Connect the Ducks — MIT Mystery Hunt 2015

The elusive Mistree Duck (Duckus konundros) celebrates nine holidays over the year. On Duck New Year, the most important holiday of all, each flock of Mistree Ducks gathers at a local pond to perform a ceremony.

For the Duck New Year ceremony, flocks ring in the New Year by reprising their celebrations of the other eight yearly holidays. The nine Mistree Duck holidays are (in order): Valentine's Will-You-Be-Mine (a day in mid-February), Spring Circling (a day in spring), May Day Heyday (a day in May), Summer Bummer (a day in Summer), Equinox Run-Amok (a day between summer and fall), Fall Ball (a day in fall), Christmas Mess (a day in either late December or early January, depending on which duck you ask), Winter Sprinter (a day in mid-January), and Duck New Year (a day in late January, known on American human calendars as Martin Luther King Jr. Day).

Legend has it that the Duck New Year celebrations are painstakingly designed to spell out messages for the universe to look down upon (assume that water can be seen through clearly) - the ducks' hopes for the new year, reflections on the past year, or existential musings.

The rules for celebrating Duck New Year are very strict and are followed instinctively by even the youngest of ducklings, although each individual duck does have individual rules that they follow, modifying some details of the proceedings.

Duck New Year always starts with the first part of the Duck New Year ritual. Then, the celebrations go through the past year's worth of holidays, in an order determined by the participants. After all eight other holidays have been remembered, and their remembrance rituals followed, the second part of the Duck New Year ritual closes the ceremony.

The local flock of Mistrees uses eight ducks as performers and one duck as the Conducktor in charge of the ceremony during their Duck New Year celebration. They follow these rules, subject to the listed limitations of the duck performers.

Overall Duck Rules:

