Tower of Hanoi - Knowledge

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pieces is odd). If there is no tower position in the chosen direction, move the piece to the opposite end, but then continue to move in the correct direction. For example, if you started with three pieces, you would move the smallest piece to the opposite end, then continue in the left direction after that. When the turn is to move the non-smallest piece, there is only one legal move. Doing this will complete the puzzle in the fewest moves.

1480: 1215: 972: 547: 319: 239: 55: 1707: 31: 3490: 2745: 3338:, a human is held prisoner on a planet where the local custom is to make the prisoner play a game until it is won or lost before his execution. The protagonist knows that a rescue ship might take a year or more to arrive, so he chooses to play Towers of Hanoi with 64 disks. This story makes reference to the legend about the Buddhist monks playing the game until the end of the world. 3217: 3197: 2731:

sequence of moves to solve this problem. A solution was proposed by Andreas Hinz and is based on the observation that in a shortest sequence of moves, the largest disk that needs to be moved (obviously one may ignore all of the largest disks that will occupy the same peg in both the initial and final configurations) will move either exactly once or exactly twice.

1455:) can be divided by 2 (i.e. the number of zero bits at the right), counting the first move as 1 and identifying the disks by the numbers 0, 1, 2, etc. in order of increasing size. This permits a very fast non-recursive computer implementation to find the positions of the disks after m moves without reference to any previous move or distribution of disks. 1751:-disk diagram—each one representing all the states and moves of the smaller disks for one particular position of the new largest disk—and joining them at the corners with three new edges, representing the only three opportunities to move the largest disk. The resulting figure thus has 3 nodes and still has three corners remaining with only two edges. 2992:{\displaystyle {\frac {466}{885}}\cdot 2^{n}-{\frac {1}{3}}-{\frac {3}{5}}\cdot \left({\frac {1}{3}}\right)^{n}+\left({\frac {12}{59}}+{\frac {18}{1003}}{\sqrt {17}}\right)\left({\frac {5+{\sqrt {17}}}{18}}\right)^{n}+\left({\frac {12}{59}}-{\frac {18}{1003}}{\sqrt {17}}\right)\left({\frac {5-{\sqrt {17}}}{18}}\right)^{n}.} 1762:. It is clear that the great majority of positions in the puzzle will never be reached when using the shortest possible solution; indeed, if the priests of the legend are using the longest possible solution (without re-visiting any position), it will take them 3 − 1 moves, or more than 10 years. 1726:

The sides of the outermost triangle represent the shortest ways of moving a tower from one peg to another one. The edge in the middle of the sides of the largest triangle represents a move of the largest disk. The edge in the middle of the sides of each next smaller triangle represents a move of each

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In Cyclic Hanoi, we are given three pegs (A, B, C), which are arranged as a circle with the clockwise and the counterclockwise directions being defined as A – B – C – A and A – C – B – A, respectively. The moving direction of the disk must be clockwise. It suffices to represent the sequence of disks

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This technique identifies which disk to move, but not where to move it to. For the smallest disk, there are always two possibilities. For the other disks there is always one possibility, except when all disks are on the same peg, but in that case either it is the smallest disk that must be moved or

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in puzzles 6, 83, and 84, but the disks had been changed to pancakes. The puzzle was based around a dilemma where the chef of a restaurant had to move a pile of pancakes from one plate to the other with the basic principles of the original puzzle (i.e. three plates that the pancakes could be moved

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A curious generalization of the original goal of the puzzle is to start from a given configuration of the disks where all disks are not necessarily on the same peg and to arrive in a minimal number of moves at another given configuration. In general, it can be quite difficult to compute a shortest

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A simple solution for the toy puzzle is to alternate moves between the smallest piece and a non-smallest piece. When moving the smallest piece, always move it to the next position in the same direction (to the right if the starting number of pieces is even, to the left if the starting number of

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The rules of the puzzle are essentially the same: disks are transferred between pegs one at a time. At no time may a bigger disk be placed on top of a smaller one. The difference is that now for every size there are two disks: one black and one white. Also, there are now two towers of disks of

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If one counts in Gray code of a bit size equal to the number of disks in a particular Tower of Hanoi, begins at zero and counts up, then the bit changed each move corresponds to the disk to move, where the least-significant bit is the smallest disk, and the most-significant bit is the largest.

