[Math] Suggestions for mathematical solitaire against boredom

recreational-mathematicssoft-question

I sometimes have to endure my parents' or teachers' endless scolding, and sometimes endure endless lectures on boring things in school, and occasionally endure really long trips.

One way for me to deal with them is to do mental arithmetic, such as calculating squares and cubes, and approximating square roots and cube roots. Another is to do "make 24", which is just think up 4 numbers, and make 24 using add, subtract, multiply, and divide. Yet another (especially good if I'm in a room with wallpapers and carpets) is to stare at any symmetry pattern and visualize all the symmetries of the pattern like this (source)

(For example, I once classified all 17 wallpaper groups over the course of 20 hours of flight. The thought of classifying all 230 space groups is imposing, but if I ever need to take such a long flight again, I might work on it.)

Another one, which is less "systematic", but still fun, is doing some topological visualizations. For example, visualizing why the fundamental group of torus is abelian. Visualizing turning a 2-crosscap to a Klein bottle. Driving a Ferrari on a Sudanese Möbius band. And so on.

(I have yet to succeed in fully visualizing the Császár polyhedron. The Szilassi polyhedron, I can only barely visualize. Boy's surface is very difficult. And visualizing the eversion of sphere feels impossible.)

So, are there any other games I can use for such occasions? Such games should be possible to play without big calculations. Geometric things are usually good in this regard.

Edit: I remembered something called Conway's Soldiers. Apparently the protagonist in The curious incident of the dog in the night-time uses it for the same purpose… I should try it sometime.

Someone suggested playing chess or go in my mind. They are fine, but a bit complicated to hold in my head. I'm not good at chess or go. I'm ok at Hex, and even with Hex, I can't play a game in my head.

The game of Sprouts, however, is quite easy to hold in my mind. There are probably more games from Conway's Winning Ways for Your Mathematical Plays, that would be great as mental solitaire. I should read it one day.

Best Answer

Here are some of my all-time favorites:

  • If there is a clock in the vicinity, try to work out the angle between its hands
  • The "overlapping polygons" game - see this puzzling SE page for an explanation
  • Think of two strange (but similar) objects, for example:
  • sphere
  • torus
  • double torus
  • n-torus
  • horned sphere
  • etc.

and try to determine if they are "topologically equivalent" - that is, can one be "morphed" into the other by deformation without cutting or pasting. It can be really entertaining to try and visualize this.

  • Think of the old cake-cutting problem (how many cuts with $n$ slices?) and try it in your head with very oddly-shaped cakes (like donuts or fractals)
  • Try doing a geometry construction in your head. It is very difficult, but also fun!

EDIT: Here are some more:

  • try to construct a magic square in your head
  • coin weighing problems, or "balance puzzles"
  • think about what life would be like living on a planet in some strange shape, and how (without going into space and looking at it) you would be able to tell the shape of the planet you were living on
  • think about the fourth dimension
  • think of two numbers that seem close together, and try to determine which one is biggest. One famous example is $\max(e^\pi, \pi^e)$. Or, if you want to challenge yourself, try $$\max\big(\sqrt{5}^{\sqrt{3}^\sqrt{2}}, \sqrt{3}^{\sqrt{5}^\sqrt{2}}\big)$$

EDIT: One more, now that I'm getting into complex analysis:

  • pick a non-injective complex analytic function (a polynomial, say) and try to visualize its Riemann Surface in your head. Trippy stuff!

I hope this helped!

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