Problem of The Month From Math Central Uregina (4)

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.

Solution to May 2004 problem

MP41: April 2004

For which positive integers is n⁴ + 4ⁿ a prime number?

Solution to April 2004 problem

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:

1. The displayed colour of the light depends only on the positions of the switches.
   
2. 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

You flick switch A to position 1, and the light changes to yellow.

Switch A

Switch B

Switch C

What will happen if you now move the second switch to position 2?

Switch A

Switch B

Switch C

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.

Solution to March 2004 problem

MP39: February 2004

The February problem has two parts to it:

1. Show how to divide a square into n nonoverlapping triangles of equal area when n is an even number.
   
2. Prove that it is impossible to divide a square into 3 nonoverlapping triangles of equal area.

Solution to February 2004 problem

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.

Solution to January 2004 problem

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,

1. 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?
2. 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.

Solution to December 2003 problem

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+y_1. Is that necessarily also the best assignment for the cooks as well?'

Solution to November 2003 problem

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?

Solution to October 2003 problem

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.

Solution to September 2003 problem