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The Problem of the Fortnight
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Suppose A = (a,a3) is any point on the curve y=x3 other than the origin. The tangent line at A meets the curve again at a point B = (b,b3). If mA and mB are the slopes of the tangents at A and B, what is the ratio of slopes mB/mA ? To earn colloquium credit, your answer must be a single number and your solution should show that the number mB/mA is what results no matter where the point A is on the curve y = x3 (aside from the origin). Showing the result for only one point A is not sufficient for colloquium credit. Write your solution -- showing all your work, please! -- on a tan gent and drop it by the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, April 19. As always, be sure to include your name and the name(s) of your professor(s) -- e.g. Carmine Dioxide, Dr. Jean Poole -- on your solution. |
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A classical mathematics question involving chess is the non-attacking rooks problem: What is the largest number of rooks that can be placed on an 8 x 8 chess board so that none of the rooks can attack another rook? The problem becomes more interesting if queens are used instead of rooks. For background on this problem, see http://mathworld.wolfram.com/RooksProblem.html When bees play chess, they play on a hexagonal board in the shape of a triangle, so their rooks can move in six directions rather than four. What is the largest number of non-attacking rooks that can be placed on a triangular shaped bee chess board with n hexagons on a side? |
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Consider the graph of y = 5x - 7x3 in the first quadrant shown in the figure. What is the constant k so that the areas of the two shaded regions are equal when sliced by the line y = k? |
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A sequence of nested squares is formed by connecting midpoints of the sides of a square to form another square as shown in the figure. Assuming that the nested squares and shaded regions in the figure continue indefinitely, what is the total area of the shaded region if the original largest square has side length one? |
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A man is pulling a heavy sled along level ground by a rope of fixed length L. Assume the man begins walking at the origin of a coordinate plane and that the sled is initially at the point (0,L). The man walks to the right along the x-axis dragging the sled behind him, as shown in the figure. Find an equation for the path of the sled. |
Starting with the vertices of a square, P1 = (0,1), P2 = (1,1), P3 = (1,0), P4 = (0,0), we construct the following points as shown in the figure on the right (click on the figure to enlarge it). In the figure, P5 is the midpoint of the line segment P1P2, P6 is the midpoint of the line segment P2P3, etc. The infinite sequence of points P1, P2, P3, P4, P5, P6, P7, ... approaches a point P inside the square.
A light bulb at point B on the vertical line x = -2 illuminates a region that is partially occluded by an arc of the circle x2 + y2 = 1. How high must the light bulb be if the point (1.25, 0) is on the edge of the illuminated region? (Click on the picture to the left to see an enlarged picture of this situation.)
The line y = x + 6 intersects the parabola y = x2 at points A and B in the figure shown on the left. (You can click on the figure to enlarge it.) Find the point C on the arc of the parabola between A and B that maximizes the area of the triangle ABC.
This Problem of the Fortnight involves a little geometry and some ideas from Calculus 1 -- and nothing more! -- so everyone should be able to take a crack at it. It's a great problem, and we hope you enjoy working on it!
Suppose that circles of equal diameter are packed tightly in n rows inside an equilateral triangle. (The figure at left illustrates the case n = 3.)
If A is the area of the triangle and An is the total area occupied by the n rows of circles, find the limit of the ratio An / A as n goes to infinity; i.e. find
In the figure shown, the blue triangle POQ is isosceles since sides OP and OQ both have length 4, theta is the angle POQ at the bottom of the triangle, and a pink semicircle sits atop the isosceles triangle.
There are 1000 bottles, one of which cotains poison. You have 10 lab rats which will die after one week if they drink any poison. How can you find out which bottle contains the poison if you have only one week of time?
A train leaves Holland headed for Kalamazoo at a constant speed of 8.6 mi/hr. At the same time, 49.7 miles away in Kalamazoo, another train on the same track leaves the station and heads for Holland at a constant speed of 9.4 mi/hr. At the instant the trains simultaneously depart, a dragonfly at the front of the train in Holland takes flight and travels along the tracks toward the train coming from Kalamazoo. Upon reaching the train from Kalamazoo, the dragonfly turns around in an instant and heads toward the train from Holland. The dragonfly continues to bounce back and forth between the trains, flying at a constant speed of 11.3 mi/hr until the trains collide. What was the total distance of the dragonfly's flight?
As you take breaks from preparing for the last round of exams and finals, you can think about our last Problem of the Fortnight for the school year. . . .
"You know, the part of summer camp I enjoyed best was probably Ms. Oddball's intelligence tests," said April. "'Seventeeners' she called them. We had a lot of them during the fortnight we were in camp."
"Why 'seventeeners'?" Mae asked.
"Well, for some oddball reason, the person who scored best on each test won 17 cents. Anybody who received a lower score, though, had to pay a penny."
"At that rate," said Mae, "some of you might have had to fork out quite a few pennies."
"We did," said April. "I, for instance, lost 30 cents on balance. But at the end of camp, Ms. Oddball paid us all back for any losses we incurred in taking her tests, so nobody really lost any money at all -- except for Ms. Oddball, that is. Overall she paid out $3.60. It was a lot of fun! All of us at camp took each test that was given, and each of us got the top score on at least one test, but no two of us had the same number of top-score tests."
On how many tests did April get the top score?
Last week's weather had us at Off on a Tangent, America's premiere fortnightly electronic mathematics department newsletter, thinking about planting our gardens. And so, from the fertile soil of our brains comes this fortnight's problem.
Amy, Ben, Carl, and Deb all have plots at the community garden. All four plots are rectangular; each side of each plot is an exact number of feet; and the diagonal of each plot, oddly enough, is exactly 221 feet. They were recently comparing the areas of their "cabbage patches." Amy's exceeds Carl's by 3,660 square feet; while Ben has 12,720 square feet more than Deb to weed and water.
