Towers of Hanoi Diamond and Gold 2 Disc Solution (Fewest Moves)
0Bicolor Towers of Hanoi (Diamond and Gold)
2 Disc Solution in the Fewest Moves
A variation on the classic Tower of Hanoi puzzle. The objective is to recreate the Diamond Tower on the diamond peg and the Gold Tower on the gold peg.
(1) A move is defined as lifting the upper disc from one of the stacks and placing it on top of another stack or empty peg.
(2) No disc may be placed on top of a disc that is smaller than itself.
(3) Identically sized discs can stack.
(4) Only one disc may be moved at a time.
Each disc in the diamond tower has the same dimensions as the corresponding disc on the gold tower. This means, for example, that the smallest diamond disc can stack on the smallest gold disc, and the smallest gold disc can stack on the smallest diamond disc. In setting up the puzzle, the discs are arranged alternately with the bottommost gold disc on the diamond peg and the bottommost diamond disc on the gold peg.
I worked on the solution for the 2, 3, 4 and 5 disc versions of this puzzle for fun, trying my best to solve them in the fewest moves possible, until I was happy enough to go looking for an equation that could tell me if I’d managed to do that. There is surprisingly little information online, however I was able to find a really good article by Nathan Bowler posted on cut-the-knot.org in 2004. This confirmed that my draft solutions for 3 discs and 4 discs were in the fewest moves at 30 and 71 respectively, but my solution for 5 discs was 3 moves more than the 156 moves it should have been. It took a couple more attempts and then I got it. There is probably more than one way of solving this in the fewest moves, as I used different approaches when trying to solve each version.
L = Least number of moves
n = number of discs
Ln = 11 x ((2 to the power of n–1) –1) – 3 x (n–2)
L1 = 11 x ((2 to the power of 1–1) –1) – 3 x (1–2)
L1 = 11 x ((2 to the power of 0) –1) – 3 x (–1)
L1 = 11 x (1–1) – 3 x (–1)
L1 = 11 x 0 – (–3)
L1 = 0 – (–3)
L1 = 3
Ln = 11 x ((2 to the power of n–1) –1) – 3 x (n–2)
L2 = 11 x ((2 to the power of 2–1) –1) – 3 x (2–2)
L2 = 11 x ((2 to the power of 1) –1) – 3 x 0
L2 = 11 x (2–1) – 0
L2 = 11 x 1
L2 = 11
Ln = 11 x ((2 to the power of n–1) –1) – 3 x (n–2)
L3 = 11 x ((2 to the power of 3–1) –1) – 3 x (3–2)
L3 = 11 x ((2 to the power of 2) –1) – 3 x 1
L3 = 11 x ((2x2) –1) – 3
L3 = 11 x (4–1) – 3
L3 = 11 x 3 – 3
L3 = 33 – 3
L3 = 30
Ln = 11 x ((2 to the power of n–1) –1) – 3 x (n–2)
L4 = 11 x ((2 to the power of 4–1) –1) – 3 x (4–2)
L4 = 11 x ((2 to the power of 3) –1) – 3 x 2
L4 = 11 x ((2x2x2) –1) – 6
L4 = 11 x (8–1) – 6
L4 = 11 x 7 – 6
L4 = 77 – 6
L4 = 71
Ln = 11 x ((2 to the power of n–1) –1) – 3 x (n–2)
L5 = 11 x ((2 to the power of 5–1) –1) – 3 x (5–2)
L5 = 11 x ((2 to the power of 4) –1) – 3 x 3
L5 = 11 x ((2x2x2x2) –1) – 9
L5 = 11 x (16–1) – 9
L5 = 11 x 15 – 9
L5 = 165 – 9
L5 = 156
The background photo is taken from a video called “Arc Shot of Diamond Flower Tower” by Housee Housee, courtesy of Pexels (treated with a filter). The Diamond Flower Tower (Tòa nhà hỗn hợp) is in the city of Hanoi (Hà Nội) in Vietnam (Việt Nam). The diamond and gold textures are by Hamid Eshafah and Peter Olexa, courtesy of Pexels (treated with a filter).
Music:
Song of Sadhana by Jesse Gallagher
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