Number of Disk Moves
Where "n" represents the number of disks in the Tower,
the minimum number of moves required to move the Tower is 2n - 1,
a frequently seen formula.
Disk Moves From The Top
The top disk moves every 2nd turn, 2n-1 times.
The second from the top moves every 4th turn, 2n-2 times.
The third from the top moves every 8th turn, 2n-3 times, and so on.
The disk at the bottom moves only once.
Disk Moves From The Bottom
Counting now from the bottom,
each odd-numbered disk always moves counter-clockwise, and
each even-numbered disk always moves clockwise.
How They Sit
Each odd-numbered disk always sits on an even-numbered disk, and
each even-numbered disk always sits on an odd-numbered disk. The odd-numbered disks never touch each other, nor do the
The Ternary Structure Of The Board
Following the moves of each disk, starting with Disk #10 on the bottom, will allow us to see more clearly the number
of times a disk goes completely around the triangle of the pegs (3 moves), with "A" indicating the anticlockwise direction
and "C" denoting the clockwise direction.
#10 - A1 = 1 move
#9 - C2 = 2 moves
#8 - A1 + 1x3 = 4 moves
#7 - C2 + 2x3 = 8 moves
#6 - A1 + 1x3 + 4x3 = 16 moves
#5 - C2 + 2x3 + 8X3 = 32 moves
#4 - A1 + 1x3 + 4x3 + 16x3 = 64 moves
#3 - C2 + 2x3 + 8X3 + 32x3 = 128 moves
#2 - A1 + 1x3 + 4x3 + 16x3 + 64x3 = 256 moves
#1 - C2 + 2x3 + 8X3 + 32x3 + 128x3 = 512 moves
for a total of 1023 moves (1024 - 1)
The Little Waltz
The ternary structure of the board also creates an interesting blend with the binary movements. In a pile of disk 1 over disk
2, two moves of disk 1, in one direction, will bring it to the same position as one move of disk 2, in the other direction,
creating a ternary whole. This is how the pile of 2 is moved : disk 1 gets off (in its direction) ; disk 2 is free to move
(in its opposite direction) ; disk 1 gets back on. An "empty" (for disks 1 and 2) beat will be added to square out the binary
rhythm and to allow a larger disk to move in its turn.
In a pile of 4 disks, the movement of each disk is represented by a horizontal bar -
disk 1 is on top with 8 black bars, then disk 2 with 4 red bars, disk 3 with 2 black bars, disk 4 with 1 red bar.
(a) Disks 1 and 2 dance their little 3-move waltz (disk 1 gets off, disk 2 is free to move, disk 1 gets back on),
indicated by a red saucer-shaped semi-hexagon which occurs four times.
(b) Disks 2 and 3 dance their little 3-move waltz (disk 2 gets off, disk 3 is free to move, disk 2 gets back on),
indicated by a black saucer-shaped semi-hexagon which occurs twice.
(c) Disks 3 and 4 dance their little 3-move waltz (disk 3 gets off, disk 4 is free to move, disk 3 gets back on),
indicated by a red saucer-shaped semi-hexagon which occurs only once.
A magnificent example of the law of HIERARCHY which tightly holds the entire process together.