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Are there any existing sources for topics or projects which are:

  • Suitable for first and second year undergraduates taking introductory math classes (possibly with some prompting or a couple pages of background to point in the right direction).
  • Have sufficient low-hanging fruit that a group of 3-4 students will typically discover something meaningful within 3 or so hours of group work.
  • Are open ended enough that a group won't run out of things to find.

The second condition seems to be the tricky one; with more time, students can spend a long time messing around with data collection and hunting for patterns and finding some false starts. This is a relatively short program (45 min a week for 4 weeks), so we need things which are a bit more directed.

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  • $\begingroup$ What does introductory math classes mean? I don't know the students' level well enough from that to be helpful... $\endgroup$
    – Sue VanHattum ♦
    Jun 18, 2015 at 16:57
  • $\begingroup$ @SueVanHattum: Ah, good point. In this case I mean calculus, but I don't want the projects to be calculus oriented, which is why I was trying to be vague. What I really mean is early undergraduate mathematical maturity and very little specific knowledge beyond maybe algebra. $\endgroup$
    – Henry Towsner
    Jun 18, 2015 at 17:00
  • $\begingroup$ Why don't you want calculus oriented projects? What are your goals? $\endgroup$
    – Sue VanHattum ♦
    Jun 18, 2015 at 17:21
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    $\begingroup$ Since it's not listed as "related," maybe it's worth linking back to MESE 1042: Ideas for high school pure maths projects $\endgroup$
    – Benjamin Dickman
    Jun 18, 2015 at 18:22

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Pythagorean triples may be open-ended enough. There are certainly challenges enough, and I think there is some low-hanging fruit. Depends on how willing to persist your students are. It's very algebraic, and may need a math circle environment, where the leader can encourage them to keep trying. (Also, the typical approach, with a = m^2 - n^2, b = 2mn, c = m^2 + n^2, is not necessary. There are more natural ways to find patterns.)

Analyzing the game of Spot It is an interesting problem. I've posted about using it many times on my blog.

There are lots of interesting problems on the Julia Robinson Mathematics Festival site. I think many of them would work for your purposes.

You might also enjoy Conway's Rational Tangles, available on Tom Davis' site (under Miscellaneous), along with lots of other possibly great stuff.

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  • $\begingroup$ I was trying to keep the question general, but maybe that was a mistake. This is for a program to expose students who've only seen algebra and calculus to topics they probably don't even realize are part of math in the hopes of recruiting some as majors. In the first part of the program they work together on puzzle-y problems on topics (probability, graph theory, etc.). We want to add a more exploratory project where groups of students build on a topic to work something out for themselves and give a presentation to the broader group. (This means we do have leaders around to support them.) $\endgroup$
    – Henry Towsner
    Jun 19, 2015 at 15:10
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Art Gallery Theorems are an accessible topic with many unsolved problems. Here is one:

How many $2$-transmitters are sometimes necessary, and how many always suffice, to cover a polygon of $n$ vertices?

A $2$-transmitter is a point that whose broadcasts/vision can penetrate $2$ walls.

J. O'Rourke. "Computational Geometry Column 52" SIGACT News, Vol. 43, No. 1, Mar. 2012. (Link.) (PDF download.)

          Fig.2

This problem remains wide open. For background, see

T.S. Michael, How to Guard an Art Gallery and Other Discrete Mathematical Adventures. Johns Hopkins Univ Press, 2009. (JHU link.)

                    TSMichael

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You could pick and choose from problem books. "Elementary Number Theory, A problem-oriented approach" by Joe Roberts is essentially a theme-arranged book of questions, with a little exposition and an answer section. This has the benefit that you can predict some of the discoveries, and gauge how long they will take.

You can also try something that draws connections between various parts of mathematics. I am looking at additive permutations which seems to have ties to group theory (orthomorphisms) and combinatorics (Langford sequences and transversals of certain Latin squares). Just playing with Skolem sequences (a variant of Langford) will give some opportunities for (re-)discovery.

Gerhard "Not Showing My Connections Yet" Paseman, 2015.06.18

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I would probably try simply constructed, yet far reaching sequences like Fibonacci, or Colatz. There are many things you can "discover" on your own and some really challenging conjectures if you go deeper. Another idea might be to start here: https://projecteuler.net/.

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