Past Fortnight Problems

We call an integer *lucky* if it is a sum of positive integers (not
necessarily distinct) whose reciprocals sum to 1. For example, 4 and
11 are lucky: 4=2+2, 1/2 + 1/2 = 1, and 11=2+3+6, 1/2 + 1/3 + 1/6=1.
However, 2, 3, and 5 are unlucky. Find all unlucky numbers.

Problem Number 60

(undergraduate) Donald Adams
Lamia Mekha
Marzhel Pinto
Marco Fernandez
Spencer Williams
(graduate) Renee Thompson
Vince Dayes
Rong Zeng

Problem Number 59

Fibonacci Nim is a two player game, played as follows. Players take turns removing stones from a pile (and discarding the removed stones). They must remove at least one stone, but no more than twice what their opponent removed immediately before. Whoever removes the last stone(s) wins, and the first player may remove any number of stones (except all of them).

Here's a sample game, starting with 10 stones:

You are playing with your friend, starting with 40 stones. Your friend went first, and removed nine (which was a mistake). There are now 31 stones. What is the only move you can make, to guarantee victory?

Problem Number 58

Six chess knights are on a 3x3 board, as pictured.
Find a sequence of legal chess knight moves that end up with the diagram reversed,
(i.e. the black knights on the bottom and the white knights on the top).

Problem Number 57

A solved mini-Sudoku is a 4x4 grid where every row, column,
and 2x2 corner contains each of the numbers 1,2,3,4 exactly once.
How many solved mini-Sudokus are possible?

Problem Number 56

A positive integer is written on a chalkboard.
We repeatedly erase its unit digit and add five times that digit to what remains.
Starting with 7^2007, can we ever end up at 2007^7?

[note: 7 is raised to the 2007th power, and 2007 is raised to the 7th power.]

Problem Number 55

Find three real numbers x, y, z that satisfy:

(or prove that no such numbers exist).

Problem Number 54

Problem Number 53

Winners: Matt Gonzer and Vince Dayes (undergrad, math solution). Kristen Maskell, Matthias Pelster

Part 1: A woman is 74 and her daughter is 47 at an instant in time. Are there any other ages when this woman and daughter will have, or have had, exactly reversed digits for their ages at some instant?

Part 2: In general, describe the conditions under which two people (not necessarily mother and daughter) can have ages whose digits are exactly reversed from each other more than once in their lifetime.

Note: For both parts, assume ages less than 100 and further assume that 00,01, 02, 03, ... are the ages 0, 1, 2, ...

Problem Number 52

What is the minimum number of people in a group so that you would be certain that 3 of them were born in the same month on the same day of the week? Explain your answer.

Problem Number 51

Winners: Matthias Pelster, Tony Tam (undergraduate) and John Manor (graduate)

In this border town, the shopkeepers indicate the prices in both francs and deutsche marks, and they give change in euros. This morning, at an antique shop, I found two beautiful chairs. To my surprise, the price in marks and the price in francs for the pair of chairs are two integers with the same four digits, only in a different order. Today the exchange rate was 10 francs for 3 marks and 2 marks for 1 euro. If I pay for the chairs with 500-franc bills, how many euros will I get as change? (at the named exchange rates)

Problem Number 50

Winners: Evelyn Wilroy & Matt Gonzer (best undergraduate math solutions), Taylon Tom.

Constance enjoys playing with numbers. Her favorite game is to take an integer and work out the product of its digits, then do the same with the result until she ends up with a one digit number. For example:

Define the persistence of an integer as the number of steps needed to get a one digit number. Thus the persistence of 6 is 0, the persistence of 23 is 1, for 54 it is 2 and for 999 it is 4.

Problem Number 49

A cardboard tetrahedron is cut along the three edges from the top vertex and the triangular faces are folded down. If the resulting plane polygon is a perfect square of side 30 cm, what was the volume of the tetrahedron?

