100 Interesting Mathematical Problems, Exercises, Puzzles, and Diversions

Rudbekia Hirta has worries about problems that turn her students off math:

Learning Curves: Contest Problems: I've also been given a book produced at the University of Delaware in 1986 called "Resource Problems to Enhance the Teaching of Mathematics." Here's a randomly selected problem:

In a 5 by 12 rectangle, one of the diagonals is drawn and circles are inscribed in both right triangles thus formed. Find the distance between the centers of the two circles.

I can't use problems like this. Most of the problems in the book are what I would categorize as contest problems for general-population high school students. These might be fine for high school students who already like math, but those are not my students. I'll definitely have a bunch of pre-engineers who need to learn an ambitious amount of calculus in a short period of time (dictated by the departmental syllabus), and the other bunch is an unknown. The regular gen-ed class would mutiny over this sort of problem (these are the students so averse to doing unfamiliar problems that they will leave questions blank on the exam and leave early). The honors version of the gen-ed class does not have a math placement prerequisite -- just an overall standardized test score -- so I have no information about how good they are at math. I'm guessing, though, that if I were to give them puzzle problems that they'd be more appreciative of the LSAT puzzle problems or problems from the quantitative section of the GMAT.

This book also features word problems about completely fanciful nonsense situations:

Sally is having a party. The first time the doorbell rings, one guest enters. If on each successive ring a group enters that has two more persons than the group that entered on the previous ring, how many guests will have arrived after the 20th ring?

It's problems like this that make my weaker students hate word problems....

So what are some problems that have clear and immediate payoffs, or that will be of interest to somebody with a normal level of curiosity? This question is of interest to me as well, for two reasons:

1. I want the sophomores and juniors I teach to understand that math is a useful tool--which means assigning them problems that they can do and understand the substantive payoff.
2. I want to persuade the kids that the payoff to math is high.

I used to have a wiki that I hoped would become a place where people could contribute and edit interesting math problems. But, alas, it got caught in the great wiki-crash of 2004, as the spam-robots became more and more sophisticated. I had to shut it down.

This is as far as I had gotten:

One Hundred Interesting Math Calculations: How do you convince adolescents that there is a big long-run payoff from math? Teaching them (mine at least) that there is a huge short-run payoff from reading and a huge medium-run payoff to writing is easy. But math is harder.

1. [World War II Bomber Pilot Survival Odds]
2. [How Many Extraterrestrial Civilizations Are There?]
3. [Gravity and "Weighing the Earth"]
4. [Economic Growth Since 1500]
5. [Exponential Growth and Human Populations]
6. [How Much Blood Is There in the World?]
7. [Julius Caesar's Last Breath]
8. [The Birthday Fact]
9. [The False-Positive Problem]
10. [The Grass-Is-Greener Paradox]
11. [The All-Knowing Alien Paradox]
12. [Repeating Decimals]
13. [Introduction to Compound Growth]
14. [Elementary Ballistics: The Kinematics of Falling Bodies]
15. [Elementary Ballistics: What Goes Up Must Come Down]
16. [How Rich Is Fitzwilliam Darcy?]
17. [The Clock Hands Problem]
18. [Sunscreen, or the Freak Mutant Near-Albino Problem]
19. [The Distributive Law, or the Get-Out-of-the-Way Problem]
20. [The Federal Reserve Problem]
21. [The Ancestor Problem]
22. [Strategy Secrets of ENRON]
23. [The Muddy Parent Problem]
24. [The Kissing Problem]
25. [Understanding "Risk Arbitrage"]
26. [Orbiting the Earth]

And here are some suggestions from others for problems it would be interesting to write up:

Suggestions For Entries:

