**Fleens!** is the first puzzle in the Deep, Dark Forest region of Logical Journey of the Zoombinis.

Geographically, it is south of the Who's Bayou.

## Premise Edit

The Zoombinis have to cross a clearing but there are Fleens that will attack the Zoombinis. You have to lure the three Fleens that are on the tree branch off the tree branch in order for the bees to be disturbed and attack the Fleens. You have to use Zoombinis with certain features to get the Fleens off the branch. After each Zoombini distracts a Fleen, they escape to the tree branch on the left of the screen. Unfortunately, only six Zoombinis can fit on the branch and when a seventh Zoombini climbs on the branch, a Zoombini falls off the branch and returns to the checkpoint.

## In-game Help Text Edit

### Not so easy Edit

*What's this! The Fleens are blocking the progress of the Zoombinis. Lure the three fleens off of the beehive branch to free the Zoombinis. This will bother the bees and chase the Fleens away.*

*Each Fleen will chase a Zoombini with hair, eyes, nose, and feet that go with its hair, eyes, nose, and feet. A wrong guess will send a retreating Zoombini onto another branch, which can only hold a maximum of six Zoombinis. Once the branch is full, each wrong guess will send a Zoombini back to Shelter Rock.*

### Oh, so hardEdit

*What's this! The Fleens are blocking the progress of the Zoombinis. Lure the three fleens off of the beehive branch to free the Zoombinis. This will bother the bees and chase the Fleens away.*

*Each Fleen will chase a Zoombini with hair, eyes, nose, and feet that go with its hair, eyes, nose, and feet. For a greater challenge, the matches between Zoombinis and Fleens change each time you play this puzzle. For example, if Fleen rocket feet go with Zoombini spring feet, rocket feet may go with propeller feet the next time you play.*

*A wrong guess will send a retreating Zoombini onto another branch, which can only hold a maximum of six Zoombinis. Once the branch is full, each wrong guess will send a Zoombini back to Shelter Rock.*

### Very hardEdit

*What's this! The Fleens are blocking the progress of the Zoombinis. Lure the three fleens off of the beehive branch to free the Zoombinis. This will bother the bees and chase the Fleens away.*

*Each Fleen will chase a Zoombini with a certain type of hair, eyes, nose, and feet. However, unlike level 1 and 2, the associations are not completely hair to hair, feet to feet, and so on. There is one like match, such as feet to feet, but the other three matches are mixed up, such as Fleen hair to Zoombini eyes, Fleen eyes to Zoombini nose, Fleen nose to Zoombini hair.*

*A wrong guess will send a retreating Zoombini onto another branch, which can only hold a maximum of six Zoombinis. Once the branch is full, each wrong guess will send a Zoombini back to Shelter Rock.*

### Very, very hardEdit

*Each Fleen will chase a Zoombini with a certain type of hair, eyes, nose, and feet. However, unlike level 3, the associations are all mixed up, such as Fleen hair to Zoombini feet, Fleen eyes to Zoombini nose, Fleen nose to Zoombini eyes, and Fleen feet to Zoombini hair.*

*A wrong guess will send a retreating Zoombini onto another branch, which can only hold a maximum of six Zoombinis. Once the branch is full, each wrong guess will send a Zoombini back to Shelter Rock.*

## Detailed Mechanics and Rules Edit

## Mathematics Edit

The number of mappings of Zoombini traits to Fleen traits for a particular attribute is 5!=120; since there are four attributes, this gives 120^4 = 207 360 000 different mappings from Zoombinis to Fleens, even at the Oh So Hard level. At the Very Hard level, we multiply this by the number of ways to mix up the attributes, 8, and at the Very Very Hard level, 9.

Despite the large number of possibilities, for a random bunch of Zoombinis, the exact mapping can almost always be uniquely determined. We can phrase this problem more rigorously with graphs. Let G be a graph with each Zoombini of the party as a node, and an edge between two Zoombinis for each trait they have in common, colored according to that attribute. This 4-edge-colored graph G must be isomorphic to the same graph formed by the Fleens. If G has no nontrivial automorphisms, then the isomorphism between G and the Fleen graph is unique, and so the corresponding mapping is unique as well.

While the graph formulation is precise and exact, it is often difficult to find such an isomorphism directly by hand (as it is difficult to visualize and notate these graphs). We may borrow an idea from statistics to help identify the mapping used: marginal distributions. While throwing away information, this method is much easier to use and is often enough to almost completely determine the correct mapping (good enough to complete the game). We create the marginal distribution of each attribute for the Zoombinis and Fleens, that is, how many of each Zoombini/Fleen has a particular trait of this attribute. In the harder difficulties, one can use this distributions to find the mappings between Zoombini and Fleen attributes, since their marginal distributions must be identical. Once that is determined, the traits within each distribution for the Zoombinis and Fleens must also match by number. If there are no repeated frequencies for a trait in a given attribute, this mapping is uniquely determined. Otherwise, some uncertainty still exists that may be eliminated by looking at the marginal distributions of two attributes.

## Strategy for Solving the PuzzleEdit

To solve the puzzle, the player must find out which Zoombini features relate to which Fleen features. (i.e. Zoombinis with certain feet will distract Fleens with certain eyes). This should be done with the first three Fleens. Try to make the first three Zoombinis have varied features (i.e. one has one eye, another has glasses, and the third has sleepy eyes) but not have an entirely unique feature. After the first three Zoombinis, try to deduce which Zoombinis will distract the Fleens on the branch. This is only a process of data analysis because the number of Zoombinis with a certain feature will correspond to the number of Fleens with a certain feature (i.e. the number of Zoombinis that have sleepy eyes will be equal to the number of Fleens with brown shoes depending on the data).