By how much does the area of Amy's plot exceed the area of Deb's?
A kayaker was paddling across the still surface of Lake Macatawa early one morning when she saw a beautiful large-mouth bass jump in the water directly in front of her kayak. She counted twelve strokes until she came to the first ever-widening circle the fish had made, and twelve strokes later she broke through the circle on the opposite side. For some time she occupied her thoughts by trying to calculate how many strokes away from her the fish had been when it jumped, but this mathematical diversion was interrupted when she noticed a Bald Eagle swoop down from a nearby tree in an attempt to nab a fish breakfast. Can you complete her calculation?
How many strokes away from her kayak was the fish when it jumped?
"Well, I've come out second best in my battle with the union," said Noah van Ark.
"How so?" asked his sister Joan.
"Well, I needed to have the union workers move thousands of crates. The exact number," said Noah, consulting his notebook, "was
69,489. The job took nine working days. I didn't think the union workers were putting all they had into it, but the union leaders
thought otherwise. Every day after the first day, I put six more workers on the job; and every day after the first day, each of the
workers -- by arrangement -- shifted five fewer crates than was the quota for the day before. The result was that, during the latter
part of the period, the number of crates being moved actually began to go down."
What was the largest number of crates moved on any one day?
"Valentine's Day is coming up soon," thought Mrs. Hartsema. "I should get a little something for my grandchildren." She decided to give each of her 31 grandchildren a number of candy hearts along with their Valentines. After counting her candy hearts and finding 470 of them, Mrs. Hartsema figured that each girl would get 7 more candy hearts than each boy. She gave 74 candy hearts to the children of her eldest son Art. How many girls did Art have?
Write your solution (not just the answer!) on the back of a Valentine, and drop it in the Problem of the Fortnight slot outside Professor Pearson's office (VWF 212) by 3:00 p.m. on Friday, February 17. As always, be sure to include your name and the name(s) of your math professor(s) -- e.g. Val N. Tyne, Professor Hallmark -- on your solution.
"I hear some kids playing in the backyard," said Suzie Smartsema. "Are they all yours?"
"Heavens, no," replied Professor Von Den Two, the eminent number theorist. "My children are playing with friends from three other families in the neighborhood, although our family happens to be the largest. The Carlsons have fewer children, the Bensons fewer still, and the Andersons have the fewest of all."
"How many children are there altogether?" asked Suzie.
"Let me put it this way," said Professor Von Den Two. "There are fewer than 18 children, and the product of the numbers in the four families happens to be my house number, which you saw when you arrived."
Suzie took her notebook out of her bag and began scribbling. A few moments later, she said, "I need more information. Is there more than one child in the Anderson family?"
As soon as Professor Von Den Two replied, Suzie smiled and correctly stated the number of children in each family. How many children are in each family?
Imagine, if you will, you have three boxes, one containing two black marbles, one containing two white marbles, and one containing one black marble and one white marble. The boxes were labeled for their contents---BB, WW and BW---but someone has switched the labels so that every box is now incorrectly labeled. You are allowed to take one marble at a time out of any box, without looking inside, and by this process of sampling you are to determine the contents of all three boxes. What is the smallest number of drawings needed to do this? Make sure you give a thorough explanation of your answer.
When the celebrated German mathematician Karl Gauss (1777-1855) was nine years old, he was asked to add all the integers from 1 through 100. He quickly added 1 and 100, 2 and 99, and so on for 50 pairs of numbers each adding in 101.
His answer was 50 · 101 = 5, 050.
Now find the sum of all the digits in the integers from 1 through 1,000,000 (i.e. all the digits in those numbers, not the numbers themselves).
Four bugs, (A, B, C & D) occupy the corners of a square 10 inches on a side. Simultaneously, A crawls directly toward B, B toward C, C toward D, and D toward A. If all four bugs crawl at the same constant rate, they will describe four congruent logarithmic spirals that meet at the center of the square.
How far does each bug travel before they meet? Note: The problem can be solved without calculus.
You have ten stacks of coins, each consisting of 10 new dollar coins. One entire stack of coins is counterfeit, but you do not know which one. However, you do know the weight of a genuine dollar coin, and you are also told that each counterfeit coin weighs one gram more than it should. Your kind chemistry professor agrees to let you weigh the coins on one of the electronic scale in the chem lab.
What is the smallest number of weighings necessary to determine which stack is counterfeit, and how do you do it?
Sally begins to solve a problem at the time between 4:00 and 5:00 p.m. when the clock's hands are together. She finishes when the minute hand is opposite the hour hand. How many minutes does it take her to solve the problem, and when does she finish it? Give exact answers in terms of fractions of minutes (i.e. no decimal approximations).
An absent-minded bank teller switched the dollars and cents when he cashed a check for Mrs. Brown, giving her dollars instead of cents, and cents instead of dollars. After buying a five cent Tootsie Roll, Mrs. Brown discovered that she had left exactly twice as much as her original check. What was the amount of the check?
At the last Pi Mu Epsilon induction ceremony, the Math Department served several pies cut into eighths. Professor Pennings notably favored of the snicker’s pie. Trying to wait politely for all to be served before taking seconds for himself, anxiety overtook him when he noticed that there was only one slice left and an unsuspecting student was moving toward it. Professor Pennings swooped in and magnanimously offered to share the coveted delicacy.
He told the student to cut the piece in half, but not into two congruent wedge shapes, oh no… he told the student to make a cut perpendicular to the cut that most people would make (see picture on the left). How far from the vertex should the cut have been made so that the two resulting pieces would have the same area? Give your answer as the ratio between this distance and the radius of the pie, using only a reasonable number of significant digits.