1. How long can Moore's Law go on? Starting from the average distance between atoms in a silicon crystal, find the time when chip features will be (supposedly) one atom wide.
2. Intro to counting and combinatorics. Suppose there were 14 (or 12) cards in one suit. Suppose there were 5 (or 3) suits in a standard deck of playing cards. How would the relative ranking of poker hands change? They don't all scale the same way. Do most of the work by cancellation, so you don't have to perform a lot of the (tedious, error-prone) multiplication.
3. Simple bits of probability, especially conditional probability from games -- card games (poker), dice games (craps), whatever. For example, understanding why it is harder to make your point the hard way (with a pair) when it is 8 rather than 4. As a grad student, I spent a lot of time teaching basic concepts to undergrads (at MIT!) that I mastered in middle school because i thought about the games i spent my time playing.
4. Consider the mathematics of triage versus parity policies as described by Garrett Hardin in Chapter 4 of "Promethean Ethics," University of Washington Press, 1980.
5. If you try something unlikely a few times, you might fail every time -- but it's commonly said "Even if the odds are a thousand to one against you, you try it a thousand times, you're sure to get it." Right? Wrong. If the odds are 50-50, and you can have two tries, you've got a 75% chance of a win. But it's downhill from there. One in a thou chance with up to a thousand tries? Only a 63.23% chance of a win. One in a million over a million tries? You're down to 63.21%, and it keeps dropping from there. How low can it go?
6. Xeno's paradox came about because the Ancient Greeks did not know how to sum an infinite series. I've always used it to illustrate the concept of limits approaching infinity, because it puts the complex math on the side of common sense.
7. For "Exponential Growth and Human Populations" set the end point as filling up the Americas by a colonizing group of 100 people, it's more interesting.
8. Richard Dawkins has an interesting calculation on human ancestors in the book "River out of Eden" (1995). Figuring 20 years per generation, calculate the number of ancestors you have 2000 generations ago if none of your ancestors appear more than once in your family tree (no inbreeding).
9. If the kids are into science fiction, have them work out dimensions of their favorite space ship based on extrapolation from the sizes of particular features (e.g. if the bridge of the Enterprise is so many feet across, how long is the whole ship?). Have them make upper and lower bound estimates to teach them error margins. It's not so great for web page presentation, though...
10. Small business economics. Next time you and the kids are at the ice cream shop or other restaurant, have them work out the typical number of customers per hour (from typical customers-in-store and customer-visit-time). From this and the amount spent by a typical customer you get typical revenue. Guess at employee wages and commercial space leasing costs. Ask them why the place closes at 9 instead of staying open all night.
11. Bridges of Konigsberg. Requires an illustration. The fundamental problem of graph theory.
12. Predator/Prey? balancing over time.
13. The different coin problem. N coins or objects of the same weight, one object of a different weight (in the simpler form lighter or heavier is known, in the slightly more difficult, just that it is "different"), a scale, and a limited number of weighings. Teaches binary group comparions. (Similarly, the switchback problem - You are at a fork in the road. You know your destination lies an unknown distance from the fork down one fork. What is the fastest way to surely find your destination?)
14. Gabriel's Horn. A mathematical object with finite volume, but infinite surface area. Thus you can conclude that if you wish to paint Gabriel's horn, it's much wiser (and less costly) to fill the horn with paint than to try to coat the outside. Full appreciation will require calculus experience. [Link] [Link]

Gabriel's Horn, alas! is too sophisticated for my purposes--but it is wonderful:

Gabriel's Horn - Wikipedia, the free encyclopedia: Gabriel's Horn (also called Torricelli's trumpet) is a figure invented by Evangelista Torricelli which has infinite surface area, but finite volume. Gabriel's horn is formed by taking the graph of y = 1/x, with the range x ≥ 1 (thus avoiding the asymptote at x = 0), and rotating it in three dimensions about the x axis...

Comments

[Alan]

"The regular gen-ed class would mutiny over this sort of problem (these are the students so averse to doing unfamiliar problems that they will leave questions blank on the exam and leave early)."

Then of course they will go home and play with XBOX or PS or whatever. My mom was a teacher in Oakland schools up to about 3 years before she passed away. The stories she would tell about experiences there led me to believe that the problems teachers face at this level are rooted far earlier in time before the kids get to jr. high school.