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in task design. They demonstrated an impact on user performance by changing the way that the rules of the game are represented, using variations in the physical design of the game components. This knowledge has impacted on the development of the TURF framework for the representation of

162:", consisting of sixty-four golden disks, according to the same rules as in the game, and that the completion of the tower would lead to the end of the world. Numerous variations on this legend regarding the ancient and mystical nature of the puzzle popped up almost immediately. 2738:

number of moves in a shortest sequence of moves between two initial and final disk configurations that are chosen at random. Hinz and Chan Tat-Hung independently discovered (see also ) that the average number of moves in an n-disk Tower is given by the following exact formula:

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In Magnetic Tower of Hanoi, each disk has two distinct sides North and South (typically colored "red" and "blue"). Disks must not be placed with the similar poles together—magnets in each disk prevent this illegal move. Also, each disk must be flipped as it is moved.

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Thailand in 2002 but rather than rings, the pieces were made to resemble a temple. Sook Jai threw the challenge to get rid of Jed even though Shii-Ann knew full well how to complete the puzzle. The problem is featured as part of a reward challenge in a

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nodes in the graph; every node has three edges to other nodes, except the three corner nodes, which have two: it is always possible to move the smallest disk to one of the two other pegs, and it is possible to move one disk between those two pegs

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disks from peg A to peg C takes 3 − 1 moves. The solution uses all 3 valid positions, always taking the unique move that does not undo the previous move. The position with all disks at peg B is reached halfway, i.e. after (3 − 1) / 2 moves.

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which involves moving all the disks from one peg to another. An alternative explanation for the appearance of the constant 466/885, as well as a new and somewhat improved algorithm for computing the shortest path, was given by Romik.

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as representing the ratio of the labor one has to perform when going from a randomly chosen configuration to another randomly chosen configuration, relative to the difficulty of having to cross the "most difficult" path of length

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The list of moves for a tower being carried from one peg onto another one, as produced by the recursive algorithm, has many regularities. When counting the moves starting from 1, the ordinal of the disk to be moved during move

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Although the three-peg version has a simple recursive solution long been known, the optimal solution for the Tower of Hanoi problem with four pegs (called Reve's puzzle) was not verified until 2014, by Bousch.

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represents the largest disk. A value of 0 indicates that the largest disk is on the initial peg, while a 1 indicates that it is on the final peg (right peg if number of disks is odd and middle peg otherwise).

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A bit with a different value to the previous one means that the corresponding disk is one position to the left or right of the previous one. Whether it is left or right is determined by this rule:

142:, first presented in 1883 as a game discovered by "N. Claus (de Siam)" (an anagram of "Lucas d'Amiens"), and later published as a booklet in 1889 and in a posthumously-published volume of Lucas' 2711:. For example, in the UPenn CIS 194 course on Haskell, the first assignment page lists the optimal solution for the 15-disk and 4-peg case as 129 steps, which is obtained for the above value of 649:, then, by renaming the pegs, the same solution can be used for every other choice of starting and destination peg. If there is only one disk (or even none at all), the problem is trivial. If 2705: 165:

If the legend were true, and if the priests were able to move disks at a rate of one per second, using the smallest number of moves, it would take them 2 − 1 seconds or roughly 585

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For the formal derivation of the exact number of minimal moves required to solve the problem by applying the Frame–Stewart algorithm (and other equivalent methods), see the following paper.

