Problem of The Month From Math Central Uregina (3)

MP33: May 2003

Given a triangle and an infinite blue and white checkerboard, show that the triangle can be placed on the board with all its vertices strictly in the blue.

Solution to May 2003 problem

MP32: April 2003

Below is a list of ten statements, numbered from 1 to 10, that enable you to determine a number n. Unfortunately, some of these statements happen to be false.

1. At least one of the last two statements on this list is true.
2. Either this is the first true statement, or it is the first false statement on the list.
3. This list contains at least three consecutive false statements.
4. The difference between the number of the last true statement and the
   number of the first true statement is a divisor of n.
5. The sum of the numbers of the true statements equals n.
6. This is not the last true statement.
7. The number of each of the true statements is a divisor of n.
8. Exactly n% of the statements on this list are true.
9. The number of divisors of n that are bigger than 1 and less than n is
   greater than the sum of the numbers of the true statements.
10. This list does not contain three consecutive true statements.

The problem for April: Find n

Solution to April 2003 problem

MP31: March 2003

The faculty of the Even University of Regina considers serving on committees to be their main duty. Their faculty association passed a rule requiring that

1. each committee have an EVEN number of members,
2. each pair of committees share an EVEN number of members, and
3. no two committees are allowed to have identical membership.

Their cross-town rival is the Odd University of Regina whose faculty likewise spends most of their day serving on committees. Their rules are a bit different:

1. each committee has an ODD number of members, and
2. each pair of committees share an EVEN number of members.

The problem for March: Even University has only 20 faculty members, while Odd University has 1000 members. Odd University tries to form as many committees as it can following its own rule, but still comes short of the number of committees at Even University. How is this possible?

Solution to March 2003 problem

MP30: February 2003

Note that 3 = 1+2, 5 = 2+3, 6 = 1+2+3, 7 = 3+4, but neither 4 nor 8 can be written as the sum of two or more CONSECUTIVE positive integers. Your task for February is to show that this pattern continues forever:

1. Show that no power of 2 can be written as the sum of two or more consecutive positive integers.
   
2. Show that any integer that is not a power of 2 can be written as the
   sum of consecutive integers.

Solution to February 2003 problem

MP29: January 2003

An L-shaped tromino is a piece consisting of three squares, two of which are attached to adjacent sides of the third.

You are given a 2ⁿ by 2ⁿ grid of squares, and uncle Albert places a penny on one of them.

Your job is to cover the remaining 2²ⁿ – 1 squares with (2²ⁿ – 1)/3 trominoes. Can you do it?

Solution to January 2003 problem

MP28: December 2002

Complete the following sentence by filling in each blank with a numeral of one or more digits.

In this sentence, the number of occurrences of 0 is ____, of 1 is ____, of 2 is ____, of 3 is ____, of 4 is ____, of 5 is ____, of 6 is ____, of 7 is ____ of 8 is ____, and of 9 is ____.

The sentence was devised by Raphael Robinson, who proved that there are exactly two correct ways to fill in the blanks.

Solution to December 2002 problem

MP27: November 2002

The Kolakoski sequence 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, ... is an example of what people call a self-reading sequence: {a_n} is defined to be the sequence of 1's and 2's whose first term is 1, and each subsequent term an is the length of the nth run (of ones or twos). In more detail,

sequence	1,	2,2,1,1,	2,1,	2,2,	1,	2,2,	...
run length	1	22	11	2	1	2

Start with a₁ = 1. The rule says that the first run (which is a single 1) has length 1.

Thus a₂ must be different, so that a₂ = 2. The second run therefore has length 2, which forces the third term, a₃, to be a two also. This completes the second run, so the third run begins with a 1; since its length is 2 we have a₄ = 1 and a₅ = 1. The fourth and fifth runs are consequently the singletons 2 then 1. And so on.

Problem for November: Prove that 0.122112122122112... is irrational. (Recall that a number in decimal form is irrational if its digits are nonrepeating.)

Solution to November 2002 problem

 
MP26: October 2002

Given a set of points in the plane such that

1. the distance between any two of them is an integer, and
2. infinitely many of them lie on the same line,

prove that all points in the set lie on that line.

What happens if we ask only that 5 points lie on the same line?

Solution to October 2002 problem

 
MP25: September 2002

Each person in a group of seven people speaks at most two languages, while among every three of them there are at least two who can communicate. Prove that there are three people who speak a common language.

Solution to September 2002 problem