I admire Ms. Hirta for doing what she can but it still strikes me as akin to polishing the nose of the Columbia.

[Rudbeckia Hirta]

A substantial minority of the students that I teach in any given year are planning careers as elementary school teachers. They struggle with questions on my state's NCLB test for 8th graders. Up until VERY recently my state's standard "elementary" teaching license has been a K-8 all subjects license.

The only word problem that has been consistently engaging has been in precalc in the exponential functions chapter. I looked up the tolerance levels of various drug tests and the half-life in the body of the chemicals that the test looks for and my students had to calculate how long they had to stay off drugs in order to pass a pre-employment drug test.

[nk]

i've not had a chance to look through all the math. here are some ideas:

1. forgetting curve- do you have a math problem on the forgetting curve?
http://www.gbn.com/ArticleDisplayServlet.srv?aid=1100

2. chaos theory - the math\theory behind chaos theory (intro, intermediate and advanced)

3. fat tails - www.trendfollowing.com/whitepaper/mauboussin.pdf

[vincible]

There are several good probability problems in here, about condom use and AIDS (though it is an essay and not written in "problem" form):

http://www.winkola.com/writings/math-aids.html

[masaccio]

I really don't envy parents today. Their ability to control the intellectual environment of their children is much lower than even ours was 20-25 years ago. We did not get cable, and when we started with computers, I was able to show them that the things were tools, because I was able to program them in several different languages. These two simple steps, coupled with a vast array of books, set up an environment of reading and puzzle solving that they found entertaining.

I do not know what we would do if we faced an environment full of cable, iPods, internet, dvd, video games, and other forms of essentially passive pasttimes.

[masaccio]

I really don't envy parents today. Their ability to control the intellectual environment of their children is much lower than even ours was 20-25 years ago. We did not get cable, and when we started with computers, I was able to show them that the things were tools, because I was able to program them in several different languages. These two simple steps, coupled with a vast array of books, set up an environment of reading and puzzle solving that they found entertaining.

I do not know what we would do if we faced an environment full of cable, iPods, internet, dvd, video games, and other forms of essentially passive pasttimes.

[quixote]

I am not a math person, and that was almost entirely due to word problems. Stuff like the nonsense about how many people go to a party, or the two stupid trains that are always meeting in some place and time where I've never been. And the coin toss, dice-rolling, card-game stuff wasn't any better because none of those meant a thing to me. I could feel my brain shut down after the first couple of words.

My whole life would have been different with the vastly improved word problems suggested. It wouldn't have taken me twelve years to realize I didn't want to major in chinese philosophy. I wanted to major in biology, even though I would have to take--gasp--a couple of math and stats classes.

(Having won through to a Ph. D. in biology, I thought I'd link to a picture of the beautiful plant that equals Rudbekia's name, although with an added "c."
http://biology.smsu.edu/Herbarium/Plants%20of%20the%20Interior%20Highlands/Flowers/Rudbeckia%20hirta%20var.%20pulcherrima%20-%203.jpg )

Suggestions for other interesting word problems: instead of finding solutions, how about critiquing/correcting the math of politicians in their speeches?

[nk]

is there something on S-Shape adoption curves (diffusion)?

[rilkefan]

Tiny url for quixote's flower: http://tinyurl.com/9j2th

[Jim S]

As others have hinted at what we really need are good problems for the younger ages so they're not completely alienated by math at a young age. Some suggestions?

How about calculating the materials needed to build a tree house? Teach the teachers how to involve even urban students with this one, something along the lines of a contest to design one for a nearby park. Calculations involving getting to design their own room in a new house might involve some students. How many posters can fit on a bedroom wall if they are different sizes? Show videos of something like an airplane flying before you ask them to do calculations on distances, speeds and vectors. Come up with things to make problems as concrete as possible before approaching them on an abstract basis.

