Pastanatorics: Optimizing Olive Garden’s Pasta Pass Menu

So you decided to punish yourself with a pasta pass and you want to make the most out of this deal. First thoughts are probably: “How many plates could I eat?”

I dare you to eat zero.

There are 7 pastas x 6 sauces x 6 toppings = 252 combinations. The 2018 $100 pass runs from 9/24/18 – 11/18/18 so there are 56 days. Clearly not enough time to reasonably try every single one. To make sure we get the fullest experience possible within our time frame, we need to figure out a way to prioritize somewhat unique combos. Meaning we’re not eating meatball and marinara seven times in a row with different noodles.

Something to Start With

Here are the options they have:

We already know the full number of unique combinations is 252. Let’s consider the flip side – eating each option only once. Let’s see how many possible combinations I have left after each plate I’ve eaten. This illustrates the multiplication principle.

First plate: 7 pastas x 6 sauces x 6 toppings = 252 combos to choose from
Second plate: 6 pastas x 5 sauces x 5 toppings = 150 combos
Third plate: 5 pastas x 4 sauces x 4 toppings = 80 combos
Fourth plate: 4 pastas x 3 sauces x 3 toppings = 36 combos
Fifth plate: 3 pastas x 2 sauces x 2 toppings = 12 combos
Sixth plate: 2 pastas x 1 sauce x 1 topping = 2 combos
There are no more possible combinations without repeating past items.

This gives us the extreme case of choosing only unique items. And now I know my boundaries! The minimum number of plates I need to eat to try the most individual items is the minimum of the number of options from the given categories. This is obvious as shown above. The maximum number of plates I need to eat to try each individual combination is 252 from the multiplication principle.

Dinner is Served

To be the most efficient, I need to eat 6 plates of pasta. To get the full experience, I need to eat 252 plates of pasta. I want an in-between approach that will allow me to try as many “different” combinations as I can given my time constraint.

But how do we define “different”? In our problem we have three categories so three levels of uniqueness. For the unique items approach where we avoid past items we’ve eaten, that is level 1 uniqueness – each single option must be a newly unique item that has not shown up before in the past. Level 3 uniqueness would be for each combination of 3 items, and that combination of 3 items must be unique. Now we see that we want level 2 uniqueness, where each combination of two items is unique.

Level 1 Uniqueness: 6 plates
Level 2 Uniqueness: ???
Level 3 Uniqueness: 252 plates

In other words, you cannot choose any combinations that contain subsets, with size equal to the level of uniqueness, from past combinations. Let’s just brute force it.

I’ve found 36 combinations that do not repeat subsets of size 2. Since my first plate is a rotini with traditional meat sauce and crispy chicken fritta, I cannot choose any next plates that contain the combinations of {rotini, traditional meat sauce}, {rotini, crispy chicken fritta}, or {traditional meat sauce, crispy chicken fritta}.

Eating this list of 36 combinations will give you the best assortment of pastas in the smallest number of plates! The next step is to consider which order to eat these in, and that’s up to you. I already sorted the list above as a suggested ordering.

Note that the wheat linguine is not on the list. Even if you throw it into the mix, the combinations of sauces and toppings cannot be repeated so you are still limited to 36 combinations. Any categories with a larger number of options becomes irrelevant. Which is actually fine in our case because everyone says the wheat linguine sucks. Anyway, we’re on to something here…we want the minimum sizes of these categories.

Taking this a step further – if we removed one of the options, say the crispy chicken fritta because that’s probably last night’s grilled chicken, we get 30 possible combos. So all this means that we’re only interested in the smallest sized categories and taking the product of them, according to how unique we want our combinations to be!

The Pasta Pass results are here:
Level 1 Uniqueness: 6 plates
Level 2 Uniqueness: 36 plates
Level 3 Uniqueness: 252 plates

Here’s my conjecture:

Yeah, I don’t know how to write math. If someone out there could help clean this up or prove (or disprove!) it, I would appreciate it.

Also, you don’t have to follow my exact list. You can make your own list of 36 combos – the point is that you are limited to 36 combos that do not repeat any combination of 2 items. The constraint is on the smallest two categories which are the sauces and toppings. Choose the pasta any way you want with those sauce + topping combos.

Here’s the Dessert Menu

What if you could choose multiple items from the same category?
What if each item had a price – how would that change the way you selected your combos?
What if you read this and ordered chicken alfredo 36 times anyway?