# 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
01
12
23
35
46
57
611
716
819
925
1038
1151
1263
1388
14125
15160
16201
17276
18357
19427
20510
21633
22773
23868
24959
251,107
261,255
271,374
281,550
291,802
302,032
312,223
322,433
332,667
342,914
Moves# Positions
353,216
363,551
373,864
384,217
394,587
404,920
415,257
425,653
436,100
446,595
457,083
467,577
478,149
488,672
499,155
509,736
5110,333
5210,881
5311,573
5412,267
5512,807
5613,409
5714,008
5814,468
5915,008
6015,579
6116,049
6216,609
6317,182
6417,608
6518,035
6618,419
6718,638
6818,949
6919,230
Moves# Positions
7019,405
7119,642
7219,790
7319,746
7419,786
7519,695
7619,317
7719,006
7818,666
7917,985
8016,927
8115,444
8213,636
8311,909
8410,202
858,570
867,261
876,061
884,941
894,021
903,230
912,657
922,155
931,686
941,306
95932
96639
97428
98268
99164
10094
10144
10217
1035
Total720,000
Avg Depth64.793
Moves# Positions
01
13
24
36
48
512
616
724
832
948
1064
1196
12126
13183
14242
15357
16444
17564
18698
19921
201,092
211,317
221,486
231,740
242,000
252,403
262,733
273,195
283,509
293,900
304,356
Moves# Positions
315,040
325,581
336,225
346,634
357,326
368,041
378,946
389,646
3910,563
4011,240
4111,964
4212,710
4313,701
4414,458
4515,060
4615,551
4716,416
4817,129
4917,712
5018,178
5118,759
5219,220
5319,680
5419,963
5520,277
5620,654
5720,706
5820,522
5920,724
6020,779
6120,280
Moves# Positions
6219,977
6319,665
6419,218
6518,498
6617,577
6716,548
6815,933
6914,964
7013,771
7112,612
7211,750
7310,392
749,137
757,932
766,979
775,610
784,511
793,324
802,520
811,518
821,029
83606
84390
85153
8674
8724
8815
896
902
Total720,000
Avg Depth54.478

## 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:

1. Move frame to the bottom-left position, so that single piece in the top row is at the left.
2. 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
3. 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.
4. Do RR to move frame to the bottom-right position. Note that the black dial should now show 0 as required.
5. Repeat the sequence LURD until the green dial is in the centre of the bottom row.
6. 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.
7. 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.
8. 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.
9. 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.