to solve the 3 x 3 x 3 cube.
We'll learn how to solve the 2x2x2
cube!!! Here is a
sketch of the solution.