How many words are required for What3Words? This is just curiosity / a personal exercise.
What3Words allocates every 3m x 3m square on the Earth a unique set of 3 words. I tried to work out how many words are required, but got a bit stuck.
$$
Area
    = 510 \times 10^6 km^2 
    = 5.1 \times 10^{14} m^2
    => ~ 5.4 \times 10^{14} m^2
$$
(rounding up to make the next step easier!)
And so there are ~ $6\times10^{13}$ 3m x 3m squares.
I assumed I could use the equation to calculate number of combinations to find the number of words needed:
$$
_nC_r = \frac{n!}{r! (n - r)!}
$$
where $r$ is 3, and total number of combinations is the number of squares: $6\times10^{13}$
$$
6\times10^{13} = \frac{n!}{3! (n - 3)!}
$$
$$
6\times10^{13} = \frac{(n)(n-1)(n-2)(n-3)!}{3! (n - 3)!}
$$
$$
n^3 - 3n^2 + 2n - (36\times10^{13}) = 0
$$
... and then, I can't work out the first factor to use to solve the cubic equation, I'm not sure I've ever had to solve a cubic eqtn with a non-integer factor and none of the tutorials I've found have helped.
(And, my stats is also not good enough for me to be convinced this is the correct equation to use anyway!)
Any hints as to the next step would be appreciated.

Seems simpler to say, if the whole area of the globe is about $5\times10^{14}$ square metres, then you need about $6 \times 10^{13}$ squares.
If the number of squares is large, the number of words required will be very near to the cube root of this number.  This means about $39,000$ words.
OK?
