I’m definitely playing too much of The Simpsons: Tapped Out these days. Eventually I’ll move onto other obsessions, but since I did some interesting math (and I hate throwing anything away that can be preserved digitally), I thought I’d share the results of a mathematical solution I came up with.

Donuts are the “premium” currency in the game – i.e. they’re what you can spend real money on, and they enable you to progress through quests and tasks more quickly as well as let you buy premium items, characters, and decorations. As you progress through levels, you earn bonus donuts until you reach level 939. After that, each “level up” gives you a choose-a-door option where you can win 1, 2, or 3 donuts. You can always spend $50k (in-game currency) to open up another package if you don’t like your first option, and there have been several lines of thought as to what the best strategy is to go for here. I went for a different tack – I already had more in-game dollars than I could ever spend, and placing a few bloodmobiles (the free-donut engine of choice among high-level TSTO-ers) would bring up several of these level-up bonuses in a row. Because I know time is my most precious commodity, I set out to figure what was the *fastest* method of getting donuts, regardless of the price in dollars.

After some highly precise measurements, I settled on the timing of 3 seconds per box opening (which, as you’ll see later, works out pretty well in the comparison tables when you’re dealing with 1-3 donuts!). So, if you click through every level-up screen and take the first pick you get, you’ll spend 3 seconds per award, and earn 1, 2, or 3 donuts. Or more statistically, you’ll earn an average of 2 donuts every 3 seconds. Or rather – 2/3 donuts per second or 40 donuts per minute (DPM), our measurement unit of choice!

All right – let’s see how that compares to the spendthrift nature of TSTO billionaires who want as many donuts and always spend their money, so they earn 3 donuts with every level-up. The probability starts getting hairy here, so it’s time for some tables!

**Always Upgrade**

Starting Donuts | 3 | 2 | 2 | 1 | 1 |

Ending Donuts | 3 | 3 | 3 | 3 | 3 |

Probability | 1/3 | 1/6 | 1/6 | 1/6 | 1/6 |

Time Spent | 3 sec | 6 sec | 9 sec | 6 sec | 9 sec |

A 3rd of the time you’ll get 3 donuts right off the bat and only spend 3 seconds doing so. The other two thirds when you start with 1 or 2 donuts, you sometimes open one package (taking a total of 6 seconds) and sometimes open 2 packages (taking a total of 9 seconds). Wait… that’s *worse* than just taking the first pick every time. Can we be smarter? Let’s try shaving off some time by speeding up that last 9 second column; we’ll only upgrade 1 a single time.

**Upgrade 1 Once, Upgrade 2 Fully**

Starting Donuts | 3 | 2 | 2 | 1 | 1 |

Ending Donuts | 3 | 3 | 3 | 3 | 2 |

Probability | 1/3 | 1/6 | 1/6 | 1/6 | 1/6 |

Time Spent | 3 sec | 6 sec | 9 sec | 6 sec | 6 sec |

Hmm. We got the same result! Though looking at the math, it’s easy to see why. We just changed the last column from 3/6/9 to 2/6/6 which is the same value! In fact, if we try another optimization (only upgrade 1 and 2 a single time), we land at the same result 36.7 donuts per minute. The same goes for upgrading 1 fully and only upgrading 2 once (note – you don’t get penalized if you try to upgrade 2 and open a box with 1 donut; the game still awards you 2).

So is there any way to beat the 40 donuts per minute rate you get by just accepting the first box all the time? Indeed there is! Always take 2 or 3 donuts, but if you get one, go ahead and upgrade it a single time.

**Upgrade 1 Once, Leave 2 alone**

Starting Donuts | 3 | 2 | 1 | 1 |

Ending Donuts | 3 | 2 | 3 | 2 |

Probability | 1/3 | 1/3 | 1/6 | 1/6 |

Time Spent | 3 sec | 3 sec | 6 sec | 6 sec |

There you have it – mathematical proof of the fastest way to progress through multiple level-up boxes in TSTO. And also proof that I wasted *way* too much time figuring this out (certainly more time than I ever saved in tapping on donut upgrades!)

(If you *really* want to upgrade 1 donut to 3, you can, but it just changes the last column from 2/6/6 to 3/6/9 which is the same rate, but means more tapping… theoretically…at least in my mind)