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Tower of Hanoi – Inverted Solution:

If the tower consisted of only 1 disk, the puzzle could be completed in only 1 move.  If the tower consisted of 2 disks, the puzzle could be completed in 3 moves.  if the tower consisted of 3 disks, the puzzle could be completed in 7 moves.  For each of these cases, the number of moves is  .  Thus for the 5 disk model, the puzzle could be completed in  moves.

 

The Travelling Salesman Solution:

The distances between the various cities, as measured in centimeters on the model, are shown on the table below.

 

  Olympia Salem Boise Helena Sacramento Carson City Salt Lake City Cheyenne Denver Santa Fe Phoenix
Olympia 14.1 29.8 42.2 51.2 48.3 60.2 85.1 90.4 105 98.3
Salem 14.1 29.5 49.1 38.1 36.9 56.8 86.2 90.3 101.5 89.7
Boise 29.8 29.5 24.9 41.6 34 30.5 56.7 61.1 75.1 72.2
Helena 42.2 49.1 24.9 66.5 58.4 39.6 47.8 55 76.8 85.8
Sacramento 51.2 38.1 41.6 66.5 9.3 46.9 82.4 82.7 82.9 59.1
Carson City 48.3 36.9 34 58.4 9.3 37.7 73.1 73.6 75.2 54.7
Salt Lake City 60.2 56.8 30.5 39.6 46.9 37.7 35.3 36.2 44.7 46.2
Cheyenne 85.1 86.2 56.7 47.8 82.4 73.1 35.3 8.7 36.2 64
Denver 90.4 90.3 61.1 55 82.7 73.6 36.2 8.7 27.5 57.9
Santa Fe 105 101.5 75.1 76.8 82.9 75.2 44.7 36.2 27.5 38.8
Phoenix 98.3 89.7 72.2 85.8 59.1 54.7 46.2 64 57.9 38.8

 

Selecting Salt Lake City as the starting city, a Mathematica computer program ran for 2 days to arrive at the path that generated the shortest distance of 276.7 cm. That path is

Salt Lake City – Carson City – Sacramento – Salem – Olympia – Boise – Helena – Cheyenne – Denver – Santa Fe – Phoenix

 

The path from Salt Lake City with the second shortest length of 281.2 cm is

Salt Lake City – Cheyenne – Denver – Santa Fe – Phoenix – Carson City – Sacramento – Salem – Olympia – Boise – Helena

 

If the starting city is Boise, the shortest path of 276.6 cm is

Boise – Helena – Olympia – Salem – Sacramento – Carson City – Salt Lake City – Cheyenne – Denver – Santa Fe – Phoenix

 

The path from Boise with the second shortest length of 284.6 cm is

Boise – Helena – Olympia – Salem – Carson City – Sacramento – Salt Lake City – Cheyenne – Denver – Santa Fe – Phoenix