From (2, 2) we may only get to (1, 2) and (3, 2) by moving the small disk from peg 2 to either peg 1 or peg 3. However, from a given node, not all configurations are reachable with a single move. Each of the 9 possible vectors corresponds to a single legitimate puzzle configuration. The only legitimate interpretation for (2, 2) is that the small disk is being on top, the bigger one at the bottom of the stack. We'll tacitly focus on the legitimate disks positions. (2, 2) means that both disks are located at the second peg. For a puzzle with two disks, there are 9 possible configurations. To avoid ambiguity the disks are ordered according to their sizes, with the smallest being the first.ДЇor a puzzle with a single disk, only three configurations are possible: (1), (2), (3), where, for example, (2) denotes the configuration in which the lone disk is located at the peg #2. A puzzle configuration is completely defined by a list of such peg numbers associated with every disk. Every disk may be said to be at one of the pegs and, for a specific configuration, be associated with one of the numbers: 1, 2, 3. Disks are all of different sizes and it's forbidden to place a bigger disk on top of a smaller one. In the Tower of Hanoi puzzle, disks stacked on one peg are to be moved to another with, perhaps an intermediate stop at a third, auxiliary peg. As was discovered by Ian Stewart, puz( Tower of Hanoi) has a surprising relationship to the Sierpinski gasket (also known as the Sierpinski triangle) and, therefore, to Pascal's triangle. If no such path exists the problem is unsolvable.ДЇor a given puzzle G, puz(G) designates the associated puzzle graph. To solve the puzzle, find dots corresponding to the initial and the target configuration and the path (of moves) from one to the other. Otherwise, it's a plain curve connecting the two dots. If the move is not reversible, the edge must be directed.

For every move from one configuration to another, connect the corresponding dots with an edge.

To represent a puzzle graphically, associate various (legitimate) configurations of the puzzle with dots in some space (practically, it's a piece of paper or a computer screen we'll be working with). We are looking into a vector (an ordered set) of parameters whose values uniquely describe appearance (or configuration, or state) of the puzzle. With every move, parameter values change and the puzzle looks a little different. More generally the idea applies to every puzzle, game, or problem (not necessarily mathematical) that may be described by a set of parameters. In the discussion on Sam Loyd's Fifteen puzzle we introduced the notion of puzzle graph.

Towers of hanoi towers of hanoi