192:. In some versions, other elements are introduced, such as the fact that the tower was created at the beginning of the world, or that the priests or monks may make only one move per day. 3461:, this puzzle is shown in Faruzan's hangout quest, "Early Learning Mechanism", where she mentions seeing it as a mechanism and uses it to make a toy prototype for children. She calls it 3078: 4570: 4257: 833:, it is easily proven that the above procedure requires the minimum number of moves possible and that the produced solution is the only one with this minimal number of moves. Using 200:

The puzzle can be played with any number of disks, although many toy versions have around 7 to 9 of them. The minimal number of moves required to solve a Tower of Hanoi puzzle with

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The operation, which counts the number of consecutive zeros at the end of a binary number, gives a simple solution to the problem: the disks are numbered from zero, and at move

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way from those sub-problems' solutions. Each of these created sub-problems being "smaller" guarantees that the base case(s) will eventually be reached. For the Towers of Hanoi:

956: 3104: 2274: 2451: 1271:(base-2) representation of the move number (the initial state being move #0, with all digits 0, and the final state being with all digits 1), using the following rules: 3225:

alternating colors. The goal of the puzzle is to make the towers monochrome (same color). The biggest disks at the bottom of the towers are assumed to swap positions.

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Examine the smallest top disk that is not disk 0, and note what its only (legal) move would be: if there is no such disk, then we are either at the first or last move.

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moves to complete. If an especially dedicated player does click through to the end of the puzzle, it is revealed that completing the puzzle does not unlock the door.

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The position of the bit change in the Gray code solution gives the size of the disk moved at each step: 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, ... (sequence

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gives an alternative way of solving the puzzle. In the Gray system, numbers are expressed in a binary combination of 0s and 1s, but rather than being a standard

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in the situation where all disks are stacked on one peg. The corner nodes represent the three cases where all the disks are stacked on one peg. The diagram for

3385: 3292: 101:, which can slide onto any rod. The puzzle begins with the disks stacked on one rod in order of decreasing size, the smallest at the top, thus approximating a 4383: 1187:

With this knowledge, a set of disks in the middle of an optimal solution can be recovered with no more state information than the positions of each disk:

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Yin, Xi; Liu, Xinhong; Pan, Yung-Tin; Walsh, Kathleen A.; Yang, Hong (November 4, 2014). "Hanoi Tower-like Multilayered Ultrathin Palladium Nanosheets".

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Counting moves from 1 and identifying the disks by numbers starting from 0 in order of increasing size, the ordinal of the disk to be moved during move

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From every arbitrary distribution of disks, there are one or two different longest non-self-crossing paths to move all disks to one of the three pegs.

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as a lock to a tomb. The player has the option to click through each move of the puzzle in order to solve it, but the game notes that it will take

1440:, where the disks begin on peg 0 and finish on peg 1 or 2 according as whether the number of disks is even or odd. Another formulation is from peg 1313:

be the number of greater disks that are located on the same peg as their first greater disk and add 1 if the largest disk is on the left peg. If

1081:, etc. for even height of the tower. This provides the following algorithm, which is easier, carried out by hand, than the recursive algorithm. 4519: 1906:

If all moves must be between adjacent pegs (i.e. given pegs A, B, C, one cannot move directly between pegs A and C), then moving a stack of

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A bit with the same value as the previous one means that the corresponding disk is stacked on top of the previous disk on the same peg.

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can be divided by 2. Hence every odd move involves the smallest disk. It can also be observed that the smallest disk traverses the pegs

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If that move is the disk's "natural" move, then the disk has not been moved since the last disk 0 move, and that move should be taken.

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As in many mathematical puzzles, finding a solution is made easier by solving a slightly more general problem: how to move a tower of

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Reid, C.R.; Sumpter, D.J.; Beekman, M. (January 2011). "Optimisation in a natural system: Argentine ants solve the Towers of Hanoi".

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were successfully able to solve the 3-disk version of the Tower of Hanoi problem through non-linear dynamics and pheromone signals.

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With three disks, the puzzle can be solved in seven moves. The minimal number of moves required to solve a Tower of Hanoi puzzle is

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games. Since it is easy to implement, and easily recognised, it is well suited to use as a puzzle in a larger graphical game (e.g.

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All other disks are 1 as well, so they are stacked on top of it. Hence all disks are on the final peg and the puzzle is complete.

