Hope College | Department of Mathematics

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. Be sure to include all your work, and as always, please write your name, the name(s) of your professor(s), and you math class(es) on your solution -- e.g. Otto B. Werkin, Professor Pal N. Drome, Math 343. Drop your solutions in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, November 20.

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Previous Problems

November 6, 2009

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.

Write your solution (not just the answer) on a bag full of Halloween candy and drop it off in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 on Friday, Nov 6. As always, show all your work (answers without work will not be considered), and write your name, the name(s) of your professor(s), and your math course(s) -- e.g. Rush Inuit, Professor Sloan Cranky, Math 314.

October 16, 2009

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.)

Write your solution (not just the answer) on the back of a CIS program, and drop it off in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 on Friday, October 16. As always, be sure to show all your work (answers without work will not be considered), and be sure to write your name, the name(s) of your professor(s), and your math course(s) -- e.g. I.M. Stuck, Professor U.R. Doin-Fine, Math 242.

October 2 , 2009

What is the total number of squares (of all sizes) on a 40 x 40 checkerboard?

Write your solution on a square piece of paper and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, October 2. As always, be sure to include your name, your math class(es), and the name(s) of your professor(s) -- e.g. I.M. Countin, Math 123 & 345, Professors I. Tally and E. Numerate. Good luck and have fun!

Ten (not necessarily distinct) integers have the property that if all but one of them are added, the possible results are:

Friday, April 17

The last Problem of the Fortnight for the 2008 - 09 academic year:

Determine the last two digits of

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).

Write your solution (not just the answer!) on the back of two NCAA Final Four tickets and drop it off at Dr. Pearson's office (VWF 212) by noon on Friday, April 3. 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. Factor Fiction, Professor Count M. Up). Good luck, and have fun!

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.

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!

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?

Write your solution (not just the answer!) on a sanitized square piece of paper and drop it by Dr. Pearson's office (VWF 212) by noon on Wednesday, November 26. 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. Trap E. Zoid, Prof. Hy Potenuse, Math 225). Good luck, and have fun!

How many of the positive factors of 36,000,000 are not perfect squares?

Write your solution (not just the answer!) on a not perfectly square piece of paper and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, November 14. 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. Ima Rhombus, Prof. Ernest Try, Math 289). Good luck, and have fun!

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.)

Carve your solution (not just the answer!) on a pumpkin and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, October 31. 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. Frank N. Stein, Profs. Jeckel and Hyde, Math 371). Good luck, and have fun!

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?

Write your solution (not just the answer!) on the back of a picture of Foghorn Leghorn, and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, October 10. 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. Art Van Delay, Prof. Van Axle, Math 371). Good cluck, and have fun!

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?

Write your solution (not just the answer!) on the back of a picture of your dream car, and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, September 26. 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. Vince Van Go, Prof. Van Delay, Math 371). Good luck, and have fun!

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?

Affix your solution to an old egg carton and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, September 12. 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. Egg Zactly, Prof. Van der Number, Math 345). Good luck, and have fun!

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

[To be clear, the function is x to the x to the x.] Write your solution on the back of a picture of a famous tower and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, April 7. Be sure to write your name, the name(s) of your professor(s), and your math class(es) on your solution (e.g. Marge N. L. Growth, Prof. Cal DeRivative, Math 172). Good luck, and have fun!

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.

Write your solution on a piece of paper that is cut to resemble your probability density function, and drop it by Dr. Pearson's office (VWF 212) by noon on Thursday, March 13 (the day before Spring Break). Be sure to write your name, the name(s) of your professor(s), and your math class(es) on your solution (e.g. Marge N. Averra, Prof. N. T. Grate, Math 172). Good luck, and have fun!

Friday, February 29

Solve the equation

(ln x)² - 2.5(ln x)(ln(4x-5)) + (ln(4x-5))² = 0

where x and all expressions in the equation are real.

Attach your solution (not just the answer!) to a small natual log (i.e. a stick) and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, February 29. Be sure to write your name, the name(s) of your professor(s), and your math class(es) on your solution (e.g. Al G. Bragh, Prof. Basey, Math 172). Good luck, and have fun!

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.

Write your solution on a parallelogram and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, January 25. Be sure to write your name, the name(s) of your professor(s), and your math class(es) on your solution (e.g. Mario Quadratini, Profs. Square and Squarer, Math 121 and 225).

Friday, December 7

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?

Write your solution (not just the answer!) on a slip of paper and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, December 7. Be sure to write your name, the name(s) of your professor(s), and your math class(es) on your solution (e.g. Even Steven, Profs. Odd and Odder, Math 123 and 699).

Friday, November 16

Sequences: A sequence of numbers {a_n} has a₁ = 7 as its first term, and every other term after the first is defined as follows:

- if a term a_n is even, then the next term is a_n/ 2
- if a term a_n is odd, then the next term is 3a_n + 1

What is the 1000th term in this sequence?

Write your solution (not just the answer!) on your favorite picture of Leonhard Euler and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, November 16. Be sure to write your name, the name(s) of your professor(s), and your math class(es) on your solution (e.g. Even Steven, Profs. Odd and Odder, Math 232 and 295).

