# Good Examples of Questions to have Students Ponder Over Without Paper

A few times now I've found myself in a situation where I want to give a precalculus-level undergraduate student something to think about that's kinda fun and that's well outside of their coursework. But at the same time, I want to ask them a question they can figure out just by thinking about it, or pondering over it, rather than having to work out some calculations or necessarily write things down.

So far my go-to question has been this one:

Suppose you take a pen and mark five points on a ball. I claim that no matter where you draw those points, I can find a closed hemisphere of the ball that contains exactly four of those points. Is this true?

I like this question because there's very little you can write down, and the geometric set up is simple enough that a student should be able to play with it in their head (or use an actual ball). But I'm not perfectly happy with this question because I have to explain to them about what I mean by a closed hemisphere, and I usually have to discuss some ideas of spherical geometry with them. But I feel like this gives them the impression that this problem would never have been tractable without my guidance. I want a problem that they evidently can think about and figure out without some necessary guidance, and possibly without having to write anything down or draw out any examples.

Does anyone have better ideas for questions like this I can ask students?

The reason I'm asking for such questions is that I want students to get over their fear of "doing the wrong thing," so I want to give them problems where there is nothing to write down and they have to mentally grapple with the question. This fear seems to spring up more easily if figuring out a question involves writing (maybe because I can see and judge their work?). They'll often sit there with a blank page and just look to me for guidance: they want me to tell them what the "right thing to do" is.

• I like to ask how many times the minute hand and hour hand of an analog clock overlap in a day. I first learned of this problem in The Tokyo Puzzles by Fujimura (edited by Martin Gardner). I actually keep a book of puzzles, and a few physical puzzles (e.g., a wooden sphere that can be taken apart; a Rubik's cube) on my desk. I like Norton Starr's L-tromino puzzle, too (link). Not sure if any of these fits the bill... – Benjamin Dickman Mar 30 '17 at 1:10
• @BenjaminDickman, I like that problem. And I'm sure many Martin Gardner style problems fit what I'm after too. In fact, when I have a bored student and there's a whiteboard available, I'll often ask them the Gardner problem about the planes on a small island that can refuel each other and need to make a trip around the world. Or if the student is in my office, I'll give them this puzzle, where you are supposed to put it back together into a cube. – Mike Pierce Mar 30 '17 at 1:24
• I have the same cube puzzle on my desk! The sphere one is here; students routinely solve the cube you linked as well as the Rubik's cube, but none of them has successfully defeated the sphere. Yet. – Benjamin Dickman Mar 30 '17 at 1:28
• @BenjaminDickman Really, students can tackle a Rubik's cube but not the sphere? That sounds like a challenge. ;) I'll be on the look out for it. And I'll bet one day a student will figure out how to reassemble it into two balls that are each identical to the original. :P – Mike Pierce Mar 30 '17 at 1:37
• One more: You walk up a mountain between noon and 3pm, and walk back down along the same path the next day between noon and 3pm. Must there be a location that you occupied at exactly the same time on both days? [Yes: Imagine overlaying two videos of these walks; clearly the you-going-up and the you-going-down cross paths at some point.] – Benjamin Dickman Mar 30 '17 at 6:57

A possibility, requiring one definition: What is a tiling of the plane with an infinite supply of congruent copies of a single tile (technically, a monohedral tiling). This can go as deep as you'd like, perhaps in stringing together several mini-sessions.

1. Can every triangle tile the plane? (Yes.) Form parallelograms, then argue that a parallelogram tiles the plane.

(Image from Mathematica Demo.)

2. Can every quadrilateral tile the plane? (Yes.) This is not as easy to see. Easier if restricted to convex quads.

(Image from Mathematica Demo.)
Cited in Mathworld, with this reference: Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 177-179, 208, and 211, 1991.

3. Can every pentagon tile the plane? (No.) The regular pentagon cannot. This is at least easy to see if the tiling is edge-to-edge.

4. You get to mention a recent advance at the frontiers of math:

(Image from Wikipedia.)

5. You get to mention an unsolved problem, beyond the frontier: Is there an algorithm for deciding if a given, single tile can tile the plane?

Update (11 Jun 2017). The pentagon tiling problem is just settled: The list of $15$ types is complete. See the Natalie Wolchover article in Quanta.