Problem Number 48

Winners: Stephen Payne (best undergrad math solution), Jeremy Burrell, Celesta Cates, An Huynh, Stephanie Kiper, Kolina Koleva, Michael Lay, Dion Moffatt, Cuong Nguyen, Joseph Penafuente, Darren Russell, Tony Tam, Tera Tran, Ian Ward

Correct answers with partial or missing solutions were also truned in by the following undergraduates: Ashley Bobey, Norberto Castellanos, Jerry Chan, William Disman, Brian De los Santos, Jeff Hagarty, Heather Hochrein, Christopher Murr, Matt Poliakoff, Erin Reed, Alexandra Strukhoff, Philip Tabares, Brenda Thinnes, Michelle Torok, Johnna Trabucco

Of six boys, exactly two were known to have been stealing apples. Who are they?
Harry said, "Charlie and George".
James said, "Donald and Tom".
Donald said, "Tom and Charlie".
George said, "Harry and Charlie".
Charlie said, "Donald and James".
Tom couldn't be found.
Four of the five boys interrogated had named one of the thieves correctly and lied about the other.
The fifth boy had lied outright. Who stole the apples?

Problem Number 47

Winners: Kevin Kirby, Stephen Payne and Jules Symons (undergraduates); Philip Etheridge, Andreas Rupp (graduate).

Jeremiah has been asked to color the four corners of a square piece of glass. He has five colors of paint from which to choose: red, white, yellow, blue and green. How many distinct colorings are there?

Note: To be distinct, two colorings must look different no matter how one flips or rotates the squares. In other words, if someone can pick up the piece of glass, and set it back down so that it matches a previously counted coloring, it does not count as distinct.

Problem Number 46

Jack and Jill are playing the nearest fraction game. They begin by choosing a fraction; this time it is 225/157. The game is played by finding integers p and q such that the ratio p/q is close to (but not equal to) the chosen fraction. For her turn, a player must either concede the hand or come up with a pair (p,q) that approximates the given fraction closer than her opponent's fraction did. To keep each hand from getting too long, they impose the condition that the difference between the numerator and the denominator be less than or equal to 1995. What is the unbeatable choice for p and q?

Problem Number 45

A semicircular sponge (shown from above in the figure) has a diameter of 20 centimeters. The sponge, full of detergent, slides without deformation on the floor in the corner of a room, keeping both ends of its diameter (A and B in the figure) in contact with the walls at all times.

What is the area that is washed as the sponge moves from the left position with a vertical diameter to the right position with a horizontal diameter?

Problem Number 44

Mr. Speeder travels on a busy highway that has the same rate of traffic flow in both directions. Except for Mr. Speeder, all other cars are traveling exactly at the speed limit. If Mr. Speeder passes one car going in the same direction as he, for every nine cars he passes going in the opposite direction, by what percentage is he exceeding the speed limit?

Problem Number 43

Problem Number 42

Two mathematicians frequent the same coffee purveyor and arrive at his establishment at a random time between 9 and 10 AM each day. Each stays exactly m minutes at the establishment. If they are in the establishment simultaneously on the average 40% of the days, what is the value of m?

Problem Number 41

Two men are walking towards each other along the side of a railway. A freight train overtakes one of them in 20 seconds, i.e., it takes 20 seconds for the train, engine to caboose, to pass him. Exactly 10 minutes after just reaching the first man, the train meets the other man coming in the opposite direction. The train passes this man in 18 seconds. How long after the train has passed the second man will the two men meet?

Problem Number 40

Winners: Mary "Lizlo" Conner (Math solution) and Tony Tam (Computer Solution)

A square of side 2 lies in the first quadrant of the xy-plane such that two adjacent vertices are on the x and y axes respectively. Find the locus of points where the center of the square lies.

