Optimizing New Horizons Money Trees

The money trees in Animal Crossing New Horizons are a great way to continuously gain bells with pretty much no effort. Just bury 10K Bells in the glowing spot each day and reap 30k Bells from the spot when the tree fully grows in 3 days. 

But according to the Animal Crossing Wiki, there's actually a 30% chance that, upon burying more than 10K Bells, your money will be tripled when the tree eventually grows. 

Nookpedia.com chart of money tree return rates in acnh

 So would the optimal method for planting money trees actually be to plant 30k each day? You wouldn't make a profit every time a tree grows, but when you do make a profit it would be 40k Bells greater. I wanted to quickly explore this thought. 

The Methods

Let's quickly define our 2 methods of Bell yield mathematically. 

 

Guaranteed (Linear) Yield Method

 This is a linear function where, each day a tree grows, the player is able to gain 20k Bells. 

   y = 20x 

(we can multiply the result by 1000, I do not want to keep typing out that many zeros.)

 

Random (30k) method

This is a cumulative binomial function!  The player plants 30k Bells each day and has a 30% chance to profit 60K Bells or a 70% chance to make nothing (never losing any Bells). 

 

To clarify, 30k bells is the minimum number of bells that can be returned if a player plants 10k or more bells (see chart above). If a player buried more than 30k bells each day they would incur losses each day they did not succeed in getting the 3x return on buried bells.

 

For this experiment let's initially look at a window of 100 days. We can define our function then as

    X ~ Bin(100, 0.3)
 

Outcomes!  

I threw a simple simulation together in R and plotted the cumulative profits of both methods over the 100 days in our experiment. 

 

...and yeah it doesn't look great for the random method. 

bell yield over 100 simulated days

After simulating 100 days of both discussed methods, the first linear method ends squarely at 2,000,000 bells. The Random Method reaches 100 days with 1,560,000 bells. 440,000 Bells short! 

 

With this simulation it's pretty easy to recommend burying 10k Bells each day and not worrying about anything else. 

 

Finally I will leave you with a simulation of 15k, 20k, and 25k bells compared to the linear method just to see how much worse they are than even the 30k method.

 

all much worse! yikes!