There are 27 locations the ducks can be in, arranged in a 3x3x3 cubical underwater grid. (See art for a top-down view of the uppermost 3x3 of points on the surface of the water.)
The three vertical levels are the surface of the pond, the middle of the pond (depth-wise), and the bottom of the pond.
The Conducktor is stationary. He does not enter the pond or switch sides of the pond. He is not included in determinations of the alphaduckical order of the ducks; only the eight participants are. Letters formed in the pond are viewed from above the pond, from his point of view.
The Conducktor only reads letters using fourteen-segment display.
Diving or rising does not change the way the duck is facing. Neither does swimming or flying forward, right, left, or backwards.
Ducks can stay underwater indefinitely.
Instructions to swim or fly forward, right, left, or backwards are determined from the perspective of the moving duck. Instructions to turn right or left are also determined from the perspective of the moving duck.
Left-wing side and right-wing side are determined from the perspective of the Conducktor.
The ordering of ducks, holidays, etc. is always done by alphaduckical order. For instance, "the third duck" means the third duck in the current alphaduckical order.
The current alphaduckical order is the based on the current order of the alphaducklet, which the Conducktor is in charge of keeping and modifying. Alphaduckical order always ignores spaces and punctuation.
The initial alphaducklet is the same as the human alphabet.
Alphaduckical order is always evaluated reading from right to left (i.e. XXY comes before AYY when alphaduckical order is in the same order as human alphabetical order).
At the end of a holiday's instructions, the alphaduckical order changes based on the Conducktor's whim. Directly after the end of a holiday's instructions, the next lead duck is determined to be the duck first in the new alphaduckical order. The next holiday is also chosen to be the holiday first in the new alphaduckical order.
No instruction is visited more than once. No duck is the lead duck for more than one holiday (not counting either part of Duck New Year's instructions). No holiday is visited more than once except Duck New Year. The first part of Duck New Year is the very first act; the second part of Duck New Year is the finale.
After one holiday's reprise, the next reprise is never the chronologically next holiday in a calendar year.
Individual Rules override Overall Duck Rules.
No two ducks can occupy the same dot on the 3d grid. A duck will ignore an order that would cause himself or herself to pass through or end on the same point as another duck, but will follow all other instructions in the step that do not cause the duck to occupy the same square as another duck.
"Adjacent" does not count diagonals.
Unless otherwise specified, all ducks enter the pond by flying to the point nearest the Conducktor on the side of the pond, facing away from the Conducktor. (The Conducktor takes up his spot at the center of one of the sides. See diagram.)
A reference to another step of the instructions is always referencing a step within the same holiday.
Ducks do not move between the end of one holiday reprise and the beginning of the next.
Swimming, diving, rising, and flying are all considered moving.
"Turn left" or "turn right" are 90° turns unless otherwise specified. "Turn around" is a 180° turn unless otherwise specified.
If a duck is facing the side of the shore the Conducktor is standing on, the duck is considered to be facing the Conducktor.
If a duck is told to swim (forward, backward, sideways, or diagonally) in a way that would make him or her leave the pond, the duck turns right instead (and doesn't move out of the pond). If a duck is at the edge, facing out, and told to swim forward one unit, the duck won't move but instead will turn to the right. If a duck is told to swim straight n units, which would make him or her leave the pond, the duck will swim to the edge of the grid, then turn right. If at the edge of the pond and told to swim multiple units, this is equivalent to being at the edge of the pond and told to swim one unit.
If a duck is told to fly in a way that would make him or her leave the pond, leave the pond. (Outside the pond, the duck's exact position does not matter.)
At any one time, a duck may be moving straight or turning, but not both at once.
When entering the pond from a side of the grid, face away from shore. If on a corner, face away from shore (diagonally) and then immediately turn 45° to the right.
A duck cannot execute a fly instruction while underwater (either at the bottom of the pond or in the middle).
A duck can swim sideways, backwards, forwards, and diagonally. Swimming diagonally is always along the same depth and never dives or rises.
Unless otherwise specified, ducks move in the direction that they are facing.
A duck can look diagonally when instructed to, but as soon as the move is complete, the duck will turn 45° to the right.
If a duck is on the surface and told to rise/swim up, the duck does nothing. If a duck is on the bottom and are told to dive/swim down, the duck does nothing. If a duck is on shore and given either instruction, the duck does nothing.
Shortest path is determined in units on the grid.
All manipulations on the letters for the universe kept by the Conducktor are done using the human alphabet, not the alphaducklet.

Individual Rules:

The nine ducks participating in the Duck New Year performance this year are:

Percy Duck (male)

Percy loves food. If he hears the dinner, lunch, or breakfast bell, he will immediately rise to the surface (ducks can hear while underwater), fly to shore, and partake of the comestibles.

Albert Duck (male)

Stubborn to the last in the autumn, if instructed to move or turn during Equinox Run-Amok, Albert will ignore the instruction and just stay in place.

Jenny Duck (female)

When swimming to the edge of the pond, coming to the edge causes Jenny to turn left at any point at which any other duck would turn right.
Jenny is the youngest.

Sarah Duck (female)

A motherly duck, Sarah likes to make sure that all ducks participate in the Christmas Mess, as it is a time of celebration for all ages. Immediately at the beginning of Christmas Mess, she will shepherd all ducks on shore, including herself, into the pond, entering one from each of the sides of the pond (in alphaduckical order, of course), starting from the space directly in front of the Conducktor and continuing clockwise around the pond.

Lemon Duck (???)

Lemon loves the holiday May Day Heyday and will refuse to leave the pond during the May Day Heyday reprise.

Fred Duck (male)

Once he enters the pond, he won't leave the pond until the finale.
He won't go underwater.

Gabrielle Duck (female)

She enjoys her time in the spotlight. If instructed to dive fancily, Gabrielle will: turn left, dive as instructed, and then swim forward two units.

Alex Duck (???)

Alex's favorite holiday is Fall Ball. Alex will be in the pond and on the surface for all of the Fall Ball reprise. Alex will move to do this even if not instructed to. This will start the instant the Fall Ball section starts, before any instructions take place.