Allyson, Bethany and Corey have a window washing business and are regularly hired by a strip mall owner. They have found that together they can finish the job in 6 hours less time than when Allyson works alone, 1 hour less than when Bethany works alone, and in ½ the time than when Corey works alone. The next time they need to wash the windows, only Allyson and Bethany will be able to work. How long will it take them?
A certain dodecahedron has edges of length 10 cm. If a fly lands on a vertex of this dodecahedron and then walks along only the edges, what is the greatest distance the fly could walk before coming to a vertex a second time and without retracing an edge? Justify that your solution is optimal.
A new elementary school teacher wants to stock her in-class library, but, since she is a recent graduate and therefore somewhat destitute, she has very little capital. Imagine her delight when she comes across a plethora of children’s books at a church rummage sale! They have boxes and boxes of paperbacks with a price tag of 4 for $1. Then there are many short hard cover books on a shelf tagged $1 each. Finally, she sighs wistfully at the shelf containing beautiful, early edition hard covers for $15 each. She withdraws $100 from the cash machine and gets busy with her selections. Determine all of the ways she can buy exactly 100 books with exactly $100.
Arrange the numerals 1 to 9 (each numeral once) as a proper fraction equivalent to 1/3. More than one solution is possible; please include the thought process you used to determine your answer.
To decorate my scrapbook, I cut a 4-inch wide parallelogram out of a square piece of paper as shown in the diagram on the left. Surprisingly enough, the parallelogram and each of the two leftover pieces of paper all had exactly the same area! What are the exact dimensions of the original square?
Over the break I decided to finally organize some of the books on my office shelves. In particular, there was a stack of six books on the edge of one shelf that were threatening to topple onto my monitor. From the clues give, determine the titles, authors, color of the spine, and position in the stack (top of stack is book #1).
Clues:
Suppose a cubic polynomial with leading coefficient of one and with inflection point at the origin passes through (c,0) and (a,b), where a>c>0. A translated copy of the cubic has its inflection point at (a,b) and passes through the origin. Show that twice the area between the two cubic polynomial curves equals a4.
October 10, 2010 garnered a lot of attention because of its representation as a calendar date by 10/10/10. Suppose instead that we represented month, day, and year in base 12. Thus in base 12, December would be indicated by 10 because 10twelve = 12 in base 10.
Determine which year(s), if any, during the 21st century will have a 10/10/10 date, where each "10" is base 12 representation. In particular, this 10/10/10 date would represent December 12 of a year (or years) for which base 12 representation of the year that ends in the digits 10.
For which natural numbers, n, is (2n+1)(2n-1) divisible by 3?
For what values of n is n3 - 9n2 + 20n divisible by 6?
In a round-robin tournament, each team plays all of the other teams exactly once. Consider the following multiple round-robin tournament setup:
N teams play a round-robin tournament and exactly one team is eliminated from further play. The remaining N-1 teams play another round-robin tournament with a second team then eliminated. Round-robin tournaments continue, with exactly one team eliminated at the conclusion of each round-robin, until only two teams remain. The last two teams in contention play a final game (which would constitute a round-robin with two teams) to determine the champion.
What percentage of total games played in the multiple round-robin tournament does the champion play?
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A digital clock shows 2:35. This is the first time after midnight when all three digits are different prime numbers. What is the last time before noon when all three digits on the clock are different prime numbers? Write your solution (not just an answer) on a piece of prime rib or your favorite prime minister and drop it off in the Official Problem of the Fortnight Slot outside VWF 212 by 3:00 pm on Wednesday, September 29. As always, be sure to include your name, the name(s) of your professor(s), and your math class(es) -- e.g. Pry M. Number, Dr. Com Posite, Math 235 -- on your solution. |
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The pages of a book are consecutively number from 1 through 384. How many times does the digit 8 appear in this numbering? |
Each U.S. $1 bill contains an 8-digit string between two letters as a serial number. For example, E12345678A might be such a serial number. Suppose that the two letters of the serial number are given, say E at the beginning and A at the end. Under the assumption that every 8-digit string is equally likely to occur between the E and the A, what is the probability that a serial number contains five or more of the same digit?
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Your sock drawer has 25 electric yellow socks, 30 blue striped socks, 17 orange socks, 13 magnetic socks, 33 pale purple socks, 30 royal red socks, 11 gruesome green socks, 14 midnight black socks, and 23 bruin brown socks! If you reach into the drawer in the dark, how many socks do you need to pull out to be sure you have a matching pair? |
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Determine the UNITS digit of the following number: 12010 + 22010 + 32010 + ... + 20092010 + 20102010 |
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Consider the following alphametic PEOPLE + COUNT ------------ CENSUS |
Delia, Tracy, and Bella are going cross-country skiing with a school group. They are carrying packs and each pack can weigh no more than 10 lbs. The packs are weighed 2 at a time. Delia and Tracy weigh their packs together and the total is 24 lbs. When Delia and Bella weigh their packs, the total is 20 lbs. Tracy’s and Bella’s packs together weigh 18 lbs. Which skiers have packs that are too heavy, and by how much?
Suppose that you have an unlimited supply of 4 cent stamps and 9 cent stamps What amounts can’t you make with these stamps?
This is the final Problem of the Fortnight for this semester. The Problem of the Fortnight will be traveling over the holidays but will return in time for the start of Spring Semester.
Abby and Becca were full-time students at the State University of Michigan (SUM) during the fall and spring semesters of 2008-09; full-time means each one took at least 12 credits each semester, and assume a maximum of 20 credits per semester for each student.
In Fall 2008, Abby's GPA was greater than Becca's. In Spring 2009, Abby's GPA was also greater than Becca's.
Therefore, we conclude that Abby's combined GPA for Fall and Spring 2008-09 was greater than Becca's combined GPA for Fall and Spring 2008-09.