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).

Write your solution on your favorite root and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, November 2. Be sure to write your name, the name(s) of your professor(s), and your math class(es) on your solution. (e.g. Root E. Bagga, Profs. Klump and Kelp, Math 232 and 295.

Friday, October 12

The Problem of the Fortnight has gone national! More precisely, the Problem of the Fortnight will now be part of the national Problem Solving Competition. The Hope College student who submits the greatest number of solutions from now until the end of the year will be eligible for a prize!

The problem for the upcoming fortnight involves a regular tetrahedron, a solid whose four sides are congruent equilateral triangles. The problem is:

A regular tetrahedron has edges 36 units in length. What is the altitude of the tetrahedron? Give your answer in simplest radical form (i.e. x √ y form) and make sure you justify your answer.

Write your solution on a piece of paper, fold it into a tetrahedron and drop it by Dr. Pearson's office (VWF 212) by noon on Friday, October 12. As always, be sure to write your name, the name(s) of your professor(s), and your math class(es) -- e.g. Al T. Tude, Professors Newton and Leibniz, Math 131 and 132 -- on your solution.

Friday, September 28

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?

Write your solution on a blank (or almost blank) page from your favorite book folded accordingly and drop it on the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by noon on Friday, September 28. As always, please be sure to include your name, your math class(es), and your professor(s) (e.g. I. M. Smart, Math 321, Professor Moore-Smarter) on your solution.

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?

Write your solution on the back of your favorite scene from the "Rabbit of Seville" (you know, the Looney Tunes cartoon where Elmer Fudd chases Bugs Bunny into the stage of Hollywood Bowl, whereupon they then reenact their own version of Rosini's "Barber of Seville"), and turn it in to your professor by noon on Friday, September 14. Be sure to write your name on your solution, and if you are taking more than one math class, make a copy of your solution to give to each of your math professors.

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 A_n is the total area occupied by the n rows of circles, find the limit of the ratio A_n / A as n goes to infinity; i.e. find

lim _{n →∞} A_n / A

Write your solution -- showing all your work, please! -- on the back of your favorite picture of your favorite mathematician 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 20. As always, be sure to include your name, you math class(es) and the name(s) of your professor(s) -- e.g. Dave Hilbert, Professor Lindemann, Calculus 1 with Early Transcendentals -- on your solution.

Friday, April 5

Find the exact value of the continued fraction [1, 2, 3, 1, 2, 3, 1, 2, 3, . . . ].

Write your answer on the back of your favorite picture of Gauss and drop it by the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Thursday, April 5. As always, be sure to include your name, you math class(es) and the name(s) of your professor(s) -- e.g. B. Riemann, Professor Gauss, Calculus 1 -- on your solution.

Friday, March 30

Let {a_n} 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, . . . ].

Write your answer on the back of your favorite picture of Einstein and drop it by the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, March 30. As always, be sure to include your name, you math class(es) and the name(s) of your professor(s) -- e.g. Isaac Newton, Professor Barrow, Calculus 1 -- on your solution.

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?

Write your solution on a circular sheet of paper and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3 p.m. on Friday, March 9. As always, please be sure to include you name, the name(s) of your math professor(s), and the math class(es) you are taking -- e.g. David B. David, Circular Logic, Professor Roundabout -- on your solution.

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?

Write your solution to the back of a picture of a red-tailed hawk and drop it on the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 pm on Friday, February 9. As always, please be sure to include your name, your math class(es), and your professor(s) -- e.g. Carrie D. One, Math 101 (Longhand Multiplication), Professor Abacus -- on your solution.

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:

n the figure below angle AOB has a measure of 15 degrees and the length of segment A₁B₁ is 4. Segment A_iB_i is perpendicular to OB for each i = 1, 2, 3, ... The lengths of segment A_iB_i is the same as A_i+1B_i for each i = 1, 2, 3, ... Find the total length of the zigzag path A₁B₂A₂B₂A₃B₃ A₄B₄ ... Give your answer in closed form.

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.

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.

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.

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.

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
	5		2					6
		2	3	6
								3
							1	2
2	1		7					9
			9		3		8
	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.)

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

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:

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.

Friday, March 4

This Problem of the Fortnight involves a little probability and comes from a Hope College professor who has devised an interesting system for collecting homework. The question was originally send to Elvis to answer, but he thought it made a good problem of you to answer. Professor Collectsumup writes:

"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.

Friday, February 11

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.

Friday, January 28

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:

Submit your rhyming proof to Dr. Pearson (VWF 212) by 3:00 p.m. on Friday, December 10. Happy holidays!

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 3:00 p.m. on Wednesday, November 24. Happy Thanksgiving!

Friday, November 12

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 3:00 p.m. on Friday, November 12.

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 3 p.m. on Friday, September 17. The Problem Solvers of the Fortnight and their just desserts (!) will be announced in the next issue.

The Problem of the Fortnight

November 20, 2009

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Previous Problems

November 6, 2009

October 16, 2009

October 2 , 2009

Friday, April 17