# Suggestions for mathematical solitaire against boredom

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.

• Oh my goodness, I do the same thing. Jul 27 '17 at 23:53
• This problem set is good. Especially the first nine (which are less technical than the rest): people.cs.uchicago.edu/~laci/REU12/puzzles.pdf Jul 28 '17 at 0:59
• As someone who does research on crystallographic groups, here's a tip: don't be surprised if you only find 219 groups. Crystallographers claim there's 230, but what do they know? They wouldn't recognise themselves when looking in a mirror ;) Jul 28 '17 at 8:01
• Logic puzzles and translating sentences you hear into logic are a good pastime. They are usually easy to do, and you can prove theorems from them if they are particularly interesting. Jul 28 '17 at 10:14
• "I have a friend who, in High School, used to kill time by writing down a random integer and then trying to factor it into primes. By the time he was in Grad School, he was killing time by writing down a random monic polynomial with integer coefficients, and trying to compute the class number of its splitting field..." - Arturo Magidin Jul 28 '17 at 11:00

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:

• 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!

• I think the horned sphere is something that's the same as the sphere in one way, but not the other way...? Jul 28 '17 at 0:18
• Also, your talk about cutting a donut reminds me of this little fact: it's possible to cut a donut to 13 pieces using only 3 cuts. images.huffingtonpost.com/2013-10-20-myfirstMGbook1.JPG Jul 28 '17 at 0:20
• Here's a specific topology visualization puzzle: i.imgur.com/VZVP30S.jpg You have three hexagons placed on top of each other in 3D space, with six vertical rectangular faces holding them together (alternating between top and bottom). This must be homeomorphic to an $n$-genus torus with some holes punched out, but which and how many? Can you visualize the deformation that takes one to the other? Jul 28 '17 at 1:06
• (To clarify, that surface should include three 2D hexagons, which means they include their interior. Hence "+hexagons" on the drawing) Jul 28 '17 at 1:17
• The part about living on strange planet reminds me. When I'm bored and inside a building, I may imagine rotating the gravity around and imagine running on the walls so that I don't fall, Inception style. Jul 28 '17 at 21:10

I don't know quite why, but I love doing matrix multiplication; I find it rather calming. I also find it enjoyable to take a matrix and try to figure out what it 'does' by multiplying it by a bunch of different things. I do prefer to have paper for this though, but in your head should be fine if you use small enough matrices. You could also try to visualize the transformation the matrix is causing to the grid lines of the plane.

Another fun thing is the Collatz conjecture - playing with numbers, and discovering little rules (such as how for any $x$ which can also be represented as $2^n$, it's easily provable that $x$ will reach $1$). In a similar vein, I enjoy playing with different aspects of complicated problems, not to solve, but just to explore and see what I can figure out by myself.

You could also try figuring out if knots are equivalent, or creating knots and categorizing them. (This goes along with Nilknarf's suggestion of trying to figure out via mental deformation of a shape whether it is topologically equivalent to another.)

Prime factorization is always really fun. I enjoy doing it by creating factor trees, because I can then connect those trees to other larger trees and so forth, much like you can do for the collatz conjecture. I also enjoy programming in my head sometimes, that is, trying to sketch out the general method for a program. You could do this with the Euler Problems, which are very math focused (or I guess, if you had a ton of time, you could do them by hand) or with general problems like thinking up a program to find the prime factorization of numbers.

• Maybe we have somewhat different mindsets... if I had a pen and paper and need to deal with boredom, I'd draw a Sierpinski Triangle, or some ponygon in perspective (try stella octangula! It's easy to draw by starting with a cube and it looks pretty after shading). Matrix algebra is pretty boring to me. The Collatz conjecture is a very good idea - just doing squaring and cubing starts to get old now. Jul 28 '17 at 0:26
• @MaudPieTheRocktorate, you should watch the doodling and mathematics video series on Khan Academy - sounds like it'd be right up your alley (I enjoyed it, it's fairly geometrically focused as well) and it's super entertaining. Plus, you can draw an infinite series of elephants =D Jul 28 '17 at 0:33
• lol those by ViHart? I've watched all her videos! Even ViHartViHart videos! Jul 28 '17 at 3:37
• @MaudPieTheRocktorate same! She's great! Jul 28 '17 at 15:04

Other than whats been mentioned by the others i have couple additional things i do when traveling.

The first is not exactly geometric but i frequently am bored while traveling as well (especially while driving) i also happen to have a non-existent short term memory one method i have employed since i was a child to compensate for this deficiency is to use a "sphere" inside of my mind, i am inside of it i guess but not a physical entity. i usually take licence plate numbers from passing cars and find the most efficient way to transmute the numbers to multiply them together while driving by writing the multiplications out inside of my head and storing them in different locations inside of the sphere which i can access at any time by simply visualizing where i wrote said information and then locating its location on the sphere and simply reading the information from the inside of it. You can also do this with long division but dividing primes is usually more interesting. When im not driving and am able to focus more intense i frequently play out symmetries and other patterns in a board game called "Go" or perhaps simply playing against myself in my mind. (if i am feeling particularly good as far as concentration i may use only the one colour stone for both sides though this requires a very large amount of mental focus.) the game involves alot of patterns and symmetries and has many many more variations then something like chess and is great for training mental focus and passing the time.

There is a book you should get, "Dead Reckoning" by Ron Doerfler. You can use this to increase your brainware calculating power and thereby magnify your abilities. However, do not take it out and study it while your father or the school principal is scolding you; it might make a bad impression on them.