Problem Number 39

Ten poker chips have numbers written on them so that one chip is numbered "1", two chips are numbered "2", three chips are numbered "3", and four chips are numbered "4". The chips are placed in a bag and three chips are drawn at random and without replacement. What is the probability that the sum of the numbers on the chips drawn is divisible by 5

Problem Number 38

Nathaniel Coil Computer Engineering
Lizlo Conner Math- Single Subject BA
Jeff Cooper Math-Emphasis in Computational Science (after a previous stint as an Art major).
John Manor Math-Emphasis in Computational Science
Jules Symons Math- BA
Tony Tam Math- BA (Imperial Valley)

N people start out at the top vertex of a hexagon with one internal diagonal as shown. (Vertices 1 and 3 have the only internal connection). They pursue a random walk by each person choosing randomly among the edges leaving the vertex where he/she currently resides. (All choices have equal probability.) After very many steps and in the limit N -> infinity, what fraction of the people is at each of the vertices of the hexagon?

Problem Number 37

Define a lattice point to be any point in the plane with both coordinates (x,y) integers. Define a lattice line to be a straight line connecting at least two lattice points. Prove or disprove: Lattice lines cover the plane.

Problem Number 36

There are n points on a circle. A straight line segment is drawn between every pair. If no 3 lines are concurrent, how many intersections are there inside the circle?

Problem Number 35

While still a sizeable distance from the Pentagon building, a man catches sight of it. Is he more likely to see two sides or three?

Problem Number 34

A swimmer stands at one corner of a square swimming pool and wishes to reach the opposite corner in minimum time. If his swimming speed is s and his walking speed is w, find his path for the shortest time.

Problem Number 33

Given a pre-sliced equilateral triangle as shown, are you able to make finitely many additional slices so that the ultimate pieces are all triangles?

A "slice" is a doubly infinite line (and so must pass completely through the triangle).Think of a pizza that we need to slice into triangular pieces using a slicer that cuts all the way across. You can work on this at your favorite pizza parlor.

The configuration is obtained as follows: Take any equilateral triangle. Place a small equilateral triangle with same center and in the same orientation as the large triangle. Then extend all sides of the small triangle until they reach the big triangle.

Problem Number 32

Zilla's math class begins every morning with a multiple choice quiz. She starts the year off a little slow and so far has only answered 50% of the quiz problems correctly. Is it possible for her to raise her average to 90% correct without having exactly an 80% average for at least one day along the way?

Problem Number 31

The hands of a clock are 5 inches and 3 inches long respectively. What is the distance between the tips of the hands when this distance is changing at the maximum rate?

Problem Number 30

Show that a positive integer N can be written as a sum of more than two consecutive positive integers if and only if N is neither a prime nor a nonnegative integer power of 2.

Problem Number 29

f(n)= the number of times the digit zero appears in the list of numbers 0, 1, 2, ..., n-1, n.

Problem Number 28

A Chamber of Commerce has a scheduling problem. A group of 2n people must meet pairwise for 5 minutes to discuss their respective businesses. Each person must meet with each other person. The room has n 2-person tables numbered 1 to n. Of course, the meetings should be concurrent and, except if necessary at the end, no one should have to sit a round out. Develop a formula or an algorithm or a program to assign each person to the correct table for each of the 5-minute sessions. Provide the output for 2n=18 people.

Problem Number 27

"Tom was talking about old Dobson's Will," said John. "I wouldn't care to be his executor."

"Eccentric as they come, I know." Len nodded. "But you would only have to do what he said."

John laughed. "Exactly! He stipulated a grave-stone cut as a square base pyramid, with all its edges and its vertical height whole numbers of feet."

"Maybe, but listen. He also insisted that the precise center of his casket should lie exactly nine feet from all five corners of the stone."

Problem Number 26

"Behold now the great ball of gold that thou hast made solid and true from the offerings of the people. Twenty kebals in diameter is it, and verily has its fame spread throughout the land. Take then that ball and from all of it make three balls of gold, solid and true, and no gold shalt thou waste in the making. And each diameter shall be a whole number of kebals, and thirty eight kebals shall be the measure of the three taken together."

And Kylis did as he was bidden, and it came to pass that the greatest of the three balls of gold was taken to the temple of Kaal where it rests to this very day.