The Conducktor (male)

He must never enter the pond.
He is in charge of communicating with the heavens, and interpreting the movements of birds.
He is in charge of maintaining the current alphaduckical order.
He is in charge of keeping the current state of the message to the universe, such as the letters spelled by the ducks' movements.

Here are instructions for this year's Duck New Year proceedings, commemorating all Duck holidays of the year. While they all must be honored during the Duck New Year celebration, they need not be celebrated in order. A holiday may be celebrated at any point in the cycle that meets its conditions except immediately after the conclusion of the previous chronological holiday. The celebration must begin and end with Duck New Year. At the end of the performance of a holiday, The Conducktor flies up overhead and looks straight down upon the pond without turning to read the next letter of the message to the universe, then returns to his post.

Duck New Year (beginning & end condition):

Beginning:

The New Year starts with a blank slate, much like the calm waters of the empty pond. The first duck enters the water embodying a child still exploring what the world and this New Year have to offer.

The youngest enters the water. This duck immediately swims forward two units.
The lead duck enters the water, and then turns left and swims forward one unit.
The next duck skips this turn to preen. The following duck enters the water instead, one unit to Jenny's right, facing away from the Conducktor.
The next two ducks enter together, on opposite sides of the pond, such that each is adjacent to another duck of the same gender, and then turn to the left. Then they both swim forward two squares, and then fly one unit to the left.
Percy enters the pond at the point that was most recently occupied by Lemon. Jenny flies one unit to her left without turning.
The Conducktor flies up and sees lines between Sarah and Percy, Percy and Fred, and Fred and Jenny. He notates that letter. The Conducktor also sees fit to change the alphaducklet, moving the first six letters of the alphaducklet to the end of the alphaducklet for determining alphaduckical order.
At this point, the ducks have ended the first part of the Duck New Year ritual and begin the commemorations of the past year's holidays, beginning with the one that is alphaduckically first.

Ending:

The two ducks on the surface fly to shore.
All ducks rise to the surface.
The alphaduckically last of the ducks in the pond flies to shore.
The newly alphaduckically last of the ducks in the pond flies to shore.
Albert and Lemon fly to shore.
The last in the pond, the lead duck, completes a circuit of the pond and then flies to shore.
Finally, the celebration comes to an end when the Conducktor applies a rot 17 to his letters to expose this Duck New Year's message to the universe.

Valentine Will-You-Be-Mine:

Will she or won't she? The answer is hidden in the mists of time. Pray to the Oracles for a vision of your future spouse!

In alphaduckical order, each duck in the pond in turn swims two units forward.
Three ducks fly down into the pond facing the Conducktor and dive down to the middle of the pond. They enter in a line that reaches straight across the pond. Albert enters the left-wing side; the alphaduckically last duck, out of the ducks on shore, enters the right-wing side, and the alphaduckically first duck, out of the ducks on shore, enters the center of the pond, directly in front of the Conducktor.
The remaining duck on shore is feeling lonely, so that duck enters the pond.
The two ducks furthest away from the Conducktor laterally turn right and then swim one unit forward.
Every duck on the left-wing side of the pond does a mating dance. This consists of, in unison: rising to the surface (if not already at the surface), then turning 180°, then swimming one unit forward, then diving down to the bottom of the pond.
All ducks turn left 90°.
The Conducktor flies overhead and sees lines between each pair of ducks on the surface of the pond and each pair of ducks in the middle of the pond. He also sees on the bottom the shortest non-overlapping path from Albert to Sarah that also goes through every duck on the bottom. The Conducktor also sees fit to move the first sixteen letters to the end of the alphaducklet for determining alphaduckical order.

Spring Circling:

Every Spring, the crop circles appear overnight, mysterious and arcane. Gather in the fields and dance among the strange patterns of flattened crops. Aliens? Gods? Bored human teenagers? Who knows! All we know is that if you don't properly observe Spring Circling, your nest will be purged.