Give a counterexample to show that the above conclusion is false; that is, give an example where Becca's combined GPA is higher than Abby's even though Abby's GPA is higher for both the Fall and the Spring.
Note: The State University of Michigan (SUM) does not have a grade of A+, and the point values for grades are as follows: A = 4.0, A- = 3.7, B+ = 3.3, B = 3.0, B- = 2.7, and so on.
Suppose that an automobile's odometer can represent mileage up to 999,999, and that leading 0s are not shown. For example, 2511 miles would not be shown as 002511. Some of these mileage readings are palindromes: a palindrome is a number that reads the same right to left as they do left to right. For example, 212, 55355, and 198891 are all palindromes.
How many mileage readings from 1 to 999,999 are palindromes, and what's the smallest difference between two consecutive odometer palindromes of 6 digits?
Write your solution on the back of a picture of Irv Gordon's Volvo P1800, a car in which he has logged 2.5 million miles since purchasing it new in 1966 for $4,150.
For f(x) = x3 + 6x2 - 15x + k, the absolute maximum and absolute minimum values on the interval [-10,2] have the same absolute value. Find the value of k.
This problem has a theme inspired by this year's Critical Issues Symposium on water.
Your cabin is two miles due north of a stream that runs east-west. Your grandmother's cabin is located 12 miles west and one mile north of your cabin. Every day, you go from your cabin to Grandma's, but first visit the stream (to get fresh water for Grandma). What is the length of the route with minimum distance? (No calculus is required to solve this problem -- just some geometry and a little creativity.)
What is the total number of squares (of all sizes) on a 40 x 40 checkerboard?
April 18, 2009
Ten (not necessarily distinct) integers have the property that if all but one of them are added, the possible results are:
The last Problem of the Fortnight for the 2008 - 09 academic year:
Determine the last two digits of
Friday, April 3
The Problem of the Fortnight has two parts this time:
1. Which of the five numbers 2007, 2008, 2009, 2010, and 2011 has the largest number of factors, and which one has the fewest number of factors?
2. Determine the total number of factors for the number
For example, 21 has 4 factors (1, 3, 7, 21) and 20 has 6 factors (1, 2, 4, 5, 10, 20).
Friday, March 12
An n x n matrix is called a Latin square if each of the integers 1, 2, ..., n occurs exactly once in each row and each column. Find the number of distinct 4 x 4 Latin squares.
Friday, February 27
Maya would like you to help her out. She needs to integrate the following.
Be sure to show all your work. You may use a computer algebra program to check your answer, but you must solve the integral by hand; computer solutions will not be accepted.
Friday, February 6
Integrate
Friday, January 23
Suppose you have 8 straight metal rods with lengths 1, 2, 3, 4, 5, 6, 7, and 8 inches. Suppose that 3 rods are selected at random. What is the probability that a triangle can be constructed from the 3 selected rods?
Write your solution (not just the answer!) on a triangular piece of paper and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, January 23. As always, be sure to write your name, the name(s) of your professor(s), and your math class(es) on your solution (e.g. Al Titude, Prof. Hy Potenuse, Math 276). Good luck, and have fun!
Wednesday, November 26
The last Problem of the Fortnight for this semester:
A square is divided into three pieces of equal area by two parallel cuts as shown. The distance between the parallel lines is 6 inches. What is the area of the square in square inches?
Friday, November 14
How many of the positive factors of 36,000,000 are not perfect squares?
Friday, October 31
Find an expression for the continued radical
C = √ ( m + √ (m + √ (m + ...)))
in terms of m that does not involve a continued radical and determine all positive integers m so that C is a positive integer. (If the nested square roots aren't clear here, check the bulletin board for a statement of the problem that is typeset more clearly.)
Friday, October 10
Farmer Jones has 65 hens. If she had one more solid-colored hen, then exactly one-third of her hens would be speckled. From her years of experience, Farmer Jones knows that one-half of the specked hens will lay speckled eggs and that each hen and a half will lay an egg and a half in a day and a half. After how many full days will Farmer Jones have four dozen speckled eggs to sell?
Friday, September 26
A minivan has two seats in front, a middle seat with spaces for three people, and a back seat with spaces for four people. Nine licensed drivers are going to ride in the van. One insists on sitting in the front seat, another insists on sitting in the middle seat, and a third insists on sitting in the back seat. How many different seating arrangements satisfy everyone?
Friday, September 12
An old woman goes to the Holland Farmer's Market and a truck runs over her basket of eggs and crushes them. The driver offers to pay for the damages and asks her how many eggs she brought. She doesn't remember the exact number, but when she had taken them out two at a time, there was one egg left. The same happened when she picked them out three, four, five and six at a time. But when she took them out seven at a time, they came out even (no eggs left) What is the smallest number of eggs she could have had?
Friday, April 7
The problem of the fortnight involves a towering exponential. We hope you construct a solution that is on a more solid foundation than the engineers at Pisa did!
Calculate the derivative with respect to x of the function
Thursday, March 13
A probability density function on [a,b] is a positive function where the area under the curve over the interval [a,b] is 1. The median, a common measure of the center of a probability density function, is the value m in [a,b] where half the area under the probability density function lies to the left of m and half lies to the right of m.
The problem this fortnight is: Find positive numbers b and k such that f(x) = kx3 is a probability density function on [0,b] with a median of 3.
Friday, February 29
Solve the equation
where x and all expressions in the equation are real.
Friday, January 25
Parallelogram ABCD has been "sliced" by diagonal AC and the segment BM, with M as the midpoint of CD. The point E is the intersection of AC and BM. If the entire parallelogram has an area of X square units, find the areas of the four pieces. Justify your answer.
