The mathematics behind the 20 win challenge

After a frustrating series of defeats in the 20 win challenge I decided to run the numbers on said challenge. What I suspected and what I later proved was that the challenge is more about how much time you're willing to sink rather than skill. As I'll show being a good player certainly increases your odds. But at the end of the day even the best players face a pretty significant time sink.

I've split this post into a few sections, if you don't care how I did my calculations just skip to the ones you're interested in.

So let's dive in

Methodology

The following probabilies were calculated using the binomial distribution. This was done with 20 trials where the probability of a loss was taken as 1-winrate. I then looked at the probability of getting less than or equal to 2 losses in that set of 20 trials.

Note: This methodology actually overestimates the true probability because losing a game means you have play another game to get to 20 wins (e.g if you lose twice you have to play 22 games to reach 20 wins). I could use a better method but I'm lazy and this is just an estimate to give us a lower bound on the amount of time you'd have to spend.

Finally I determine a win or a loss based purely on winrate, this doesn't take into account hard counters, players quitting/disconnecting etc.

Expected number of attempts

For a typical player with a 50% win rate there's a 1/5000 chance they'll win 20 times with less than 3 losses. Which means this player would need to attempt the challenge an average of 5000 times to reach 20 wins.

The same calculation for an exceedingly skilled player with a 70% win rate, which is pretty unrealistic even for the best of players, yields a ~3.5% chance of getting 20 wins with less than 3 losses. This means this hypothetical player would need to attempt the challenge an average of 28 times before they'd reach 20 wins.

Expected time sink

I couldn't find any existing data for the average length of a Clash royale match so I just took it to be about 3 minutes. Most games end at the 3 minute mark and even though overtime extends the game by 2 minutes, it's unlikely it goes the whole way. Plus games can end early if the king tower is taken so I just presumed these two effects cancel out.

If we assume every challenge attempt apart from the winning one has about 5 games played we can estimate the time sunk to get to 20 wins. This 5 games number is completely arbitrary, it just seemed about right.

For the 50% winrate player:

4999 x 5 + 20 = 25015 total games played 25015 x 3 = 75045 minutes

That's equivalent to 52 days, 6 hours and 53 minutes of pure gameplay. That's completely unachievable for the average player

For the 70% winrate player:

27 x 5 + 20 = 155 total games played 155 x 3 = 464 minutes

This is equivalent to 7 hours and 45 minutes. Which is definitely achievable but still an obscene amount of time for a mobile game. And this is also for a winrate I doubt anyone has.

Conclusion/Final thoughts

I hope you enjoyed my rabbit hole. I'm completely aware that there's a lot of estimation and guesswork involved with these calculations and my methodology is most likely flawed in some way that I haven't noticed. But I think it still gets to the point that this challenge is basically impossible for the average player to achieve (Which I think we all knew already). In no way is this meant to be a criticism of how CR handles this annual challenge, only a very small percentage of the player base is meant to reach 29 wins. This is just a light hearted post about some funny numbers I made.

I hope you enjoyed :)