[cafl]

If I understand her post, she is creating problems for college students. ("Honors gen ed", "pre-engineers") Word problems/problem solving needs to start much earlier than college. My kids had a page of word problems and puzzles assigned each week by their 5-8 grade math teacher (Calif. public charter school). Since I helped in that classroom, it was interesting to see these kids learn the skills they later used in algebra to learn to set up equations to solve algebra word problems. To Quixote, while math is beautiful in itself, the point of math to many or most of us is to be able to apply it. For that you need to be able to do "word problems". For a good time, see George Polya, "How to Solve It". :->

[anne]

Though I cannot recall precisely there was a fine math program for students begun in Mississippi by a Civil Rights Worker who had gone to Harvard Education to design a teaching program. The name escapes me, but here was a program for young Mississippi public school students that brought nice results where there had been only poor results. I am thinking....

[timfc]

Anne...

The Algebra Project, Robert Moses.

But, it's not really full of the sorts of problems that would be appropriate for RH's students.

tf

[Anita Hendersen]

Brad, man, I was looking for your list of math problems a few months ago, but couldn't find them. Interesting stuff. I'm sure I'm in a minority among your blog readers, but I actually appreciate that you have math problems.

[anonymous]

In a 5 by 12 rectangle, one of the diagonals is drawn and circles are inscribed in both right triangles thus formed. Find the distance between the centers of the two circles.

Square root of 65.

[anonymous]

Incidentally, here's a good one that I've heard has been used in Microsoft interviews: a man offers to play the following game of chance with you. He puts down $2 on a table, and you take a coin (perfectly balanced so that there's a 50/50 chance of landing heads or tails) and flip it. If it comes up heads, he gives you the money on the table, and the game ends. If it comes up tails, he doubles the money on the table, and the game continues. You keep flipping the coin until the game ends. The man asks you to pay him for the privelege of playing the game, and when you ask him how much, he tells you to make him an offer. How much should you offer to play the game?

There are at least 4 correct answers to this question, depending on how you think about it.

[anne]

Tim FC

Thank you, well remembered. But you are right, Robert Moses' approach is not directly appropriate.

Rudbeckia Hirta

I admire you, and am much interested in learning of several possible solutions.

Anita Henderson

Me too.

:)

[anne]

Everywhere about there are fledgling chasing after food. Thoroughly exciting. Carry about a bag of peanuts and stand still with some in your hand, you might even have a male cardinal carry one off and there you will see what red is all about. Oh my :) The female was too smart to come today.

[Mike Maltz]

Here's one I used to use on my students: How much do you save on a yearly basis if you consistently search out a gas station that is cheaper by 5 cents/gallon?

I get them to tell me how much they drive and what kind of mileage they get. Then, assuming 1,000 miles/month and 20 mpg (= 600 gallons/yr), I show them that you save the munificent sum of $30.00.

It always shocks them -- but you know, I still look for the lowest price gas, too! I think it has to do with the frequency of gas transactions and the continual provision of information at every turn.

[dberg]

If you have a 2000 calorie per day diet and 20% of your calories come from fat, how many tablespoons of butter can you eat a day (assuming you eat no other fat and that butter is 100% fat)?

[DoBeDo]

"these are the students so averse to doing unfamiliar problems that they will leave questions blank on the exam"

That quote certainly describes me in high school and my first two years of college. To me math was a mystery that only the cognoscenti could unravel. Or, really, unravel is the wrong word. Because, as far as I could tell, the math cognoscenti looked at a problem and by some kind of magic solved it. I had absolutely no clue as to how to solve an unfamiliar problem.

Then I was lucky enough to take Computer Science 101 at the University of North Carolina. The goal of this course was less to teach how to program and more to teach how to solve problems. In fact, the primary text for the course was a book by Polya called "How to Solve It" (for a nice summary of this book see http://www.math.utah.edu/~alfeld/math/polya.html).

Through the fifteen weeks of the course our professor showed us at each class how to use Polya's method to solve that week's programming problem. This course got me so excited about programming that I made it my career. And, having learned Polya's method, I was able to go on and really begin to learn math. In fact, I learned enough to work for statistical agencies of the U.S. government for many years.