Problem Number 25

You meet 3 witches. One always lies, one always tells the truth, and one sometimes lies and sometimes tells the truth. You get to ask 3 yes-no questions and must correctly identify which witch is which. The witches each know the truth telling policy of the other witches and each question must be addressed to only one witch who is the only one to answer. What questions should you ask and how do you interpret the replies?

Problem Number 24

Winner: Troy Mestler (Math Solution)
Winner: Michael Brumlow (Computer Solution)

For each of the digits 0 through 9, find how many times they occur in the first million integers starting at one and including 1,000,000. In other words, find the number of 0's, 1's, 2's, ... in the list 1, 2, 3, 4, ..., 999999, 1000000.

Problem Number 23

A mathematician is lost in a woods bounded by a linear beach. The fog is very dense, so that visibility is epsilon. She knows she is one mile from the beach, but does not know in which direction the beach lies. She can measure distance and walk in any path she wants to. Show that she can be certain to reach the water in less than 1+7*Pi/6+sqrt(3) miles.

Alternative statement: Show that there is a curve of length 1+7*Pi/6+sqrt(3) = 6.39724... starting at the center of the unit circle and touching or crossing all tangents to the unit circle.

Problem Number 22

Mr and Mrs. Jones are entertaining four couples. During the evening, many handshakes occur but no one shakes hands with their spouse. At the leave taking, Mr Jones asks everyone (each guest and Mrs Jones) how many hands they had shaken, and everyone tells him a different number. How many hands did Mrs Jones shake?

Problem Number 21

Two integers are selected at random between 2 and 999. What is the probability that their only common divisor is 1?

Problem Number 20

A necklace consists of pearls which increase uniformly from a weight of one carat for the end pearls to a weight of 100 carats for the middle pearl. If the necklace weighs altogether 1650 carats and the clasp and string together weigh as much (in carats) as the number of pearls, how many pearls does the necklace contain?

Problem Number 19

Start with the fraction 1/1 for the first generation and make two fractions from it by replacing either the numerator or the denominator by the sum of the numerator and denominator. Thus at generation two we have 2/1 and 1/2. Proceed in this fashion, getting 3/1 and 2/3 from 2/1 and getting 3/2 and 1/3 from 1/2 for generation three.

Show that this algorithm generates every positive rational number exactly once and in lowest terms.

Problem Number 18

Consider an acute angle theta () made by two rays A and B meeting at vertex O. Assume that the rays A and B act as mirrors and consider a ray of light starting at some point P in the interior of the angle and moving along a line L that does not pass through O. Show that the number of reflections the light has with sides A and B is finite.

Problem Number 17

There is a queue of n+m persons at a theatre box office. Admission is $5. Each of the n people in line have exactly one $5 bill. The other m have exactly one $10 bill. The ticket seller has no change to begin with. What is the probability that the ticket taker will have enough change for everyone as they buy their ticket, i.e. without having to wait past their turn?

Problem Number 16

An island contains cats, birds and snakes. Each day each cat eats a bird for breakfast, each snake has a cat for lunch and each bird has a snake for supper. There are no births or deaths except at these meals.

At a certain time on the n-th day of the month, a bird becomes the only living creature left on the island. What time of day did this happen (i.e. at what meal)? Find an expression (or write a program) to evaluate the number of birds before breakfast on the first day of the month in terms of n.

Problem Number 15

diverges. Throw out all the terms that have a 9 in them, like 1/9, 1/29, 1/911, 1/9923 etc.

Problem Number 14

Prove that the square of every prime number greater than 3 yields a remainder of 1 when divided by 12.

Problem Number 13

Suppose that Norm and Cliff (from Cheers) enter a football pool with Pete Rose. They each put $10 into the pool and then choose the winners of each of ten football games. The player with the largest number of correctly chosen winning teams wins the $30 pool. In the case of a tie, the pool is split.

