Jaap's Puzzle Page

Nautilus

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:
01
171
21301
31654
48001
512543
625673
737216
818908
941696
1037232
1131104
1239220
137700
1428240
1534584
1614984
175736
189808
195192
209800
21560
225192
235760
2419600
251120
2610352
279248
Total422496
= 326*36*36
01
17
217
342
497
5147
6193
7188
8180
9144
10128
11144
12100
13116
14104
1556
1632
1732
1824
1932
2016
2132
2240
2364
2432
2532
2616
Total2016
01
116
249
366
440
536
642
720
88
98
108
1116
1216
Total326
Including the middle layer
Turn metric: Turn metric
Twistable positions:
Twist metric:
01
171
21301
31656
48153
515853
638951
750138
862048
965948
1054328
1197552
1254430
1356710
1440568
1562284
1636776
1738680
1811468
1919620
2014992
2110360
225756
2310948
2425360
2520720
2611480
2719592
289248
Total844992
= 326*36*36*2
01
17
217
344
4117
5229
6367
7354
8356
9356
10280
11280
12254
13206
14240
15156
1688
1772
1860
1952
2048
2148
2276
23100
2496
2564
2656
278
Total4032
= 2016*2
01
116
265
3116
4106
576
678
762
828
916
1016
1124
1232
1316
Total652
= 326*2

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.

1Solution 1/ U2132
2Solution 2/ U2 D' / D6
3Solution 3/ U' / U'6
4Solution 4/ D' / U' D / U24
5Solution 5/ U' / U D' / U2 D24
6Solution 6/ U / U' D' / U2 D6
7Solution 7/ U' / U' D' / D6
8Solution 8/ D / U D' / U' D / U2 D  6