Pizza Pass is the third trial the Zoombinis must face in the Logical Journey of the Zoombinis.
Premise
The way of the Zoombinis is blocked by the Pizza Trolls (Arno, Willa and Shyler). They are hungry and will not let the Zoombinis pass until you have made each of them the specific pizza (and, in later difficulties, sundae) that each desires. To aid this production, there is a large machine, with buttons for each of the toppings. There is a display on it that shows what the completed pizza will look like, and when this button is pressed, the pizza is ejected. The machine ejects the pizza with a burping noise, and a Zoombini hastily catches it on its head and presents the pizza to the trolls.
As the difficulty increases, more trolls appear (at first there is only one) and the trolls begin to also demand more pizza toppings and sundae toppings along with their pizza.
Unsuitable pizzas fall into two categories: pizzas that have too few toppings for the trolls, and pizzas with too many toppings for the trolls. There are three stone tables behind the trolls, and a pit in front of them. If there are too many toppings for all of the trolls present, the bad pizza is cast into the pit. If there are not enough toppings for a troll, they will throw that pizza onto the stone table behind them.
The trolls will accept a certain number of bad pizzas depending on the difficulty, but after too many failures, they will grow angry and strike the Zoombini that presents the unsuitable pizza. The first hit is a warning; each subsequent hit sends the Zoombini all the way back to Zoombini Isle. Unlike the previous two puzzles, this means that you lose your Zoombinis one at a time over a long, gradual period.
Solving the Puzzle
This puzzle is very easy but very tedious because you have to listen to what the trolls say. All you have to do is present each topping one at a time. Depending on the trolls' reactions to each topping, you then serve the trolls with all the toppings they like. On the hardest difficulty, there is at least one topping that WON'T mix with two other toppings (i.e. mushrooms won't mix with a cherry sundae and a chocolate sundae). Test all the toppings except the topping that's in the pit and by process of elimination, you can figure out which troll will have that topping.
Mathematics
On the hardest difficulty, each pizza topping is used exactly once by the pizza trolls. There are 38 = 6,561 unique ways of distributing the toppings, disregarding the information provided by the four pizzas initially provided. Taking into account the information provided, there are instead 1,458 different topping combinations. Using the above strategy of testing the remaining toppings individually, the player will be left with the four original toppings, of which only two distinct pairs can coexist. At least one of those pairs must be present. At this stage, there are just 18 possible distributions of toppings. Testing each of the two untested pairs of the original four toppings is guaranteed to reduce the number of possibilities to no more than two. From there, the player may test one of the remaining four original toppings individually, solving the puzzle.