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MP42: May 2004
The positive integers d and n are
chosen so that
(a) d divides
evenly into n, and
(b) nd +
1 divides evenly into n2 + d2.
What are the possible values of d and n?
We thank Andy Liu of the University of Alberta for
this month's problem.
MP41: April 2004
For which positive integers is n4 +
4n a prime number?
MP40: March 2004
Here's the way traffic lights might
work if a mathematician were running things. The red, yellow,
and green colours of a traffic light are controlled by three 3-position
switches. The three positions of each switch are labeled 0, 1,
and 2, and the switches are subject to two rules:
-
The displayed colour of the light
depends only on the positions of the switches.
-
If you change the positions of
all three switches at the same time, then the colour of the
light will change.
Initially the traffic light is red and all three
switches are in position 0.
Switch A |
Switch B |
Switch C |
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You flick switch A to position 1,
and the light changes to yellow.
Switch A |
Switch B |
Switch C |
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What will happen if you now move the
second switch to position 2?
Switch A |
Switch B |
Switch C |
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More generally, explain which positions of the switches correspond to what
colour of the traffic light?
This month's problem is based on an
introductory problem found in Graph Products, Fourier
Analysis, and Spectral Techniques, by Alon, Dinur, Friedgut,
and Sudakov.
MP39: February 2004
The February problem has two parts to
it:
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Show how to divide a square into
n nonoverlapping triangles of equal area when n is an even number.
-
Prove that it is impossible to divide
a square into 3 nonoverlapping triangles of equal area.
MP38: January 2004
Here is another dice problem for the holiday season. With an ordinary pair
of dice the sum distribution is
| Possible sums |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
| distribution of sums |
1 |
2 |
3 |
4 |
5 |
6 |
5 |
4 |
3 |
2 |
1 |
The table tells us that the sum 2 appears
in one way (as 1 + 1), 3 appears in two ways (as 1+2 and as 2+1),
and so forth. The problem for January is to find another way to put
numbers on the dice with the same sum distribution.
MP37: December 2003
In stores one finds perpetual
calendars that consist of a pair of cubes whose faces each
show a number from 0 to 8. The numbers are arranged so that
the two blocks can be placed together to represent all 31
days of a month. One block has the numbers 0 through 5 on
its faces, while the other has the numbers 0, 1, 2, 6, 7,
8. It is obvious that the numbers 0,
1, and 2 must appear on each
block (to accommodate the single-digit days and also days
11 and 22). But what about 9, 19, and 29? One needs here
the trick that the face numbered 6 serves also for 9.
It is said that we use a base-10
system for counting because we have ten fingers. Our problem
for this month is to use numbers in bases other than 10 to
design calendars for people with fewer than ten fingers.
In particular,
-
can you place base-2 numbers
on each of the 12 available faces on the two cubes to produce
all numbers from 1 to 31 (= 111112)? If you
do it economically there will be faces left over to accommodate
advertising?
-
is it also possible using
base 9?
What are base-2 numbers, you ask?
Their story is told on the Mathworld web page,
http://mathworld.wolfram.com/Binary.html
On that page you will find the
numbers form 1 to 30 written in base-2 notation.
MP36: November 2003
The town has four meter maids. They
work both a morning and an afternoon shift, and they all work
at about the same speed. The morning routes have lengths x1 > x2 > x3 > x4,
and the afternoon routes have lengths y1 > y2 > y3 > y4.
Any meter maid who works more than H hours has the extra hours
paid in overtime.
Incidentally, they all meet for lunch
at the local doughnut shop where the four cooks experience a
similar situation: To come to work they must park their scooters
in the street, where the parking meters have a time limit of
H hours. Their workday consists of baking chores of lengths x1 > x2 > x3 > x4,
followed by cleaning chores of lengths y1 > y2 > y3 > y4.
The shop pays the parking tickets of the cooks whose workday
has more than H hours.
Show that the best way to assign
a morning and afternoon route to each meter maid in order to
minimize overtime is x1+y4, x2+y3,
x3+y2, x4+y1. Is
that necessarily also the best assignment for the cooks as well?'
MP35: October 2003
What can be said about four points in space with
the property that every sphere or plane through the first two meets
every sphere or plane through the second two?
MP34: September 2003
Four players sit in a circle on chairs numbered
clockwise from one to four. Each player has two hats, one black and one
white, wearing one and holding the other. In the centre sits a fifth player
who is blindfolded. That player designates the chair numbers of those whose
hats should be changed. The goal is to get all four wearing a hat of the
same colour, in which case the game stops. Otherwise, after each guess
the four walk clockwise past an arbitrary number of chairs (maintaining
the same cyclic order), then sit for the next guess.
What is the best strategy to get all four hats
the same colour? Include a convincing argument why your strategy uses the
fewest possible number of guesses.
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