# Fossil Completionist's Guidebook

My attempt at relieving fossil collector’s anxiety.

In *Animal Crossing: New Horizons*, every starfish-shaped crack is another chance for glory. Once you dig up a new fossil, you can then donate it to Blathers the museum curator. After an enthusiastic but rather unsolicited introduction about the fossil’s origin, it becomes a permanent part of the exhibition for you and your friends to enjoy.

If you are like me, all was well until the honeymoon phase started to wear out and at some point, you began to get nothing but duplicates. Days soon turned into weeks yet things didn’t get any better, and you are on the verge of suspecting foul play. Could Nintendo be deliberately withholding a few fossils from your reach just to keep you coming back? Could it be a marketing campaign to sell you the online membership where trading with friends is the only way to advance? Or could it be your bad luck again, as with everything else in life?

In this post, I analyze the problem of how long it takes on average to complete the fossil collection. As the answer suggests, it is still too early to be concerned. More specifically, players have at least until June 17th to start either questioning whether the system is rigged or being considered unlucky.

## General Setup

First things first, there are a few important facts to keep in mind before we get started:

- for tractability, it is assumed that each fossil has an equal chance of showing up (comment below if you disagree!)
- 4 fossils can be found on any given day
- there are 73 unique fossils in total to add to the museum’s collection

## Numerical Solution

With the help of any modern computing device, numerical simulation indicates that on average, it takes 89 days (or 356 trials) to collect all fossils. So there you go - it’s not that bad.

Furthermore, if we look at the probability distribution at the end of the 89th day, more likely than not you will have completed the collection already.

## Analytic Solution

Generally, if the probability of some random event happening is $x$, it takes $1 / x$ trials on average for it to occur at least once. Adding all expected number of trials to collect the first $n$ fossils together, we have:

\[\mathbb{E} \left[ T ^ N (n) \right] = \frac{ N }{ N+1-1 } + \frac{ N }{ N+1-2 } + \ ... \ +\frac{ N }{ N+1-(n-1) } +\frac{ N }{ N+1-n }\]where $N$ is the total number of unique collectibles. It grows almost linearly at first (as it’s fairly unlikely to get duplicates already), but the last one is expected to cost you $N$ tries.

The expected number of unique fossils as a function of the number of trials tells the other side of the story. For each new trial, the incremental contribution to the collection in terms of additional “uniqueness” equals the probability of finding something that has not been found so far:

\[\mathbb{E} \left[ F ^ N (t) \right] = \left\{ \begin{array}{c} 1, \ t = 1 \\ \mathbb{E} \left[ F ^ N (t - 1) \right] + \frac{ N - \mathbb{E} \left[ F ^ N (t - 1) \right] }{ N } , \ t \geq 2 \\ \end{array} \right.\]Again, initially it fills out almost linearly. The curve starts to flatten out eventually as you build up your collection, desperately waiting for the last few pieces to arrive. That said, all you really need is to keep calm and carry on - as previously mentioned, the odds are in your favor in 3 months’ time (and increasingly so).

## Conclusions

*Animal Crossing: New Horizons* was launched on March 20, 2020, so if you started playing on day 1 and never missed a single fossil, June 17th (i.e., day 89) will be your benchmark to beat. Should that not be the case, you can always blame it on Zipper!

*This is Part III of my Animal Crossing post series. For Part II, see here*.