• I quite like this progression of questions. :) I am concerned though, that students won't be able to convince themselves of even the case with triangles without drawing out some examples first. Like I was hoping for questions that they could keep and work-out entirely in their head. – Mike Pierce Mar 29 '17 at 19:25
• Also, I can see the construction in the case of quadrilaterals, but I can't see any way approach a proof besides "know that construction, and prove that construction works." I suppose I'm worried that if a students gets stuck on that question, what could I say to guide that student in a fruitful direction? – Mike Pierce Mar 29 '17 at 19:27
• @MikePierce: You are certainly correct that working entirely in your head would be challenging. But just doodling would make it feasible. I think if you want to keep it purely "in your head," then perhaps it has to be a logic puzzle...? (I must admit, I find the "in your head" restriction a bit unnatural.) – Joseph O'Rourke Mar 29 '17 at 19:45
• Yeah, it's kind of a weird restriction. I've edited my original question to include why I'm looking for questions like this. – Mike Pierce Mar 29 '17 at 20:01
• @MikePierce: Re quads. Yes, that would not be easy. Perhaps it is best to just skip quads, and tell them it turns out every quad does tile. They should be able to see that a regular pentagon cannot tile, just from multiples of $108^\circ$ mixed with $180^\circ$. – Joseph O'Rourke Mar 29 '17 at 21:22

Perhaps logic puzzles would work in this case. Some classic examples are:

1. You're traveling along a road and arrive at a fork. Two guides are posted, but one always lies and the other always tells the truth, but you don't know which one is which. What one question can you ask to find out which path you should take?

2. Three boxes are labeled "Apples", "Oranges" and "Apples and Oranges" but each box is labeled incorrectly. How can you tell what each box really is by only picking one fruit from one box?

3. You know you have ten white socks and ten black socks in a drawer. In the darkness of the morning, how many do you have to pull blindly out before you are guaranteed to have a pair? What if there's only two white socks and one-hundred black socks? What if there are also some grey socks? What if you are an alien and need three matching socks?

What I like about these is that they're pretty easy to understand but nontrivial to solve and guessing is worthless because the real answer comes with a reason/proof/explanation.

Since Benjamin Dickman mentioned The Tokyo Puzzles in the comments, I'll include a couple of the questions from that book here that I thought fit the prompt nicely; neither requires a pen and paper to think about, and a student can just ponder them mentally.

1. Two brothers decided to run a 100-meter race. The older brother won by 3 meters. In other words, when the older brother reached the finish, the younger brother had run 97 meters. They decided to race again, this time with the older brother starting 3 meters behind the starting line. Assuming that both boys ran the second race at the same speed as before, who do you think won?

1. Between noon today and noon tomorrow, how many times does the long hand on the clock pass the short hand? "Pass" means that one hand follows, overtakes, and goes ahead of the other. Since both hands are at the same spot at noon, the long hand does not pass the short hand at twelve o'clock, the starting time.
• +1: The latter is one of my favorites! (It periodically rears its head in some form during interviews of the google facebook quant variety...) – Benjamin Dickman Mar 13 '18 at 21:13

Ask them to give you a number between $0$ and $100$. The number they give you must be as close as possible to $2/3$ of the average of all the class' numbers.

We could argue that this is more psychology than math, but it fits the no wrong answers.

See Wikipedia for some history of this.

• This is a cool idea, but it still doesn't quite fit what I'm looking for. I want to ask them a math question, so a question that has a right answer or is either true of not true, but I just want to avoid the student treating it like every other math exercise they've seen in school previous. Problems where when presented, the student thinks, "Oh, this is a _______ kinda of problem, so I need to get out my piece of paper and do the procedure that solves a _______ kind of problem." And the way I'm trying to avoid this is by asking a question that doesn't entail anything being written down. – Mike Pierce Mar 30 '17 at 1:15

Matt Enlow recently posted a wonderful collection of problems on twitter in his tweet here.

He links to the "More Questions than Answers" pdf on dropbox here; it is a compilation of 100 math problems, which run the gamut in difficulty and range (for me) from well-known to never-before-seen.

• Any chance you have time to edit this image of text into a quote block containing actual text? – shoover Mar 14 '18 at 15:52
• I couldn't find anything in MESE meta, but maybe things are different here. Over at SO, they frown upon text-as-image. – shoover Mar 14 '18 at 22:42
• SEO is part of it, but more importantly screen readers will think that your post ends "The document begins: enter image description here". – Peter Taylor Mar 15 '18 at 16:59
• Dear Downvoter: the text image is gone. If there is another reason for dving, please let me know. – Benjamin Dickman Mar 23 '18 at 15:32

Here's another one that I like.

Suppose you and a friend are sitting at a circular table and you each have a large stack of pennies, so you start to play this game where you take turns placing one penny flat on the table. At each turn you may not place a penny over any other penny (because the pennies must be flat on the table), no pennies may lay over the edge of the table, and once a penny has been placed it cannot be moved. The last person to be able to place a penny on the table wins.

• It's your turn to go first. Do you have a winning strategy?

• (And then after the student answers the first question) What if the table isn't circular? On what other shaped tables will your strategy work?

