text: # Logi Toli
  
  Logi Toli is a simple puzzle consisting of a arrangement of tracks containing six sliding pieces. The tracks form the shape of an elongated letter H, but with a small side track in the centre. At the start three orange pieces fill the vertical section of track on the left, and three yellow pieces fill the vertical section on the right. The aim is to slide the pieces along the tracks until the orange pieces have swapped places with the yellow.
  
  From this:
  
  to this:
  
  The puzzle is easy, so you could set yourself the extra challenge of using as few moves as possible, or to minimise the total distance travelled by the pieces.
  
  This puzzle is listed in L. E. Hordern's book "Sliding Piece Puzzles", entry B33, where it says that it was called "A Motor-Car Problem" when published in the Strand magazine in 1903, and that a cardboard version called "Germans vs. Allies" was sold in 1918.
  
  If your browser supports JavaScript, then you can play the Logi Toli puzzle by clicking the link below:
  
  ## The number of positions:
  
  You could consider the puzzle to have 10 piece locations - 3 on each of the outside vertical sections, 3 more on the horizontal track, and one on the small central side track. These ten locations contain 3 orange pieces, 3 yellow pieces, and 4 empty spots. This can be arranged in 10! / (3!·3!·4!) = 4,200 ways.
  
  With the above counting of positions, a single move would be sliding a piece along a track by a distance of one piece diameter. God's Algorithm can be calculated using this definition, and it turns out that the furthest away you can get from the start position is 66 moves, though the goal position is just 62 moves away from start.
  
  Depth#positions
  01
  12
  27
  314
  426
  532
  644
  754
  867
  966
  1061
  1124
  1218
  1314
  1422
  1526
  1650
  Depth#positions
  1760
  1884
  1992
  20112
  21104
  22104
  2370
  2456
  2534
  2636
  2740
  2876
  2984
  30128
  31152
  32196
  33212
  Depth#positions
  34252
  35200
  36164
  3776
  3856
  3936
  4052
  4148
  4288
  4392
  44104
  4592
  4696
  4776
  4864
  4940
  5036
  Depth#positions
  5124
  5222
  5322
  5440
  5538
  5652
  5750
  5844
  5938
  6045
  6130
  6223
  6314
  6410
  654
  664
  Total4,200
  
  The four antipodes are:
  
  A more natural way to define a move is as a shift of one piece by any distance along any path. With this move definition, the furthest away you can get from the start position is 19 moves, though the goal position is just 17 moves away from start.
  
  Depth#positions
  01
  18
  251
  3110
  4190
  5142
  6130
  7346
  8314
  9254
  10558
  11652
  12292
  13152
  14424
  15190
  1665
  17171
  18130
  1920
  Total4,200
  
  Here are some of the antipodes:
  
  The remaining antipodes are horizontal or vertical reflections of these.
  
  ## Solution:
  
  The following solution is optimal in both metrics, i.e. pieces are moved only 17 times, and the total distance moved is 62 piece diameters. There are 8 such optimal solutions, but they are related to each other by the following symmetries:
  
  • Mirroring the moves, swapping left and right.
  • Mirroring the left vertical track, swapping the top left and bottom left locations.
  • Mirroring the right vertical track, swapping the top right and bottom right locations.
metadata: {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}