So if you are looking to teach students how to solve unfamiliar problems, I hardily recommend using Polya's book.

[John Quiggin]

Have wiki defences against spam robots improved? I'm thinking about setting one up, but I can't afford too much time on this.

[nk]

low price gas may be a good chance to teach opportunity cost and investing-

what if someone used his or her time to invest in ExxonMobil or an energy index on the "dips", vs. using time to search for low-cost gas?

[nk]

i can see how saving five cents per gallon of gas would be an interesting math opportunity.

consumer education story problems would probably be valuable (credit card fees, managing credit rating - all kinds of math, insurance premiums and deductibles, interest expense on loans, saving and investing, awareness of brand marketing and related costs)

[quixote]

Thanks to rilkefan for the improved URL for the flower. Mine messed up rather badly. To cafl and others who mentioned Polya's "How to solve it," that sounds like something well worth checking out. Why don't math teachers use that as a math text (for crying out loud!)? I noticed, as time went by, that the ability to solve word problems was useful.... Unfortunately, by then my mind had been poisoned, and it took an analog of cognitive therapy to sort it out.

[Tad Brennan]

small point:

the guy's name was "Zeno", not "xeno". (Fear of paradoxes is different from xenophobia).

bigger point:

Some aspects of Zeno's paradoxes become non-paradoxical when we are comfortable with algorithms for summing infinite sequences and series.

Some other aspects stay paradoxical.

These paradoxes don't simply dissolve when you pour math on them; some parts are still devilishly curious, all math aside. Sure, the distance across the stadium does not get any longer than a finite number, just 'cause you cut it into an infinite number of (decreasing) segments. But the number of tasks you have to complete to reach the other end is still infinite. And therein lies scope for further paradoxes.

See, e.g. J. J. Thompson on "Tasks and Supertasks", with a later piece by Benaceraff on the same. Thompson imagined a desk-lamp with a button that toggles on and off. Press it on at t-60 seconds. Press it off at t-30 seconds. Press it on at t-15 seconds. Continue in this way. When we reach t, will the lamp be on or of? This is a super-task. There was also a paper published in MIND in the middle '90s called "A Beautiful Supertask", finding analogous paradoxical behavior in atomic motion.

[Tad Brennan]

Two other suggestions:

1) A book called "The Number Devil" by H. M. Enzensberger (originally in German, Zahlenteufel), was pretty fun reading for my fifth grader, and taught me a few things about Fibonacci's sequence that I hadn't known.

2) The Platonic solids, i.e. the five regular convex Euclidean solids, have long been a thing of beauty to me. They lend themselves to hands-on arts and crafts projects, but have lots of good math in them too. They are also a good entre to understanding two excellent books: Lakatos, and Euclid himself. Also, Euclid's proof that there can be five and only five perfect solids is in some sense a perfect proof itself: so simple and clear, yet it proves something deeply non-intuitive. Seeing how it works and what it accomplishes might help with the general question "what's the good of doing proofs anyhow?", which is one of the hardest things for math people to communicate to non-math people.

[tim silman]

this is off topic, but are the answers to the problems mentioned 6.5 inches and 39 (that one I'm sure of ((19x2)+1))? just curious, mosing about and figured some quick online math problems would be a great way to not write the paper i should be working on.

[anonymous]

tim silman> this is off topic, but are the answers to the problems mentioned 6.5 inches and 39 (that one I'm sure of ((19x2)+1))? just curious, mosing about and figured some quick online math problems would be a great way to not write the paper i should be working on.

Question 1: (the inscribed circles in the two halves of the rectangle) the answer is the square root of 65. Each half of the rectangle is a right triangle with sides 5, 12, and 13. Without the ability to draw a picture, it would be hard to explain how you can figure out that the circles have radii of 2 units. But once you know that, it's easy to figure out that their centers are separated by 1 unit in one direction and 8 units in the other. The distance is the square root of 1 squared plus 8 squared, or the square root of 65.