Slips of Paper: Consider, if you will, the number 12355699. If we write each of the digits in this number on separate slips of paper, put them in a bowl, and draw three of the numbers at random, without replacement, what is the probability that the sum of the numbers drawn will be even?
Friday, November 16
Sequences: A sequence of numbers {an} has a1 = 7 as its first term, and every other term after the first is defined as follows:
Friday, November 2
Three Roots: Consider, if you will, the equation Ax3 + (2 - A)x2 - x - 1 = 0, where A is a real number for which the equation has three real roots, not necessarily distinct. For certain values of A, there is a repeated root r and a distinct root s. List all values of the triple (A, r, s).
Friday, October 12
To mark his place in the algebra book he is reading, Clint always folds the page as shown in the figure to the right so that the bottom-right corner touches the opposite side of the same page. The pages of the book are eleven inches wide. In terms of theta, what is the exact length, in inches, of line segment labeled L?
Friday, Spetember 14 
In a little town in West Michigan lives a math professor, who hears one day that the barber has three children. So, on the next visit to the barber, the professor casually inquires, "I have heard you have three children, is that right?" "Yes!" says the barber. "Well, how old are they?" "You are the math professor, aren't you? I tell you, if you multiply the ages of the three, you'll end up with 36." "All right!" the professor answers and walks home. The next day the professor comes back to the barber shop and says: "With the information you have given me, it is impossible to figure out how old your kids are." Then the barber says: "Very good, I see you are a good mathematician. If you add the ages of the three, the sum will be the number of my house." So, the professor walks out, looks at the house number and returns home. Still the professor can't find the solution. The next day, the professor tells the barber that there still must be some information that's missing. "Yes, you are very clever!" says the barber. "The next information I'm giving you is the last word I'm saying about the age of my children. Now you will have enough information. Don't come back again and ask for more. The youngest has blonde hair." The professor goes home and figures out the answer.
What are the ages of the barber's children, and how did the professor figure it out?
Friday, April 20
The final Problem of the Fortnight for the year involves a little geometry and some ideas from Calculus 1 -- but nothing more! -- and so everyone should be able to take a crack at it. It's a great problem, and we hope you enjoy working on it!
Suppose that circles of equal diameter are packed tightly in n rows inside an equilateral triangle. (The figure at left illustrates the case n = 3.)
If A is the area of the triangle and An is the total area occupied by the n rows of circles, find the limit of the ratio An / A as n goes to infinity; i.e. find
Friday, April 5
Find the exact value of the continued fraction [1, 2, 3, 1, 2, 3, 1, 2, 3, . . . ].
Friday, March 30
Let {an} be a (possibly infinite) sequence of positive integers. A creature like
is called a continued fraction and is sometimes denoted by [a0, a1, a2, a3, . . . ]. A fact that is well known by those who know it well is that π can be represented as the continued fraction [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, . . . ] The first few convergents are 3, 22/7, 333/106, and 355/113. The very large term 292 means that the convergent [3, 7, 15, 1] = 355/113 is a very good approximation to π (accurate to 6 decimal places), a fact first discovered by astronomer Tsu Ch'ung-Chih in the fifth century A.D.
While you're on spring break, ponder our problem of the fortnight: Find the exact value (no decimal approximations allowed) of the continued fraction [1, 1, 1, 1, . . . ].
Friday, March 9
Let C be a circle of radius 1. Pick two points on C at random (using a uniform distribution on the circle so that each point on the circle has an equal probability of being chosen).
What is the expected value of the length of the chord connecting the two points?
Friday, February 9
On a beautiful January afternoon a few days ago, I was sitting at my desk, trying to conjure up another problem of the fortnight. Not having much luck, I looked out the window and was amazed to see a red-tailed hawk had perched itself on a limb of the tree outside my window. It was an awe-inspiring sight! During this unexpected bird-watching, I must have been idly clicking my mechanical pencil because when the hawk flew away and I got back to business, I noticed that a piece of lead about 1 cm in length had broken off and fallen onto the notepad of lined paper on my desk. Just then I realized that the problem I had been searching for had quite literally fallen out of my pencil.
If a 1 cm piece of lead falls randomly onto a notepad of lined paper, where the lines are 1 cm apart, what is the probability that the piece of lead will intersect one of the horizontal lines?
Friday, January 26
Whether Euler actually discovered Sudoku puzzles, as Swiss Radio International claims, or their history extends deeper into history, one thing is undisputed: they're really fun! And so, we tip our hats to "The Year of Euler" by offering the following Sudoku puzzle as our first Problem of the Fortnight of the New Year. Fill in the blank cells so that each row, each column and each 3 x 3 block contains the digits 1 through 9 exactly once.
The last Problem of the Fortnight of the semester comes to us from Mr. Vern Hoekstra of Zeeland, MI. Mr. Hoekstra writes:
We have been playing golf from time to time with 16 people. In our group there are four levels of handicaps -- let's call them A, B, C, and D -- and there are four people with each handicap level -- so we could let A1, A2, A3 and A4 represent the four people with handicap level A, and so on for the other handicap levels. On the first day we might have:
| Team 1 |
A1 |
B1 |
C1 |
D1 |
| Team 2 |
A2 |
B2 |
C2 |
D2 |
| Team 3 |
A3 |
B3 |
C3 |
D3 |
| Team 4 |
A4 |
B4 |
C4 |
D4 |
Friday, November 10
n the figure below angle AOB has a measure of 15 degrees and the length of segment A1B1 is 4. Segment AiBi is perpendicular to OB for each i = 1, 2, 3, ... The lengths of segment AiBi is the same as Ai+1Bi for each i = 1, 2, 3, ... Find the total length of the zigzag path A1B2A2B2A3B3 A4B4 ... Give your answer in closed form.