There's isn't an obvious procedure to solve this problem, and since the situation just involves placing pennies on a table (so placing little circles into a bigger circle), a student should be able to fiddle with the problem in their head easily. A student should at least be able to come up with a response that will prompt a discussion without having to write something down first.

I found another one I like. It's the example in this question on MathSE. There are lots of other good ones in this post too.

Three friends Albert, Betty, and Chadwick ask the Game-Master to play a game with them. The Game-Master agrees, and proceeds to paint two colored dots on each of their foreheads. The dots are either blue or yellow, and among all the dots on their foreheads no color is used more than four times. They can each see the colors on the others' forehead, but not the colors on their own. The Game-Master asks the friends in the order Albert, Betty, Chadwick, Albert, Betty, Chadwick, ... whether or not they know the colors of their dots. If one says "no", the Game-Master asks the next friend. If one says "yes", that person has to state their guess, and if they are correct, the friends win the game, but if they are wrong, then they lose. The friends were not given any time to make up a strategy.

They start to play the game, and their answers are

No, No, No, No, Yes!

and they win the game! What are the colors of the dots on Betty's head?

There are some great puzzles here on Andrej Cherkaev's website, most of which he attributes to Vlad Mitlin. I'll list paraphrased versions of a few of them here which I thought most fit the requirement that they may be pondered over without any suggestion that they require calculations to answer. Note that I've reworded these questions rather dramatically.

• A monk needs to meditate for exactly forty-five minutes, but - living in an abbey - he doesn't have a watch or a clock with which to time himself. All he has is two incense sticks, which he knows each take exactly one hour to burn. Unfortunately, being hand-made, the incense sticks aren't identical to each other, and they're imperfectly shaped so that he can't rely on a stick burning at the same rate all the time (so if he were to break an incense stick in half, there's no guarantee that would burn for exactly half an hour). Using these incense sticks and some matches, how can the monk arrange for exactly forty-five minutes of meditation?

• You throw a party and invite only your $n$ closest friends. Everyone arrives at the party, but then you realize that none of your $n$ closest friends have ever met one another! You frantically start introducing them, when you realize that you can make a game out of this situation. Can you introduce your friends in such a way that after you've finished, no three of your friends will have met the same number of people at the party?

• The military is preparing to do a training exercise at night on a large flat open field. They need to illuminate the entire field for the exercise, so four planes fly overhead and each airdrops a giant lamps somewhere onto the field. Each lamp is very powerful, but they were designed to be placed in corners, so they're each only able to illuminate a quadrant, an area spanned by ninety degrees, of the region around them. Upon trying to move the lamps to the corners of the field though, the soldiers on the ground realize that the lamps are too heavy to move. The lamps however can be rotated in place. Is there way that the lamps can be rotated so that when they are turned on, they can still illuminate the entire field?

• "A few times now I've found myself in a situation where I want to give a precalculus-level undergraduate student something to think about that's kinda fun and that's well outside of their coursework." Why? (1) Are the students all getting As and nailing the existing work and have free time to do more? And even if so, why not just let them have true free time to work on other courses? (2) Is it really that the students will benefit from this sort of exercise or just that you enjoy doing it yourself? – guest Feb 1 '18 at 8:23
• @guest The latest actual scenario has been that I'm giving out quizes in discussion (I'm a TA), and a few students finish well before time is up, so I just put one of these sort of questions on the back of the quiz for them to think about if they like. Since there's a quiz going on, they can't really do much else. So yeah, with regards to your comment "why not just let them have true free time to work on other courses," I'm not assigning questions like these to students. That would we weird. In this case it's just to give them something to think about while they wait. – Mike Pierce Feb 1 '18 at 17:48
• @guest More generally though, I'll occasionally get a student of a certain type: they usually like math, are pretty unchallenged by the coursework, but they show up to office hours anyways ("good student" types). I'll often give questions like these to this sort of student. They appreciate the challenge. More than that though, I think they appreciate how these sorts of questions contrast with their usual coursework. They've gotten an impression of "what math is" from the lineage of classes they took as a high school student and undergrad, and questions like these show a side of math, ... – Mike Pierce Feb 1 '18 at 17:49
• ... and encourage a type of mathematical thought, that they likely had never thought existed. Just an acedote, I once gave this Martin Gardner problem to a student taking trig/precal. While working on it he would bounce ideas off me, I would ask him guiding questions, and we had real discussion about it. It's really hard to have a earnest discussion like this over the problems that students see in their typical coursework. ... – Mike Pierce Feb 1 '18 at 17:49
• ... Anyways, after he had figured it out, the student told me that at no point in his life had he ever felt so engaged in a math problem before, and that he now understood why some people could actually enjoy doing mathematics. – Mike Pierce Feb 1 '18 at 17:50