Question 2: (houseguests and doorbells) the answer is 400. After 1 ring, 1 guest has entered. After 2 rings, 1 + 3 = 4 guests have entered. After 3 rings, 1 + 3 + 5 = 9 guests have entered. After n rings, 1 + 3 + ... + (2n - 1) guests have entered. 1 + 3 + ... + (2n - 1) is equal to (2n - 1 + 1) * n/2, or n squared. So after 20 rings, 400 guests have entered.

[theorajones]

In 6th grade, I had a math teacher who gave us $100 monopoly dollars and instructed us to go to the newspaper and "buy" $100 in whatever stocks we wanted. We followed the stocks (in the newspaper) for a couple of months and graphed the prices. We could trade, but there was a cost-per-trade, etc. Kids who did the best got a prize of some kind. Some kids even brought in "secret" information-publicly available stock analyses. It was amazing, and it's the only reason I know anything about financial markets.

Something like that would be great. Also, how's about an analysis of casino games? Run through the cycles enough times, and you'll have kids begging to be the house, and refusing to play unless they can be the house. Which is good. And how's about learning stats by lotto playing? Have the kids NOT BUY TICKETS, but keep track of what numbers they would have picked and what they would have won. And at the end of the year, when nobody's won...

[Tim Silman]

whoa....should have spent more than 1 minute on those problems before posting, especially the second one. ouch. further evidence that ADD does exist

[nk]

some sort of story problem on internet would be interesting-

download\upload speeds for cable vs. dsl- when does this matter anyways?

also, story problems on population density, internet usage, and affect on download\upload speed (cable and dsl)

regarding the two trains\cars leaving from different locations and traveling at different speeds problem and headed toward the same location:

how about a problem on two pieces of information\data leaving from different locations at different speeds yet headed toward the same point --> when do they (the info\data) meet?

[nk]

hypothetical problem:

If I purchase toilet paper via the internet and have it shipped direct to my house, is the freight charge (in $s) a function of 1) the weight, 2) the volume, 3) distance shipped, or 4) some function of weight, volume and distance?

there might be a chance to work in a currency conversion calculation.

what kind of water does Professor DeLong drink?

http://njk42.blogspot.com/2005/07/obligatory-shopping.html

[Tim Gray]

What about: How many people have *ever* lived on the earth? (ie. 6bn+ currently plus all those who have died since the very first human - roughly!)

Tim Gray

[Academic Coach]

This is fabulous stuff. I want to send it to my daughters' 5th, 3rd and kindergarten teachers. Anybody know how we can find/create relevant probs for elementary school students who could care less how many apples are in the cart but might be fascinated by how many kids play neopets?

[Panic]

I tried to enter this post into the mathematical problem listing regarding the all-knowing alien paradox but was unable to so I thought I'd try sending it here.

"There is actually a fairly simple response to this situation.

The real philosphy behind this so-called paradox lies in the fear that they may lose everything.

This is an example of humans irrationality and inability to actually logically comprehend this situation without the influence of emotion. If you take the $10 then the $1,000,000 won't be there. It was a certainty. If you leave the $10 then the $1,000,000 will be there. Debating the idea will not change the outcome only prolong what was inevitable.

The idea that the $1,000,000 will be there even if you take the $10 first would never work because the paradox stated that the alien was always right.

This leaves only one real issue: Fear.

The fact is man is stupidly conceited. It's been forgotten that we are not alone.

The solution to this situation where one might not fear the outcome is (Though to be afraid that one might not profit from a situation totally undeserved is a pretty pathetic dipiction of today's ideals. We need to learn to work for what we get.):

Have someone else take the $10. Then you can take your $1,000,000. If it's not there than the alien was full of it (YOU didn't take the $10) and there's nothing that can be done about that but the free $10 has not gone to waist and you don't need to think about "what ifs" were you to have taken the $10 and found the second empty."