Suggestions for mathematical solitaire against boredom

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 $$\max\big(\sqrt{5}^{\sqrt{3}^\sqrt{2}}, \sqrt{3}^{\sqrt{5}^\sqrt{2}}\big)$$

I hope this helped!

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.