Percy, feeling antisocial, isn't feeling the joy of spring. He dives down to the bottom of the pond.
The duck fourth in alphaduckical order flies into the pond, and immediately dives down to the bottom.
On shore, Gabrielle is calling for the duck after the duck who previously moved, so that duck flies forward one unit to be nearer Gabrielle. Fred, seeing that spot open up, swims two units to occupy the recently abandoned spot. Because he is stealthy, he does this without turning.
The duck fourth in alphaduckical order is greedy and takes another turn before anyone else has a chance to. This duck turns around 180° and then swims forward one unit.
The lead duck swims diagonally two units.
The duck who moved in step 2 swims forward 1 unit, then turns around 180°.
Of the letters he currently has, the Conducktor rotates the middle one 90° clockwise. The Conducktor flies overhead and sees lines between each pair of ducks on the surface, each pair of ducks in the middle of the pond, and each pair of ducks at the bottom of the pond. Giving in to his contrary side, the Conducktor reverses alphaduckical order.

May Day Heyday:

Mayflies only live for a day, and in that time they must find a mate and lay eggs. For May Day Heyday, you emulate them in a twenty-four hour all-out party of momentous proportions! This is traditionally followed by a ritual hiding of snail shells.

It is time to fly like the mayfly. First, Sarah flies to shore. With a flurry of wings, every duck whose place in alphaduckical order is even rises to the surface and flies to shore following Sarah.
All ducks rise to the surface.
Lemon turns around, then swims forward one unit, then swims backwards one unit.
The alphaduckically first duck out of those on the shore enters the pond.
The second, third, and sixth ducks dive one level. The second and third ducks then each swim two units to their right.
Lemon and the lead duck swim backwards one unit.
The Conducktor is pleased to note that his letters currently spell a word. Looking for another letter, the Conducktor flies up and, for each level (bottom, middle, and surface) of the pond, sees connections between ducks which are the shortest paths to connect all ducks on that level. Next, the Conducktor changes the alphaducklet by deleting and reinserting one letter at a time: the seventeenth letter to the fifth position; then the seventeenth letter to the fourth position; then the seventeenth letter to the third position; then the seventeenth to the second position; then the seventeenth letter to the first position.

Summer Bummer:

Summer is a time for laziness. Celebrate this lofty ideal by dozing on the pond all day!

The last half of the ducks (alphaduckically) flies to shore.
The alphaduckically last two ducks, in alphaduckical order, swim two units to their right without turning.
The third duck swims forward two units.
Alex dives down one level, then dives down another level, then turns around. Sarah ascends one level, and then turns around. Fred just turns around.
The females in the first half of the alphaducklet swim left one unit, in order.
The lead duck swims two units forward.
The Conducktor sees a connection between the pair of ducks at the surface, and a connection between the pair of ducks at the bottom. He also sees a set of connections containing every duck in the middle of the pond; these connections are rotationally symmetric to the surface and bottom segments together. He is pleased with this. Not wanting to be predictable, the Conducktor reverses alphaduckical order... again.

Equinox Run-Amok:

On the Equinox, the day and night are equally matched. Everybody flips the ceremonial flat pebble and joins either the Day or the Night and gathers for a show brawl at dawn on the shore. Usually a few of the reckless new adults, barely out of duckling stage, get hurt, but they heal quite easily.

All ducks on shore other than Sarah and the agendered duck enter the pond in alphaduckical order.
The third duck swims forward one unit.
The lead duck dives down one unit. The other duck(s) that were on the same level (surface, middle, bottom) as the lead duck before the dive rise to the surface and fly to shore.
The dinner bell sounds for the spectators. All ducks on shore grab a snack.
Fred turns left. Then, Fred and the duck after him in alphaduckical order each fly straight two units, with Fred going first.
Gabrielle does a fancy dive down to the bottom of the pond.
The Conducktor switches the fifth letters in from each end of the alphaducklet with each other. To look for another letter, the Conducktor flies up and, for each level (bottom, middle, and surface) of the pond, sees connections between ducks that are the shortest paths to connect all ducks on that level. Then the Conducktor reverses alphaduckical order.

