Palette Mix 4

Palette Mix 4 is a puzzle with 4 square pieces. The pieces lie inside a 2×3 base, and are also held inside a cross-shaped frame. The top and bottom arms of this cross can hold only 1 piece, and the left and right arms a 1×2 square of pieces. By moving the frame horizontally or vertically, the pieces are moved about the 2×3 rectangle. To complicate matters, each piece contains a cogwheel with 10 teeth. The frame has teeth along the inside of its bottom edge, as do the tops of the square pieces themselves. As you move the frame around, the cogs engage the frame and rotate.
Each cog has a coloured arrow on it. When solved, all the arrows should be pointing upwards, with the blue arrow on the top row on its own, and the bottom row containing black, green, and red in that order.

These puzzles were invented by Douglas A. Engel, who also invented various other puzzles such as:

The number of positions:

There are 4 pieces and, as with any puzzle that essentially only uses plungers to move the pieces, only even permutations of the pieces are possible. This gives 4! / 2 = 12 positions of the colours for any frame position. Any arrow can be twisted any amount without disturbing anything else. Each arrow has 10 possible orientations, so there are 12·10⁴ positions of the pieces in the frame. The frame itself can be positioned in 2×3 = 6 ways, so there are really 6·12·10⁴ = 720,000 positions.

I have used a computer to calculate God's Algorithm. The results are in the two tables below. Each table uses a different definition of what constitutes a single move.

Sliding frame exactly one tile is one move

Sliding frame in one direction is one move

Moves	# Positions
0	1
1	2
2	3
3	5
4	6
5	7
6	11
7	16
8	19
9	25
10	38
11	51
12	63
13	88
14	125
15	160
16	201
17	276
18	357
19	427
20	510
21	633
22	773
23	868
24	959
25	1,107
26	1,255
27	1,374
28	1,550
29	1,802
30	2,032
31	2,223
32	2,433
33	2,667
34	2,914

Moves	# Positions
35	3,216
36	3,551
37	3,864
38	4,217
39	4,587
40	4,920
41	5,257
42	5,653
43	6,100
44	6,595
45	7,083
46	7,577
47	8,149
48	8,672
49	9,155
50	9,736
51	10,333
52	10,881
53	11,573
54	12,267
55	12,807
56	13,409
57	14,008
58	14,468
59	15,008
60	15,579
61	16,049
62	16,609
63	17,182
64	17,608
65	18,035
66	18,419
67	18,638
68	18,949
69	19,230

Moves	# Positions
70	19,405
71	19,642
72	19,790
73	19,746
74	19,786
75	19,695
76	19,317
77	19,006
78	18,666
79	17,985
80	16,927
81	15,444
82	13,636
83	11,909
84	10,202
85	8,570
86	7,261
87	6,061
88	4,941
89	4,021
90	3,230
91	2,657
92	2,155
93	1,686
94	1,306
95	932
96	639
97	428
98	268
99	164
100	94
101	44
102	17
103	5
Total	720,000
Avg Depth	64.793

Moves	# Positions
0	1
1	3
2	4
3	6
4	8
5	12
6	16
7	24
8	32
9	48
10	64
11	96
12	126
13	183
14	242
15	357
16	444
17	564
18	698
19	921
20	1,092
21	1,317
22	1,486
23	1,740
24	2,000
25	2,403
26	2,733
27	3,195
28	3,509
29	3,900
30	4,356

Moves	# Positions
31	5,040
32	5,581
33	6,225
34	6,634
35	7,326
36	8,041
37	8,946
38	9,646
39	10,563
40	11,240
41	11,964
42	12,710
43	13,701
44	14,458
45	15,060
46	15,551
47	16,416
48	17,129
49	17,712
50	18,178
51	18,759
52	19,220
53	19,680
54	19,963
55	20,277
56	20,654
57	20,706
58	20,522
59	20,724
60	20,779
61	20,280

Moves	# Positions
62	19,977
63	19,665
64	19,218
65	18,498
66	17,577
67	16,548
68	15,933
69	14,964
70	13,771
71	12,612
72	11,750
73	10,392
74	9,137
75	7,932
76	6,979
77	5,610
78	4,511
79	3,324
80	2,520
81	1,518
82	1,029
83	606
84	390
85	153
86	74
87	24
88	15
89	6
90	2
Total	720,000
Avg Depth	54.478

Notation:

Sliding the frame left or right by exactly one tile is denoted by L or R respectively. In the same way moving it up or down is denoted by U or D.

Solution:

Move frame to the bottom-left position, so that single piece in the top row is at the left.
Find the piece with the black arrow. Bring it to the bottom left position. This is easy, and can be done as follows:
Bottom left: Do nothing.
Bottom middle: RULD
Bottom right: RRULLD
Top left: URDL
Repeat the sequence RULD until the black dial is in the bottom left corner again and has number 4 at the top. You may have to repeat the moves up to 27 times. Note that if the black dial shows 1, 2, 5, or 8 then it will be quicker to repeat URDL instead.
Do RR to move frame to the bottom-right position. Note that the black dial should now show 0 as required.
Repeat the sequence LURD until the green dial is in the centre of the bottom row.
Repeat the sequence LURD until the green dial is in the bottom centre location again and has number 0 at the top. You may have to repeat the moves up to 27 times. Note that if the black dial shows 3, 6, 9, or 2 then it will be quicker to repeat ULDR instead.
The blue and the red pieces should be at their correct locations (blue in the top row, red at the bottom right), and only need to be twisted to the correct numbers.
To rotate the blue arrow do one of the following:
Clockwise: LLURDR (ULDR)4 LLURRD (LURD)4
Anti-clockwise: (ULDR)4 ULLDRR (LURD)4 LULDRR
Repeat until the blue dial is correct.
To rotate the red arrow do one of the following:
Clockwise: (LURD)4 LLURDR (ULDR)4 LLURRD
Anti-clockwise: ULLDRR (LURD)4 LULDRR (ULDR)4
Repeat until the red dial is correct.

Jaap's Puzzle Page

Palette Mix 4

The number of positions:

Links to other useful pages:

Notation:

Solution: