# Pentarot puzzle



## Herbert Kociemba (Feb 10, 2020)

I programmed a little 2D rotational puzzle which I named "Pentarot" with 36 pentagons which can move on 5 circles containing 10 pentagons each. I do not think it is physically realizable but I like the design with the 5 rotating interlocking pentagonal circles (use right and left mouse button to move the circles). I have the impression that it is quite difficult to solve, I did not search for any algorithms yet and hence have no idea how to solve it in the moment. There are some variants (rotate by multiples of 36°, 72° or 180°), use only 2 colors or include the orientations for the pentagons. I also do not know the size of the state space of any of these variants.... 
You can download the windows program from the link below.

kociemba.org/themen/pentarot/pentarot.zip


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## Filipe Teixeira (Feb 11, 2020)

good job!


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## Ben Whitmore (Feb 11, 2020)

God's algorithm for 2 colour, 36 degree moves (QTM)


```
Depth   New         Total
0       1           1
1       10          11
2       75          86
3       590         676
4       4540        5216
5       33785       39001
6       235870      274871
7       1433180     1708051
8       6953210     8661261
9       23926415    32587676
10      53938745    86526421
11      76503742    163030163
12      62360392    225390555
13      24419575    249810130
14      4052120     253862250
15      313619      254175869
16      10851       254186720
17      135         254186855
18      1           254186856
```

This is the depth 18 position:







Solution: U R U DR2' U R' L2' R' DR' DL L' DL DR' DL U DL


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## Herbert Kociemba (Feb 12, 2020)

Thanks, this was fast, very nice that there is a beautiful unique antipode with 10 symmetries! I would not call it QTM (quarter turn metric) but TTM (tenth turn metric) though. I also wonder about the notation of the rotation axes. I understand U, R and L, but downside there are two axes, so how do you name them and what means "D" in your solution?

With three colors the state space is probably Binomial[36, 6] Binomial[30, 10] = 58.521.439.856.880, which is too much for a breadth first search. So God's algorithm will be difficult to compute.


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## Pyjam (Feb 12, 2020)

DL and DR for Down Left and Down Right, I suppose.


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## Herbert Kociemba (Feb 12, 2020)

Pyjam said:


> DL and DR for Down Left and Down Right, I suppose.


Yes of course. How could I miss this.


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## Herbert Kociemba (Feb 15, 2020)

If you do not only rotate the 10 pentagons which make up a ring but also the inner 3 pentagons - so you rotate a disk now - depending on the rotation angle the right or left neighbor disk may lock. While there are in principle 10^5 ways to rotate the 5 disks (each disk has 10 possible positions), there are only 1301 ways which are possible and where at least 2 disks are unlocked. We may ignore the constellations where only one disk is movable because this is a sort of dead end and to proceed we have to rotate this disk into a position where another disk is unlocked.
This locking feature adds a level of difficulty to the puzzle.
In the example below U^3 has locked the R move, while all other moves are still possible.


I added this feature to my program now.


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## Herbert Kociemba (Apr 9, 2020)

I finally am able to manually solve all variants of the puzzle myself and found reasonable short optimal maneuvers for some basic operations like single pair swap or single pair twist. For my own documentation purposes I wrote some thoughts down here:
http://kociemba.org/themen/pentarot/pentarot.html


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