**Pizza Pass** is the third trial the Zoombinis must face in the Logical Journey of the Zoombinis.

## PremiseEdit

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 have the right to 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 PuzzleEdit

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 Edit

On the hardest difficulty, each pizza topping is used exactly once by the pizza trolls. There are 3^{8} = 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.

## In-game instructionsEdit

### Not so easyEdit

*Arno desperately needs a pizza. Make his favorite pizza and he'll let the Zoombinis pass. Build your pizza and be aware of what Arno likes and dislikes. If a pizza has a topping that arno hates, he'll throw it into the reject pit. He'll throw pizzas missing one or more of his favorite toppings onto his rock. Arno is a finicky eater and he won't necessarily like the same pizza the next time you see him.*

### Oh, so hardEdit

*Arno and Willa will let the Zoombinis pass if you make a special pizza and sundae for each of them. Any topping Arno likes, Willa does not. For example, if Arno likes pepperoni, Willa will not. If any pizza has a topping that Arno and Willa hate, they'll throw into the reject pit. They'll throw pizzas missing one or more of their favorite toppings onto their rocks.*

*Try making Arno's pizza and sundae first before starting on Willa's, or vice versa. Build your toppings and be aware of what each troll likes and dislikes. Arno and Willa are finicky eaters and they won't necessarily like the same pizzas the next time you see them.*

### Very hardEdit

*Arno, Willa, and Shyler will let the Zoombinis pass if you make a special pizza and sundae for each of them. Each troll only likes toppings that the others dislike. For example, if Arno likes pepperoni, Willa and Shyler will not.*

*If a pizza has a topping that the trolls hate, they'll throw it into the reject pit. They'll throw pizzas missing one or more of their favorite toppings onto their rocks. Try making one troll's pizza and sundae first before starting on the next. Make your pizzas and be aware of what each troll likes and dislikes.*

*The trolls are finicky eaters and they won't necessarily like the same pizzas the next time you see them.*

### Very, very hardEdit

*Arno, Willa, and Shyler will let the Zoombinis pass if you make a special pizza and sundae for each of them. Each troll only likes toppings that the others dislike. For example, if Arno likes pepperoni, Willa and Shyler will not.*

*If a pizza has a topping that the trolls hate, they'll throw it into the reject pit. They'll throw pizzas missing one or more of their favorite toppings onto their rocks. Try making one troll's pizza and sundae first before starting on the next. Build your toppings and be aware of what each troll likes and dislikes.*

*The trolls are finicky eaters and they won't necessarily like the same pizzas the next time you see them.*

*Four pizzas that have been rejected by the trolls are already in the pit.*