Friday, October 27
Friday, October 13
Consider the polynomial
Given that p(a) = 0 = p(b) and ab = -32, find k.
Once you've found the value for k, graph the polynomial p(x) and write your solution (not just the value of k, but how you determined it) on the back of your graph and drop it in the Problem of the Fortnight slot outside Professor Pearson's office (VWF 212) by 3 pm on Friday, October 13. Please be sure to include your name, your math class(es) and the name(s) of your professor(s) -- e.g. Ima Student, Math 131, Professor Isaac Newton -- on your solution.
Friday, September 29
The Hamilton family wanted to cross Broom Bridge at night, but they had only one lantern and the bridge was too weak for more than two to cross at a time. William, the father, could cross the bridge in 1 minute, and his wife Helen could cross in 2 minutes. Their eldest son Edwin could cross the bridge in 5 minutes, but the youngest son Archibald took 10 minutes to cross the bridge. Given that anyone crossing the bridge must have the lantern in order to see the way across, what is the fastest way for the Hamilton family to cross Broom Bridge, and how do they do it?
Write your solution on the back of a picture of Broom Bridge and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3 pm on Friday, September 29. As always, show all your work for full credit, and please put your name, the name of your professor(s) and your math class(es) -- e.g. I.M. Student, Math 972, Professor Carl Friedrich Gauss -- on the top of your solution.
Friday, September 15
Having had a relaxing and rejuvenating summer, The Problem of the Fortnight is back for another season of problem-solving fun!
Ten (not necessarily distinct) positive integers have the property that if all but one of them are added, the possible results (depending on which one is omitted) are:
82, 83, 84, 85, 87, 89, 90, 91, 92.
(This is not a misprint; there are only nine possible results.) What are the ten integers?
Write your solution on a (not necessarily regular) decagon and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, September 15. In addition to your name, please write your class and your professor's name (e.g. Math 132 - Dr. Pennings) on your solution.
Friday, April 21
Two perpendicular chords intersect in a circle. The lengths of the segments of one chord are 3 and 4. The lengths of the segments of the other chord are 6 and 2. Find the diameter of the circle.
Write your solution on an official Rawlings baseball signed by Pudge Rodriguez (actually, a picture of a baseball will suffice) and drop it in the Problem of the Fortnight slot outside Dr. Pearson’s office (VWF 212) by 3 p.m. on Friday, April 21.
Friday, April 7
On the heels of Pi Day (3-14) and the accompanying break you enjoyed celebrating this important number, we offer the following problem about the somewhat more obscure numbers 13,511, 13,903 and 14,589. (Editor's Note: Whether Hope planned its spring break in honor of Pi Day is unsubstantiated at this point.)
Determine the greatest integer that will divide 13,511, 13,903 and 14,589 and leave the same remainder.
Write your solution in whipped cream on the top of an apple pie (with a cup of coffee, please!) or write it in green ink on the back of a St. Patrick's Day card and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 on Friday, April 7.
Friday, March 16
Three 1x1 squares are joined to make a figure -- call it L.
Friday, March 3
A 2x3x4 rectangular box is constructed from unit cubes, which divide each face of the box into a grid. You have to travel from one corner of the box to the corner diagonally opposite along these grid lines, staying on the outer faces of the box. (No fair going inside! But it is fair to travel along the edges of the box.)
How many paths are there from one corner of the 2x3x4 box to the corner diagonally opposite such that the total distance traveled is 2 + 3 + 4 = 9, so that no back-tracking is allowed?
Write your solution on a piece of paper cut and folded into a 2x3x4 box, and drop it off at Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, March 3.
Friday, February 10
A long hallway has 20,000 LED lights (let's be environmentally friendly, as long as we've got so many lights and are making it up!). Each is operated by a switch that turns the LED light either on or off. As coincidence would have it, 20,000 people form a line at one end of the hallway. Initially the lights are all off. The first person walks through the hallway and turns each light on. The second person walks through the hallway and hits the switch on every second light, thereby turning all the even-numbered LEDs off. The third person walks through the hallway and hits the switch on every third LED, turning some on and others off. The fourth person hits the switch on every fourth LED, and so on.
Which LEDs are on after the 20,000th person has passed through the hallway?
Write your solution on a camping LED headlamp with elastic headband, and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office by 3:00 p.m. on Friday, February 10.
Friday, January 27
Prompted by the article on sudoku puzzles in Focus, the problem of the fortnight is the following sudoku puzzle.
4 |
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5 |
2 |
6 |
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2 |
3 |
6 |
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3 |
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1 |
2 |
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2 |
1 |
7 |
9 |
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9 |
3 |
8 |
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9 |
3 |
8 |
7 |
|||||
8 |
4 |
Friday, November 28
A bug starts from the origin on the plane and crawls one unit upwards to (0,1) after one minute. During the second minute, it crawls two units to the right, ending at (2,1). Then during the third minute, it crawls three units upward, arriving at (2,4). It makes another right turn and crawls four units during the fourth minute. From here it continues to crawl n units during minute n and then makes a 90-degree turn, either left or right. The bug continues this until after 16 minutes, it finds itself back at the origin. Its path does not intersect itself. What is the smallest possible area of the 16-gon traced out by its path?
Cut a sheet of paper to replicate the minimal 16-gon of the problem, write your solution on it, and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Monday, November 28. (Alternatively, since we received no copies of "A Bug's Life" for our earlier Problem of the Fortnight about the dance of a hundred ants, and since we have not yet seen the flick, problem solvers are invited to submit solutions on the back of "A Bug's Life" DVD; we request that problem solvers opting for this alternative format submit their solutions before Thanksgiving break so we can watch the movie in between naps and turkey sandwiches.)
