[based on a true story]
I have here some climbing holds that I've made. There are two relevant parameters:
- The angle on the top, and
- The thickness, as shown.
Now it is definitely the case that $B_1$ is easier to hold than $S_1$. If we implement an easiness score for each hold — the smaller the number, the easier to grip — it is clear that the easiness score of $B_1$ $<$ easiness score of $S_1$. We will use the function $E(x)$ to denote the easiness score of a hold. Therefore, $E(B_1) < E(S_1)$.
Now, clearly $B_2$ is easier to hold than $S_2$, so we can say: $E(B_2) < E(S_2)$. We can also extend this to $B_3$ and $S_3$: $E(B_3) < E(S_3)$.
It is also the case that $B_1$ is easier to hold than $B_2$, which is easier to hold than $B_3$ — therefore, $E(B_1) < E(B_2) < E(B_3)$. Additionally, $E(S_1) < E(S_2) < E(S_3)$.
In general,
- The thicker the hold, the easier it is to grip;
- The more the hold slopes towards the wall, the easier it is to grip.
I want to arrange a course in which they go in order of increasing difficulty. There are 5 possible ways this could be done given the above constraints:
- $B_1$, $B_2$, $B_3$, $S_1$, $S_2$, $S_3$
- $B_1$, $B_2$, $S_1$, $B_3$, $S_2$, $S_3$
- $B_1$, $S_1$, $B_2$, $B_3$, $S_2$, $S_3$
- $B_1$, $B_2$, $S_1$, $S_2$, $B_3$, $S_3$
- $B_1$, $S_1$, $B_2$, $S_2$, $B_3$, $S_3$
But, what if I have 4 different angles? Or 5 angles? What's the general number of ways of arranging the routes?
And, trickier perhaps, what if I have 3 different thicknesses ($S$, $M$, $L$) and 3 different angles? Or in general, $n$ different thicknesses and $m$ different angles?