[Math] Recreational mathematics: where to search

big-listmathematics-educationrecreational-mathematics

I am not sure I can strictly define recreational mathematics. But we all feel what it is about: puzzles, problems you can ask your mathematical friends, problems that will bother them for a couple of hours, and then they will get them (or not?). Problems to distract yourself from research. Sometimes, however, these problems lead to some ideas and concepts connected to "serious mathematics". I adore this kind of problems and, with holidays approaching, I let myself surf on Internet for some new problems to distract myself a little near the Christmas tree. I usually look at some blogs but my search is very chaotic. I'll try to formulate more precisely what kind of problems I'm searching for: there are two kinds of them.

First, good mathematical problems. So good that you can talk about the solution for an hour although the formulation could be really easy.

Example: the old Arnold's question about the perimeter of a banknote. By folding up a banknote, we decrease the area of the polytope obtained. But can we increase the perimeter? The answer is yes, and we can increase it as much as we want. The proof is beautiful and doesn't need any knowledge of higher mathematics although it is not at all trivial.

Second, problems giving some publicity for higher mathematics. I will give an example – hat puzzle.

A sultan decides to give a test to his sages (a countable number of sages actually!). He has the sages stand in a line, one behind the other, so that the person in a line sees everybody before himself. Yes, the sages are clever but also they have a very good vision. He puts the hats on them: white or black. Then, the sages cry (all at one time) the color that they think they are wearing. Everybody who is wrong will be killed. The question is, can the sages achieve the result that only the finite number of them will be killed?
I won't spoil you the pleasure giving the answer but this puzzle in some sort opens a path to some serious mathematics and can be a good pretext to explain it on the seminar for high-school student (or even to undergraduate).

I search for problems that are easy to formulate and not trivial to solve, and that could give a nice pretext to talk about them for a couple of hours for undegraduates. In other words, I search for problems that could make a good advertisement of "serious" mathematics for high-school students that love puzzles.

My question is — are there any journals (I think, Mathematical Intelligencer can be one of the possible answers..) that publish some kind of research articles on the subject? Maybe some blogs I do not know?

Any links or suggestions will be welcomed.

Best Answer

There is a journal called Journal of Recreational Mathematics, but I do not like some of the articles it contains. You might do better with The American Mathematical Monthly, which often publishes papers on what can be considered recreational mathematics. (Here I define recreational mathematics as those that can be done by amateurs. As was mentioned in the commnents above, the mathematics popularized by people such as Martin Gardner can be considered recreational mathematics.)

Edit: It seems that the Journal of Recreational Mathematics ceased publication in 2014. But another publication, the Recreational Mathematics Magazine, was started in 2014 and seems to be published regularly (twice a year) up to now.

Related Solutions

[Math] Examples of undergraduate mathematics separation from what mathematicians should know

One category of mathematical result that belongs to 1 is statements that you need to know are true and that have complicated proofs. Obviously some such proofs are worth knowing because they will help you find other, similar proofs. But not all of them fall into that category. For example, almost all mathematicians can get by just knowing that it is possible to construct a complete ordered field. And perhaps a more important example: many mathematicians use Lebesgue measure, but all most mathematicians need to know is a few basic facts about it, and not the full details of the construction and proof that it works. Another result I remember my undergraduate lecturer more or less explicitly apologizing for was the simplicial approximation theorem, which I remember disliking intensely.

Why do we teach results like this? One reason is that when we teach we are not just equipping people with the tools they need for research, but also demonstrating that we can build up the edifice of mathematics from just a few basic axioms. One can argue about whether we really do this, but I think we tend to do enough to convince any reasonable person that it can in principle be done. If we were to start leaving lots of gaps (there's this thing called Lebesgue measure ... it has the following properties ... it can be shown that these properties are consistent but the proof is tedious and I'll omit it) then this valuable aspect of a mathematics course would be in danger of being lost.

[Math] Looking for a “scientific” application of a recreational puzzle.

I don't know about scientific applications, but you can solve the puzzle as follows (I hope you consider this solution simple):

Look at the Ferrers diagram of the partition of 15 you have, and consider the sum of the Manhattan distances from each dot to the corner of the diagram. At every step of the barman's process, this sum either stays the same or decreases, and it only stays the same if the number of stacks is at least as large as the size of the largest stack minus 1. In this case, we get the new Ferrers diagram by "rotating" the first column of the Ferrers diagram to make it into the first row, and shifting the rest of the diagram diagonally. For example:

x*****      xxxxx
x***        *****
x**     ->  ***
x           **
x

If we assume the existence of a cycle of partitions, then this sum must be constant along the cycle, so the Ferrers diagrams in the cycle can be produced by simply cyclically shifting each diagonal. If there are two adjacent partially filled diagonals, then since adjacent numbers are relatively prime, eventually in this cycle we will get a Ferrers diagram where a hole in the diagonal closer to the corner is next to a dot in the further diagonal, which is not allowed. For instance, if we continue the previous example, we eventually see the hole in the fifth diagonal line up with the dot in the sixth diagonal:

******     *****     ****     *****     *****     *****     ******     ****
****       *****     ****     ***       ****      ****      ****       *****
***    ->  ***   ->  **** ->  ***   ->  **    ->  ***   ->  ***    ->  ***
*          **        **       ***       **        *         **         **
*                    *        *         **        *                    *
                                                  *

Thus, there is at most one partially filled diagonal in any partition that cycles, and the only partition of $15$ satisfying this condition is $15 = 1+2+3+4+5$. Since the number of partitions of $15$ is less than $1000$, the barman enters this cycle before the end of his process.

Best Answer

Related Solutions

[Math] Examples of undergraduate mathematics separation from what mathematicians should know

[Math] Looking for a “scientific” application of a recreational puzzle.

Related Question