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The iterative solution is equivalent to repeated execution of the following sequence of steps until the goal has been achieved:

5384: 4527: 5199: 5159: 4837: 4810: 4706: 4267: 4240: 4196: 4171: 3925: 3411:). Some implementations use straight disks, but others disguise the puzzle in some other form. There is an arcade version by 2084:

The move-patterns of transferring a tower of disks from a peg to another peg are symmetric with respect to the center points.

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Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack or on an empty rod.

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From every arbitrary distribution of disks, there is exactly one shortest way to move all disks onto one of the three pegs.

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the objective has already been achieved. Luckily, there is a rule that does say where to move the smallest disk to. Let

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The bitstring is read from left to right, and each bit can be used to determine the location of the corresponding disk.

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Between every pair of arbitrary distributions of disks there are one or two different longest non-self-crossing paths.

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shape. The objective of the puzzle is to move the entire stack to one of the other rods, obeying the following rules:

5038:"Neuropsychological study of frontal lobe function in psychotropic-naive children with obsessive-compulsive disorder" 4339: 3831: 1519: 1254: 1011: 586: 358: 278: 5297: 2106:

However, in case of four or more pegs, the Frame–Stewart algorithm is known without proof of optimality since 1941.

741:− 3, and so on until only one disk is left. This is called recursion. This algorithm can be schematized as follows. 5364: 4331: 4585: 3942: 3253:

3D AFM topographic image of multilayered palladium nanosheet on silicon wafer, with Tower of Hanoi-like structure.

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Let C(n) and A(n) represent moving n disks clockwise and counterclockwise, then we can write down both formulas:

1497: 1232: 989: 564: 336: 256: 3380: 3278: 1501: 1236: 993: 729:, but the same method can be used both times by renaming the pegs. The same strategy can be used to reduce the 568: 340: 305:

Following this approach, the stack will end up on peg B if the number of disks is odd and peg C if it is even.

260: 1540:, the Gray code operates on the premise that each value differs from its predecessor by only one bit changed. 5379: 5036:

Beers, S. R.; Rosenberg, D. R.; Dick, E. L.; Williams, T.; O'Hearn, K. M.; Birmaher, B.; Ryan, C. M. (1999).

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the remaining third peg. If the number of disks is odd, the smallest disk cycles along the pegs in the order

1295:(That is to say: a straight sequence of 1s or 0s means that the corresponding disks are all on the same peg.) 1690:

The nodes at the vertices of the outermost triangle represent distributions with all disks on the same peg.

3861: 3365:-move Tower of Hanoi game entitled The Trilogic Game with the pieces forming a pyramid shape when stacked. 3109: 1321:

is odd, the disk located one peg to the left (in case of an even number of disks and vice versa otherwise).

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Zhang and Norman used several isomorphic (equivalent) representations of the game to study the impact of

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The mathematics related to this generalized problem becomes even more interesting when one considers the

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For h + 1 disks, take the graph of h disks and replace each small triangle with the graph for two disks.

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All other disks are 0 as well, so they are stacked on top of it. Hence all disks are on the initial peg.

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is to recognize that it can be broken down into a collection of smaller sub-problems, to each of which

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Gedeon, T. D. (1996). "The Cyclic Towers of Hanoi: An Iterative Solution Produced by Transformation".

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Jeux scientifiques pour servir à l'histoire, à l'enseignement et à la pratique du calcul et du dessin

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This algorithm is presumed to be optimal for any number of pegs; its number of moves is 2 (for fixed

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Between every pair of arbitrary distributions of disks there are one or two different shortest paths.

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Identify the disks in order of increasing size by the natural numbers from 0 up to but not including

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is moved the minimal possible distance to the right (circling back around to the left as needed).

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Troshkin, M. "Doomsday Comes: A Nonrecursive Analysis of the Recursive Towers-of-Hanoi Problem".

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Also assume "wrapping"—so the right peg counts as one peg "left" of the left peg, and vice versa.

