Good Examples of Questions to have Students Ponder Over Without Paper

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

56. 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?

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90. 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.

Answer 4

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.

Answer 5

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.

Answer 6

I call these "haircut questions": questions that you can think about, and answer, while getting a haircut, i.e. just thinking, no writing.

How about:

If everyone in the world (population 8 billion) chooses a random real number between 0 and 1, one by one, approximately how many times will a world record for lowest number be set?

There is an intuitive solution that does not require writing. And the solution could lead to further questions for curious pre-calculus students.

Answer 7

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.

Answer 8

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?

Answer 9

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?