Math games for car journeys

The “What number am I?”-game. “What number am I? If you double me an add 9 you'll get 19.” A very advanced schizophrenic one: “What number am I? If you multiply me by myself you'll get me.” Be sure that the kids invent their own numbers.

Here's one I play by myself when walking long distances.

"Verifying" the Collatz' Conjecture.

Pick any natural number $n$ you dare. Using only mental arithmetic, try to "verify" that $n$ satisfies the Collatz' Conjecture by the time you reach your destination. That is, show that there is a $k$ such that for $f:\mathbb{N}\to\mathbb{N}$ given by $$f(m)=\cases{\frac{m}2 &: $m$ is even, \\ \\ 3m+1 &: $m$ is odd,}$$ we have $f^k(n)=1$ . . . before you get to where you're heading. (You must keep "verifying" until you hit $1$, so you can't, say, stop once you've found an $\ell$ such that $f^\ell(n)$ is a power of two).

It's quite fun actually. Local pedestrians might think I'm a bit weird, though, since I'm muttering numbers to myself so often . . .

Try to find the prime factorization of every integer you come across - e.g. you see a 76 gas station, so you think:

$$ 76 = 2 \cdot 38 = 2 \cdot 2 \cdot 19 = 2^2\cdot 19 $$

The Licence Plate Game.

Pick a (simple) key for the alphabet (like

$$ \operatorname{key}(\color{blue}{\alpha })=\cases{\color{blue}{0} &$\text{ if }\color{blue}{\alpha}\in\{\color{blue}A, \dots , \color{blue}M\}$ \\ \color{blue}{1} &$\text{ if }\color{blue}{\alpha}\in\{\color{blue}N, \dots , \color{blue}Z\}$,} $$

for example, for a letter $\color{blue}{\alpha}$), some (simple) operations (like addition, multiplication, exponentiation, etc.), then pick a licence plate. The first one to calculate correctly the biggest (possible) number with it wins the round. Repeat (for a different plate).

A trivial example: Let's say you've chosen the key above and all you have is addition. Someone gets to pick a licence plate - maybe the winner from a previous round or a coin flip - and that plate is, say, $$\color{blue}{Y}344\quad \color{blue}{PNA}. $$Then the winning number might be $\color{blue}{1}+3+4+4+\color{blue}{1}+\color{blue}{1}+\color{blue}{0}=\color{red}{14}$.

You could vary this however you like. For example, you could use the winning number from the first round as a goal for the second (so, for example, you race to get as close as possible to $\color{red}{14}$ with a different licence plate), alternating thereafter.

I haven't tried this yet. It's based on a silly word game where you make the funniest nonsense you can from the letters of a given licence plate, like "$\color{blue}{\text{P}}$andas $\color{blue}{\text{N}}$udge $\color{blue}{\text{A}}$liens" or whatever, aiming for a short story :)