Fall Ball:

The Fall Ball is a wooden ball carved with a pattern of oak leaves. Nobody knows where the ball came from, or what it symbolizes. Tradition states that it has always been here. Every year, there must be a great dance held to pay homage to the Fall Ball.

Alex is one of only two students privileged to participate in the Duck New Year celebration this year. In celebration of the bored young ducks going back to school in the fall, all ducks still young enough to be in school fly forward two units.
The last four ducks in alphaduckical order all rise one unit.
The lead duck dives one unit then swims forward one unit. Fred then follows suit, making those same moves.
Alex turns 180°, then rises one level, and then swims forward two units.
Lemon turns to the right, then swims forward two units, then turns to the right again.
The first and fourth in alphaduckical order both swim two units forward.
The Conducktor likes the popular pangram "the quick brown fox jumps over the lazy dog" and changes each letter that he currently has to the next letter after the first instance of it in that pangram. The Conducktor sees connections on each level of the pond (surface, middle, and bottom) of the shortest path connecting all ducks on that level. Feeling contrary, the Conducktor reverses alphaduckical order... again.

Christmas Mess:

Every late December, the ripped up wrapping paper from all the humans' houses is scavenged, gathered up by dedicated volunteers, and fashioned into a lovely cape. The Christmas Mess Honored Duck is elected from all the eligible bachelors and bachelorettes, and gets to wear the cape. There is also a rock-skipping competition, the winner of which gets to go on a date with the Honored Duck.

All ducks rise one level.
All ducks in the left-wing column ascend one unit or, for those already on the surface, descend to the bottom of the pond.
Alex is always excited for the Christmas Mess. Proud to be the only duck to move during this step, Alex turns left and then swims forward two units.
The sixth duck (alphaduckically) swims forward one unit. There is exactly one duck one unit away. The sixth duck swims diagonally such that the ending spot is still one unit away from that same duck, then swims diagonally again and ends up directly under the third duck.
Lemon dives down one level. Alex turns around. Gabrielle swims forward one unit, and then turns right.
All ducks with an odd number in the alphaduckical ordering swim forward one unit, in order.
For each letter of which there are multiple in his letters, the Conducktor changes the second copy of the letter by rot 13. The Conducktor sees connections on each level of the pond (surface, middle, and bottom) of the shortest path connecting all ducks on that level. The Conducktor extracts the consonants present in the name of the duck who was lead duck during Duck New Year's first part out of the alphaducklet and places those (in the order that they are in the name) at the beginning of the alphaducklet.

Winter Sprinter:

Everybody likes to cozy up into their nest under a blanket of leaves, but the cold is no excuse for sloth! The elders choose the day every year, to keep the ducklings on their toes, but sometime in early January, the ducklings are woken up in the middle of the night and shepherded out to the shore of the pond. They must then race to complete a mile long racecourse before going back to their nests.

The lead duck turns left and swims forward two units. Any ducks directly above or below the lead duck after the lead duck's move swim backwards one unit.
All ducks rise to the surface in alphaduckical order.
The lead duck and third duck both dive to the bottom. Then both swim forward one unit.
Alex turns around and then swims one unit forward. The duck on shore flies into the space Alex just vacated, facing the Conducktor.
Percy and Jenny swim forward one unit then dive two units.
The sixth duck turns around, then dives to the bottom, and then swims two units forward.
The Conducktor sees, for each level that has ducks, a quadrilateral missing the side closest to himself. The Conducktor also sees fit to move the first eight letters to the end of the alphaducklet for determining alphaduckical order.