0019: Crazy Elephant Dance

Crazy Elephant Dance

This puzzle took place in the "Nob Yoshigahara Puzzle Design Competition" at the 25th International Puzzle Party (IPP25) that was held in Helsinki. It was awarded by the jury with an 'Honorable Mention': silver medal

Click on the picture to play the puzzle.

(java-applet, 93 Kb in total)
The following persons successfully have solved all levels of this puzzle:
Dirk Weber, Kirk Bresniker, Matthew Urick, Lance Nathan, Christofferson Rolan, Martin Towell, Kassem Sobh, Roger Phillips, Jon Braunhut, Rob Leach, Eddy Ramirez, J.B. Gill, Jonathan Cross, Jasmin Müller, Kyle Monarch, Karst Koymans, Michael Toulouzas, Serhiy Grabarchuk Jr., Heiko Moye, Charlotte Henderson, Gerhard Schanzer, Jarno Honkanen, Tom Cutrofello, Szabó József Imre, Cary Kawamoto, Daniel Holtzman, Juozas Granskas, Fabian Lemp, Jess Paulson, Hor Guo Yi, Henk Stipdonk, John Moores, Alan Lemmon, Richard Amstutz, Brian Pletcher, Robert Stegmann, Calin Barbat, Jutta Hopp, Pantazis Houlis, Yong Hao Ng, Robert Bruce, Brandon Martin, Mortsemious, Gabriel Fernandes, Detlef Krüger, Michel van Ipenburg, Goetz Schwandtner, Davud Salakhov.

prototype

The famous circus star Maestro Mansini trains his elephants to always give an extraordinary performance. This time he invented the "Crazy Elephant Dance". While standing on their usual platforms the elephants have to turn according to the following rule: Using the snout each elephant is only allowed to scratch other elephants backs going upwards and downwards - but not across. Strictly following that rule they subsequently are allowed to leave the arena. Although he once worked out the sequence for when the elephants should turn, now he is a little bit confused. Can you help him and his elephants to gloriously dance off the stage?

Turn the elephants to make them all dance off the stage.

1.		each elephant has 3 orientations: pointing upwards, pointing to the right, pointing downwards
2.		an elephant can only be rotated at the position with the circular cut-off
3.		an elephant can be rotated from pointing upwards to the right (or vice versa) only when the elephant to its right is pointing downwards
4.		an elephant can be rotated from pointing downwards to the right (or vice versa) only when the elephant to its right is pointing upwards
5.		an elephant can only leave the stage when pointing downwards

Do you know the wonderful puzzle Spin Out by Binary Arts? It is a mechanical sliding puzzle, where you have to slide a red bar ouf of a green shell. Ontop of the red bar there are elephants attached which can be rotated. According to the mechanism the elephants all have to point to the exit before the red bar is able to be slid out completely. Below you can see a picture of that puzzle.

Spin Out puzzle by Binary Arts

Each elephant has exactly two orientations which corresponds to a binary system. My idea was to extend this puzzle using three orientations for each elephant instead of two. This extends the puzzle Spin Out from a binary version to a ternary version. On top of this page you see a mechanical prototype of the puzzle which has been lasercut by Peter Knoppers (Buttonius Puzzles & Plastics).
The mechanism for the Crazy Elephant Dance is more complex than the Spin Out puzzle and uses more layers where the locking and unlocking takes place. Each elephant has three orientations: pointing up, to the right or pointing down. In the start position all elephants are pointing up. The objective is to turn all elephants such that they are pointing down. In this position the slide can be moved out of the shell.

When an elephant is pointing to the right, it interlocks with its snout to its right neighbour. The next pictures show the layers where the most 'interesting things' happen:

In contrast to Spin Out, the puzzle Crazy Elephant Dance is more complex. While solving it, you can do 'wrong' moves that lead to dead ends. So, if you don't watch out carefully you have to go back 'some steps'. This means that you always have to think which move has to be done next.
It is interesting to know how many steps are necessary to solve the puzzle with n elephants. Therefore, I use S(n) for the number of times you have to grab an elephant and turn it (in the mechanical puzzle version). In this case it doesn't matter whether you only turn it 90 degrees or 180 degrees. Here is a table showing the numbers S(n) for n=1,...,10:

n = 1 2 3 4 5 6 7 8 9 10 ...

S(n) = 1 5 15 37 83 177 367 749 1515 3049 ...

For S(n) there is a recursive formula: S(n+1) = 3*S(n) - 2*S(n-1) + 2 , for n >= 2
but also an explicit one: S(n) = 3*(2^n - 1) - 2*n , for n >= 1.

Example: let's assume you want to make n=7 elephants dance off the stage and each second you can completely rotate an elephant (eventually sliding the elephants is included in that step). Then you will have finished the 367 'dancing steps' in 6 minutes and 7 seconds!