# Generating “random” mental calculation exercises?

I recently tried to come up with "random" mental calculation exercises in an attempt to fight traffic jam boredom. Unfortunately, I quickly got bored by the lack of creativity of the problems I can generate for myself.

My question is: Are there any standard patterns that can generate a sequence of mental calculation problems? Ideally the generation scheme

• should not require any tools like smartphones
• should not depend excessively on random number input (because e.g. in a traffic jam you are surrounded by the same number plates and it would be nice not to depend on being surrounded by numbers, which typically have a bias)
• should generate problems that cover a broad range of arithmetic challenges.

The first thing that came to my mind is a pattern like the Collatz conjecture. The problem is that the arithmetic operations are very limited and it still requires a random number starting seed, but at least the sequence can keep me busy for a while. On the other hand, other sequences that came to my mind (e.g. terms of a Taylor series) were a bit too much for a traffic jam. Is there anything in-between?

How about something that can go on indefinitely?

For example, pick a number plate near you, and then keep doubling that number until it becomes too large for you to remember all the digits?

Or maybe, computing the digits of things like $\pi$, $e$, $\sqrt x$, etc., using numerical methods?

$\pi=\frac 41 - \frac 43 + \frac 45 - \frac 47 +\cdots$

$e=\frac{1}{0!}+\frac {1}{1!} +\frac{1}{2!}+\frac{1}{3!}+\cdots$

I've been getting into mental multiplication recently, especially two digits multiplied by 2 digits. I'm not sure what the number plates look like wherever you're from, but you could try multiplying two digits from one number plate by two digits from another. Same stratey could work for three digits multiplied by three digits, which requires more memorization of past calculations.

Another one could be picking a random number, say $$x$$, between $$1$$ and $$10$$ and then selecting another random number, say $$a$$, between $$10-100$$ (from a number plate maybe) and computing $$x^a$$.

Reading both of them makes them sound kind of dull, but it could be a nice way to practice memory while performing mental math calculations.