1225: 982: 557: 329: 249: 4923: 2409: 5092: 3850: 3583: 3452:) struggled to understand how to solve the puzzle and are aided by their fellow tribe members. 3349: 3285: 2116: 2112:

For other variants of the four-peg Tower of Hanoi problem, see Paul Stockmeyer's survey paper.

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The graph for two disks is three triangles connected to form the corners of a larger triangle.

1425: 830: 530: 146:. Accompanying the game was an instruction booklet, describing the game's purported origins in 4067: 3550: 1765:

The longest non-repetitive way for three disks can be visualized by erasing the unused edges:

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A second letter is added to represent the larger disk. Clearly, it cannot initially be moved.

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Bonanome, Marianna C.; Dean, Margaret H.; Dean, Judith Putnam (2018). "Self-Similar Groups".

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to beginning programming students. A pictorial version of this puzzle is programmed into the

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next smaller disk. The sides of the smallest triangles represent moves of the smallest disk.

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A Sampling of Remarkable Groups: Thompson's, Self-similar, Lamplighter, and Baumslag-Solitar

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In 2010, researchers published the results of an experiment that found that the ant species

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Incidentally, this longest non-repetitive path can be obtained by forbidding all moves from

665:> 1, then somewhere along the sequence of moves, the largest disk must be moved from peg 4889: 3396: 3262: 1463: 1283: 631:. First, observe that the problem is symmetric for permutations of the names of the pegs ( 176:

There are many variations on this legend. For instance, in some tellings, the temple is a

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This variation of the famous Tower of Hanoi puzzle was offered to grade 3–6 students at

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Hinz, Andreas M.; Klavzar, Sandi; Milutinovic, Uros; Petr, Ciril; Stewart, Ian (2013).

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Furthermore, the disk to be moved is determined by the number of times the move count (

709:− 1 smaller disks and can be temporarily ignored. Now the problem is reduced to moving 3870:", 1953 Arthur C. Clark short story with a similar premise to the game's framing story 5327: 5195: 5155: 5110: 5057: 5018: 4981: 4943: 4905: 4833: 4806: 4781: 4702: 4554: 4335: 4263: 4236: 4192: 4167: 4022: 3962: 3921: 3679: 3564: 3545: 3540: 2115:

The so-called Towers of Bucharest and Towers of Klagenfurt game configurations yield

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The topmost small triangle now represents the one-move possibilities with two disks:

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Illustration of a recursive solution for the Towers of Hanoi puzzle with 4 disks. In

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Romik, D. (2006). "Shortest paths in the Tower of Hanoi graph and finite automata".

4410: 4095: 3384:, this puzzle, called in the film the "Lucas Tower", is used as a test to study the 2606:

should be picked for which this quantity is minimum. In the 4-peg case, the optimal

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Disks seven and eight are also 0, so they are stacked on top of it, on the left peg.

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The Tower of Hanoi is also used as a test by neuropsychologists trying to evaluate

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editor, accessed by typing M-x hanoi. There is also a sample algorithm written in

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Disks whose ordinals have even parity move in the same sense as the smallest disk.

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Philosophical Transactions of the Royal Society of London. B, Biological Sciences

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Stewart, B. M.; Frame, J. S. (March 1941). "Solution to advanced problem 3819".

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to be moved. The solution can be found using two mutually recursive procedures:

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Miller, Charles D. (2000). "Ch. 4: Binary Numbers and the Standard Gray Code".

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Hinz, Andreas M.; Klavžar, Sandi; Milutinović, Uroš; Petr, Ciril (2013-01-31).

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when performing computer data backups where multiple tapes/media are involved.

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disks (in steps 1 and 3), that is, do nothing—which does not violate the rules.

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disks are distributed in valid arrangements among the pegs; assuming there are

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As more disks are added, the graph representation of the game will resemble a

837:, the exact number of moves that this solution requires can be calculated by: 5348: 5186:. Compact Textbooks in Mathematics. Cham, Switzerland: Springer. p. 96. 4947: 4785: 4558: 3966: 3632: 3611: 3508: 2090:

Groups of the smallest disk moves alternate with single moves of other disks.

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to be the minimum number of moves required to transfer n disks using r pegs.