Assume at first that all three players' choices for each game are randomly determined, and that the actual winner of each game is also randomly determined (say by the flip of a fair coin). If each player chooses his teams independently, then each player wins, on average, $10 (i.e. they get their money back). This can be seen without doing any calculation, on the general principle that the situation is symmetric: All 3 players have the same strategy and nobody has an advantage, so the expected outcomes are the same.

But Norm and Cliff have come up with a scheme to beat Pete, namely, the Evil Twin Strategy:

Cliff (randomly) determines who he thinks will win each game, and Norm chooses the opposite team for each game. Thus if Cliff picks a game incorrectly, then his pal Norm has it right. (Pete is still picking independently and randomly.) Therefore the Cliff-Norm Axis of Evil Twins has at least one of them with at least 5 correct picks. If they split their winnings, how much will they each expect to win on average?

Problem Number 12

Consider a game similar to tic-tac-toe. It is to be played on an unlimited planar grid. Players X and O alternate moves. X goes first. The object of the game is to get four symbols in a row and diagonals do not count so these must be a row or a column of four X's or four O's. Since starting is an advantage, player O is declared the winner if the game lasts more than 10 rounds.

PROBLEM: If both players play optimally, who wins? What strategy forces the win?

Problem Number 11

Each of the rectangles in the equation below stands for one of the digits 1-9. Find a way to use each digit exactly once in the equation below so as to make it a valid equality, i.e., to write the number 1 as the sum of three fractions with the numerators one digit numbers and the denominators two digit numbers.

Problem Number 10

Problem Number 9

You have an 8 x 8 chessboard and 21 3 x 1 dominoes. Can we cover 63 squares of the chessboard with the dominoes? If this can be done, what are the possible locations for the uncovered square? Prove that these are all of the possibilities.

Problem Number Eight

Find a partition of the positive integers 1 through 52 into four bins in such a way that no number can be written as the sum of two other numbers in the same bin.

Example: For the numbers 1 through 5 into two bins the following partition works:

Problem Number Seven

The following problem is loosely based on Showcase Showdown from The Price is Right. Two contestants play a game using a fair spinner which generates an integer n, 1 < n < 20 with each value of n having the same probability p=1/20. Each player gets to spin either once or twice and ends up with a score which is the sum of the n's from his spin(s). The first player completes his turn before the second player begins. A score greater than 20 automatically loses but otherwise the larger score wins. In the event of a tie, the first player wins.

In the above simplified form of the Showcase Showdown, there is only one real choice to be made by the players: After the first spin, player one has to decide whether to spin again. Whatever his resulting score, player two has an obvious strategy - if his first spin would result in a loss, he should spin again no matter what his score.

The Problem: Find the largest result of the first spin for which the first player should spin the wheel a second time. (Equivalently, find the smallest value for which the first player should "stand" and not spin a second time.) We'll say that a strategy "should" be followed if the probability of winning with the strategy exceeds that of winning with the complementary strategy. Assuming that the first player uses the optimal strategy, is the game fair? More precisely, what is the probability that the first player will win with this strategy?

Problem Number Six

This year 2002 is a PALINDROME, as was 1991. The next is in 2112. Consider palindromes made up only of the digits 1 and 2. Find, as a function of n, P(n) =the number of such palindromes whose digits add up to n. Of course you must explain your answer. An example to get you started:

Problem Number Five

A stick is broken randomly into 3 parts. What is the probability that the three parts form a triangle?

Problem Number Four

You live on a street with N houses, where N is between 50 and 500. The houses are numbered consecutively from 1 to N and are all on one side of the street. You notice that all the numbers on one side of your house add up to exactly the same total as all the numbers on the other side of your house. What is the number of your house?

Problem Number Three

A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge.

Problem Number Two

Find at least 5 triangles with sides of integer length such that (in some units) their area and perimeter are numerically equal.

Bonus Question (Extra book awarded): Find a sixth one or prove that there are only five.

Problem Number One

Start with any triangle T. We want to place two non-overlapping rectangles inside T so as to maximize the proportion of the area of T which is covered by the rectangles. What is the maximum proportion that can be covered. Prove it.

Problems of the Fortnight