Friday, November 11
Suppose a straight stick is broken in two places. The locations where the stick is to be broken are chosen randomly and the location of the second break does not depend on the location of the first.
What is the probability that the pieces will form a triangle?
Write your solution inside your favorite Pythagorean triple triangle, and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, November 11.
Friday, October 28
There are 100 point-sized ants on a meter stick, distributed and oriented randomly so that they are directed toward one of the two ends. The ants travel at 1 meter per minute. When two ants collide, they reverse their orientations, and if they reach the end of the stick unimpeded, they fall off. What is the longest time before the meter stick is guaranteed to be free of ants? Write your solution on the back of a copy of the DVD A Bug's Life and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. Friday, October 28.
Friday, October 14
A little problem about big numbers. . . .
Find the smallest N, or show that none exists, for which the decimal representation of 
ends in exactly 2005 zeros.
Write your solution on the back of two 2005 American League Championship Series tickets, or on the back of a $2005 bill, and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 Friday, October 14.
Friday, September 30
It's only the second week of the semester, but most of you have probably already nestled into your "assigned" seats for the term. We kick off the problem solving season this year with a seating rearrangement problem.
There are 25 seats in a certain classroom, arranged in five rows of five seats per row. Each student is to change seats by going to one of the four nearest seats -- the seat directly behind, directly in front, immediately to the left or immediately to the right of the seat he or she is currently using. Sitting on the floor isn't an option -- and neither is sitting in someone's lap! Determine whether a rearrangement following these rules is possible, starting with a full class of 25 students, and explain your answer.
Write your solution on the back of one of your discarded fall schedules -- you know, the ones you filled out before the last round of schedule shuffling in the Drop-Add period -- and drop it in the Problem of the Fortnight Slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, September 16.
While we're still uncertain whether the chicken or the egg came first, we are certain that this fortnight's problem is one to crow about! Sit on it for a while and see if you can hatch a solution!
Suppose you wish to know which windows in a 36-story building are safe to drop eggs from and which will cause the eggs to break. We make a few assumptions:
* An egg that survives a fall can be used again.
* A broken egg must be discarded.
* The effect of a fall is the same for all eggs.
* If an egg breaks when dropped, then it would break if dropped from a higher window.
* If an egg survives a fall, then it would survive a shorter fall.
* It is not ruled out that the first floor windows break eggs nor that the 36th floor windows do not cause an egg to break.
If only one egg is available, then the experiment can be carried out in only one way: Drop the egg from the first floor, and if it survives the fall, drop it from the second floor; continue going up a floor at a time until the egg breaks. In the worst case, this method would require 36 droppings.
Suppose that two eggs are available. What is the least number of egg drops in the worst case scenario you need to make in order to determine with certainty which floor is the last safe floor from which you can drop an egg?
Write your solution on an egg carton and drop it (sorry -- couldn't resist) by Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, September 30.
Friday, September 16
It's only the second week of the semester, but most of you have probably already nestled into your "assigned" seats for the term. We kick off the problem solving season this year with a seating rearrangement problem.
There are 25 seats in a certain classroom, arranged in five rows of five seats per row. Each student is to change seats by going to one of the four nearest seats -- the seat directly behind, directly in front, immediately to the left or immediately to the right of the seat he or she is currently using. Sitting on the floor isn't an option -- and neither is sitting in someone's lap! Determine whether a rearrangement following these rules is possible, starting with a full class of 25 students, and explain your answer.
Write your solution on the back of one of your discarded fall schedules -- you know, the ones you filled out before the last round of schedule shuffling in the Drop-Add period -- and drop it in the Problem of the Fortnight Slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, September 16.
Friday, April 22
Drum roll, please. . . . Our final problem of the fortnight:
With all the receptions, ceremonies and other events accompanying the end of the year and graduation, it's a sure bet that a lot of handshaking will occur in the upcoming weeks. And with that in mind, we introduce you to our final problem of the problem solving season.
Marge and her husband Homer went to a party where there were four other married couples, making a total of 10 people. As people arrived, a certain amount of handshaking took place in an unpredictable way, subject only to two obvious conditions: no one shook his or her own hand, and no one shook the hand of the person to whom he or she was married. When it was all over, Marge asked everyone how many hands he or she shook and was surprised by the replies: each of the nine people she asked gave her a different answer!
How many hands did Homer shake, and how did you figure it out?
In honor of the colloquium on Escher this week, write your solution on the back of a reproduction of your favorite Escher piece and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 Friday, April 22.
Friday, March 17
In honor of this week's colloquia on puzzle games and Penrose tiles, the Problem of the Fortnight asks you to take a walk on the tiled side and see if you can piece together a solution to either of the following problems:
Friday, March 17
Last Monday snow began to fall in Holland (again!) sometime before noon and fell at a constant rate until about dinner time. At noon a snow plow started to plow River Street. The plow cleared one mile of River Street during the first hour and one-half mile during the second hour. What time did it start to snow?
(Hint: You may assume that at any instant of time the volume of snow removed is constant; i.e. the snow plow clears snow at a constant rate. What does this tell you about how the depth of the snow and the linear distance traveled by the plow are related to each other? There's a neat calculus problem buried in the snow here. Can you dig it out?)
Write your solution on the back of a paper snowflake and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 on Thursday, March 17 before you head out of town on Spring Break to bask in the sun.
"In each of my classes this semester, I begin by rolling a die to determine whether the homework assignment will be turned in. In my early morning session, if I roll a 1 or a 2, then the class turns in the assignment. In late morning session, things are more complicated; they wanted to turn in assignments more frequently than just 1 day in 3. In this class, there is an escalating chance that the assignment will be turned in for each day that it is not turned in. In particular, the first day of class there is a 1 in 6 chance that the assignment will be collected. If the assignment is not collected, then the next day there is a 2 in 6 chance. If it is still not collected, then the chance rises to 3 in 6, and so on. Whenever I do collect the assignment, the probability of collecting the assignment on the subsequent day drops back to 1 in 6 and then begins to rise again. My question is this: in the long term, what expected fraction of the total number of assignments will the late morning session turn in to be graded?"
