In graph theory and recreational mathematics, the Hanoi graphs are undirected graphs whose vertices represent the possible states of the Tower of Hanoi puzzle, and whose edges represent allowable moves between pairs of states.

The Hanoi graph


The Hanoi graph   (black discs) derived from the odd values in Pascal's triangle

The puzzle consists of a set of disks of different sizes, placed in increasing order of size on a fixed set of towers. The Hanoi graph for a puzzle with   disks on   towers is denoted  .[1][2] Each state of the puzzle is determined by the choice of one tower for each disk, so the graph has   vertices.[2]

In the moves of the puzzle, the smallest disk on one tower is moved either to an unoccupied tower or to a tower whose smallest disk is larger. If there are   unoccupied towers, the number of allowable moves is


which ranges from a maximum of   (when   is zero or one and   is zero) to   (when all disks are on one tower and   is  ). Therefore, the degrees of the vertices in the Hanoi graph range from a maximum of   to a minimum of  . The total number of edges is[3]


For   (no disks) there is only one state of the puzzle and one vertex of the graph. For  , the Hanoi graph   can be decomposed into   copies of the smaller Hanoi graph  , one for each placement of the largest disk. These copies are connected to each other only at states where the largest disk is free to move: it is the only disk in its tower, and some other tower is unoccupied.[4]

General propertiesEdit

  with 12 edges deleted to yield a Hamiltonian cycle

Every Hanoi graph contains a Hamiltonian cycle.[5]

The Hanoi graph   is a complete graph on   vertices. Because they contain complete graphs, all larger Hanoi graphs   require at least   colors in any graph coloring. They may be colored with exactly   colors by summing the indexes of the towers containing each disk, and using the sum modulo   as the color.[3]

Three towersEdit

A particular case of the Hanoi graphs that has been well studied since the work of Scorer, Grundy & Smith (1944)[1][6] is the case of the three-tower Hanoi graphs,  . These graphs have 3n vertices (OEISA000244) and 3(3n − 1)/2 edges (OEISA029858).[7] They are penny graphs (the contact graphs of non-overlapping unit disks in the plane), with an arrangement of disks that resembles the Sierpinski triangle. One way of constructing this arrangement is to arrange the numbers of Pascal's triangle on the points of a hexagonal lattice, with unit spacing, and place a unit disk on each point whose number is odd. The diameter of these graphs, and the length of the solution to the standard form of the Tower of Hanoi puzzle (in which the disks all start on one tower and must all move to one other tower) is  .[2]

More than three towersEdit

Unsolved problem in mathematics:

What is the diameter of the graphs   for  ?

For  , the structure of the Hanoi graphs is not as well understood, and the diameter of these graphs is unknown.[2] When   and   or when   and  , these graphs are nonplanar.[5]

See alsoEdit


  1. ^ a b Hinz, Andreas M.; Klavžar, Sandi; Petr, Ciril (2018), "2.3 Hanoi Graphs", The tower of Hanoi—myths and maths (2nd ed.), Cham: Birkhäuser, p. 120, doi:10.1007/978-3-319-73779-9, ISBN 978-3-319-73778-2, MR 3791459
  2. ^ a b c d Imrich, Wilfried; Klavžar, Sandi; Rall, Douglas F. (2008), "2.2 Hanoi Graphs", Topics in Graph Theory: Graphs and their Cartesian Product, Wellesley, MA: A K Peters, pp. 13–15, ISBN 978-1-56881-429-2, MR 2468851
  3. ^ a b Arett, Danielle; Dorée, Suzanne (2010), "Coloring and counting on the Tower of Hanoi graphs", Mathematics Magazine, 83 (3): 200–209, doi:10.4169/002557010X494841, MR 2668333, S2CID 120868360
  4. ^ Stewart, Ian (2003), "Chapter 1: The Lion, the Llama, and the Lettuce", Another Fine Math You've Got Me Into, Mineola, NY: Dover Publications, ISBN 0-486-43181-9, MR 2046372
  5. ^ a b Hinz, Andreas M.; Parisse, Daniele (2002), "On the planarity of Hanoi graphs", Expositiones Mathematicae, 20 (3): 263–268, doi:10.1016/S0723-0869(02)80023-8, MR 1924112
  6. ^ Scorer, R. S.; Grundy, P. M.; Smith, C. A. B. (July 1944), "Some binary games", The Mathematical Gazette, 28 (280): 96, doi:10.2307/3606393, JSTOR 3606393
  7. ^ "Hanoi Graph".