# Nautilus

Nautilus is a colourful puzzle in the shape of a Nautilus shell. It is mechanically similar to the Square-1, as it has three layers, where the middle layer is split through the middle into two halves. When solved, the top and bottom layers each consist of seven pieces arranged like slices around the centre point. The seven pieces in a layer have different angular sizes, namely 20, 30, 40, 50, 60, 70, and 90 degrees, and are coloured like a rainbow from violet to red. The outer layers can rotate about their centre point. If both layers are rotated such that the seam through the middle layer lines up with seams between the pieces, then you can turn over half the puzzle, mixing the pieces from the two outer layers together.
The puzzle was designed by Tim Selkirk, and produced by Uwe Meffert.

## The number of positions:

As this puzzle is so heavily bandaged, it is not possible to calculate the number of positions, and you have to simply enumerate them all by trying all possible move sequences (i.e. find God's algorithm). It turns out that the movements are so restricted that it is impossible to swap two pieces of the same size, and even that the arrangement of pieces in one outer layer determines the order of the pieces in the other.
I consider one half of the middle layer as fixed in space. The other half of the middle layer has two possible positions. If positions that differ by turns of the top or bottom layers are considered the same, then there are 326 possible arrangements of the outer layers, giving 652 positions. If we count twistable arrangements, those in which it is possible to twist the middle layer, then there are 2016 arrangements of the pieces, or 4032 positions all together. Note that this shows that on average a layer has sqrt(2016/326) = 2.48 orientations in which it lines up with the seam, i.e. there are on average 1.24 cuts per layer.

I have used a computer to calculate God's algorithm for the puzzle. The results in the table below show that the pieces can be solved in at most 12 twists, or 13 if you want to solve the middle layer as well. If you count all outer layer turns as moves too, then it takes 27 moves, or 28 with the middle layer.

Excluding the middle layer
Turn metric: Turn metric
Twistable positions:
Twist metric:
 0 1 71 1301 1654 8001 12543 25673 37216 18908 41696 37232 31104 39220 7700
14 28240 34584 14984 5736 9808 5192 9800 560 5192 5760 19600 1120 10352 9248
 0 1 7 17 42 97 147 193 188 180 144 128 144 100 116
14 104 56 32 32 24 32 16 32 40 64 32 32 16
0 1 16 49 66 40 36 42 20 8 8 8 16 16
Including the middle layer
Turn metric: Turn metric
Twistable positions:
Twist metric:
 0 1 71 1301 1656 8153 15853 38951 50138 62048 65948 54328 97552 54430 56710 40568
15 62284 36776 38680 11468 19620 14992 10360 5756 10948 25360 20720 11480 19592 9248
 0 1 7 17 44 117 229 367 354 356 356 280 280 254 206
14 240 156 88 72 60 52 48 48 76 100 96 64 56 8
0 1 16 65 116 106 76 78 62 28 16 16 24 32 16

## Notation:

A twist of the right hand side of the puzzle will be denoted by a slash /.
A clockwise turn of the top layer by less than a half turn to the next twistable arrangement is denoted by U. Similarly an anti-clockwise turn of the top layer is U'. A half turn of the top layer, regardless of whether there are any intermediate twistable states is denoted by U2.
In the same way, a clockwise turn, half turn, and anti-clockwise turn of the bottom layer are denoted by D, D2, and D' respectively.

## Solution:

1. Hold the puzzle so that the middle layers big red section is at the front-left, with the green or orange piece at the front-right. The middle layer seam lies between them.
2. Do whatever moves are necessary to bring the two largest (red) pieces together, side by side in the same layer. This can always be done in no more than 4 twists, but usually in one or two. You have to manoeuvre the two red pieces so that they are in different layers, one at the front and one at the back, and on opposite sides of the seam, so that a twist will bring them together. This is fairly easy and tends to happen automatically when doing random moves.
3. Put the two big red pieces in the left half of the bottom layer. This keeps them out of the way allowing you to work on the rest using twists and top layer turns only.
4. Without disturbing the big red pieces, do whatever moves are necessary to place the two smallest pieces (purple) in the top right half, one at the front directly to the right of the seam, the other at the back directly to the right of the seam. In exceptional cases, this can take up to 7 twists, but is usually much less. Usually just doing random twists and top layer turns suffice to bring the two purple pieces to the top layer, the right distance apart for a top layer turn to put them both directly to the right of the seam.
5. Do a twist move to place the two small purple pieces adjacent to the two big red pieces.
6. Rotate the top layer so that a yellow or a green pair of pieces is at the front, straddling the seam, and do the same with bottom layer red pair. There are only 8 possible positions that the puzzle can now have. Look up this position in the table below, and do the indicated move sequence to solve the outer layers.
7. If the middle layer is not yet solved, you can flip the right-hand side of the middle layer, by doing U2 / U2 / U2 /.

Here is a table with the finishing sequences for the eight goal positions. The number in the final column is the number of twistable positions with the reds adjacent to each other from which that goal position can be reached without splitting the reds. It gives a rough measure of how likely each goal position is to occur.

1 2 / U2 132 / U2 D' / D 6 / U' / U' 6 / D' / U' D / U 24 / U' / U D' / U2 D 24 / U / U' D' / U2 D 6 / U' / U' D' / D 6 / D / U D' / U' D / U2 D 6