Write your solution on the back of an old homework assignment that wasn't collected and drop it in the Problem of the Fortnight Slot outside Dr. Pearson's office (VWF 212) by 3:00 on Friday, March 4. Solutions received by that time will have a probability p = 1 of being eligible for a prize.
Punxsutawney Phil came out of his den today to determine whether winter would persist another six weeks. As he was contemplating the skies, he noticed two goats, Harry and Billy, tethered in a pasture together and engaged in a spirited debate about their grazing areas. "I wish I were tethered with a longer rope," complained Billy. "You get to so much more to eat than I because you have a longer rope. And that's especially important during these winter months, when the grass isn't growing very quickly, if at all." 
"No need to get gruff, Billy goat," countered Harry. "It's true," Harry ruminated, "that my rope is 11 feet long, while yours is only 10. Mine, however, is tied to a ring on the outside wall of this circular silo, and I can just reach the point on the wall diagonally opposite the ring, so I get no grass at all in that particular direction. You, on the other hand, can graze over a complete circle, so it seems you are much better off than I. So stop your bleating!"
"Oh," sighed Billy, "I wish I had continued my studies of math. Then I could prove to you that I'm right!"
"Well, I don't agree that math would prove you right," rejoined Harry. "But I do agree that if we had taken calculus, we could probably settle this question for ourselves." Because Punxsutawney Phil has finished calculus before going to meteorology school, he just chuckled to himself and slid back down into his den.
Settle the great goat debate once and for all by computing the area each goat has for grazing.
Tether your solution to a circular slice of goat cheese (wrapped in cellophane, of course) and drop it in the Problem of the Fortnight slot outside Dr. Pearson's den (VWF 212) by 3:00 on Friday, February 11.
We received a request for a problem in three-dimensional coordinate geometry to start off the new year. At "Off on a Tangent," we aim to please, so here goes. . . .
The two lines
L1(t) = <4, -5, 1> + t<2, 4, -3>
L2(s) = <2, -1, 0> + s<1, 3, 2>
in three-dimensional space are skew: that is, they are not parallel and do not intersect. Find the distance between L1 and L2.
Affix your solution to the end of a barbecue skewer (it's never too early to think about summer!) and drop it in the "Problem of the Fortnight" slot outside Dr. Pearson's office (VWF 212) by
Friday, December 10
As we usher out the old year and look forward to the new, we invite you to
deliberate and show that this is true:
1 - 1/2 + 1/3 -
1/4 + ... + 1/2003 - 1/2004 = 1/1003 + 1/1004 + ... + 1/2004
Submit your rhyming proof
to Dr. Pearson (VWF 212) by
Wednesday, November 24
What's the most efficient way to bisect a triangle? Well, as it
stands the question doesn't quite make sense. What do we mean by
"efficient" and "bisect"? By "bisect"
we mean bisect the area, and one bisection is more
"efficient" than another if the length of the curve it uses it
shorter. For instance, the figure below shows four ways to bisect an
isosceles triangle, and of these, the one on the left is clearly the most
efficient.
The problem this fortnight is: What is the most
efficient way to bisect an equilateral triangle? That is, what is
the shortest curve that will
bisect your equilateral piece of pumpkin pie this Thanksgiving?
Write your solution in whipped cream on top of a pumpkin pie and drop it
off at Dr. Pearson's office (VWF 212) by
For an ellipse with major axis twice as long as the minor axis, what is the
ratio of the area of the ellipse to the area of the largest inscribed
rectangle?
Write your solution on the back of a discarded campaign sign and drop it in
the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by
Friday, October 15
You have twelve coins, numbered 1 through 12, say, and you know one is
counterfeit, but you do not know whether it is heavier or lighter than the
other eleven, which are of equal weight. Using a balance scale, what
is the minimum number of weighings needed to
determine with certainty (1) which of the twelve coins is bogus and (2)
whether the counterfeit coin is heavier or lighter than the other eleven .
. . and how do you do it?
Inscribe your solution on an "Omega" counterfeit of a $20 U.S.
gold piece (see http://rg.ancients.info/bogos/ for details on how a coin dealer bought such a counterfeit for $3500 from a
fellow who looked like Newman from "Seinfeld"), or write your
solution on the back of a $12 bill and drop it in the Problem of the
Fortnight slot outside Dr. Pearson's office by 3:00 p.m. on Friday, October
15. (Over fall break, you might try using a rusty balance scale to
convince your folks that they have a bunch of bogus currency in the house and
offer to "get rid of" it for them.) Please include your
math course (number and professor) on your solutions.
Friday, October 1
Since so many of you enjoyed the first problem, here's another in a similar
vein:
You have 50 coins, one of which is counterfeit and heavier than the other
49. What is the minimum number of weighings
needed, using a balance scale, to determine which coin is bogus, and how do
you do it?
Affix your solution (containing your name and your math class, please) to
the back of a buffalo nickel or a wheat penny and drop it in the Problem of
the Fortnight slot outside Dr. Pearson's office by 3:00 p.m. on Friday,
October 1.
Friday, September 17
You have eight coins, all of which look, feel and smell identical.
One of them is counterfeit, and it is heavier than the other seven.
You also have a balance scale, on which you can put coins in the pans on
each side and compare weights. What is the minimum number of weighings you need to determine which coin is bogus,
and how do you do it?
Tape your solution to a bogus Susan B. Anthony dollar and drop it in the
Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by