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disks from the source peg A to the target peg C, using B as the spare peg.

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Chan, T. (1988). "A statistical analysis of the towers of Hanoi problem".

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Move one disk from peg B to peg C or vice versa, whichever move is legal.

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Move one disk from peg A to peg C or vice versa, whichever move is legal.

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Move one disk from peg A to peg B or vice versa, whichever move is legal.

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This article is about the mathematical disk game. For the card game, see

4656: 3006:, only the first and second terms do not converge to zero, so we get an 533:

and is often used as an example of recursion when teaching programming.

5106: 4493: 4452: 4127:(in French). Vol. 3. Librairie Albert Blanchard, 1979. p. 58. 3757: 3697: 3642: 3445: 3375:

onto, not being able to put a larger pancake onto a smaller one, etc.)

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disks to a single peg other than the start or destination pegs, taking

1504: in this section. Unsourced material may be challenged and removed. 1239: in this section. Unsourced material may be challenged and removed. 996: in this section. Unsourced material may be challenged and removed. 571: in this section. Unsourced material may be challenged and removed. 343: in this section. Unsourced material may be challenged and removed. 263: in this section. Unsourced material may be challenged and removed. 4901: 4747: 3916:

Metamagical Themas : Questing for the Essence of Mind and Pattern

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th move can also be found elegantly from the binary representation of

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Disk five is also 1, so it is stacked on top of it, on the right peg.

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Disk two is also 1, so it is stacked on top of it, on the middle peg.

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Major Ingredients: The Selected Short Stories of Eric Frank Russell

4541: 3943:"A device for demonstrating some elementary properties of integers" 3257:

The Tower of Hanoi is frequently used in psychological research on

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The solution for the Cyclic Hanoi has some interesting properties:

705:. The presence of the largest disk does not impede any move of the 189: 151: 98: 38: 1722:

list disk positions from left to right in order of increasing size

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peg, which is guaranteed to be a valid move, by the assumptions —

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The number of disks moves specified by C(n) and A(n) are minimal.

1755: 1638:, or one more than the power of 2 within the move number. In the 155: 102: 5150:

Russell, Eric Frank (2000). "Now Inhale". In Katze, Rick (ed.).

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Warren, Henry S. (2003). "Section 5-4: Counting Trailing 0's.".

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In 2007, the concept of the Towers Of Hanoi problem was used in

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If that move is not the disk's "natural" move, then move disk 0.

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Move the smallest disk to the peg it has not recently come from.

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As mentioned above, the Tower of Hanoi is popular for teaching

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for neuropsychological diagnosis and treatment of disorders of

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be the number of non-self-crossing paths for moving a tower of

1747: + 1 disks is obtained by taking three copies of the 1091:

Move another disk legally (there will be only one possibility).

673:. The only situation that allows this move is when all smaller 169:

years to finish, which is about 42 times the estimated current

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No disk may be placed on top of a disk that is smaller than it.

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Birtwistle, Graham (January 1985). "The coroutines of Hanoi".

1597:, etc. If the number of disks is even, this must be reversed: 529:

This approach can be given a rigorous mathematical proof with

465:. Rules are not violated, by assumption. This leaves the disk 184:. The temple or monastery may be in various locales including 4303:"Question Corner -- Generalizing the Towers of Hanoi Problem" 3296: 3211:

2ème Championnat de France des Jeux Mathématiques et Logiques

1403:= 2), the disk is one peg to the right, i.e. on the left peg. 536: 185: 2672:{\displaystyle n-\left\lfloor {\sqrt {2n+1}}\right\rceil +1} 1128:

is even, the remaining third peg during successive moves is

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Disks whose ordinals have odd parity move in opposite sense.

870:. This result is obtained by noting that steps 1 and 3 take 158:

have been carrying out the movement of the "Sacred Tower of

4520:"Loopless Gray Code Enumeration and the Tower of Bucharest" 3888: 3638: 3412: 1885: 1626: 1159:

is odd, the remaining third peg during successive moves is

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1000 playthinks: puzzles, paradoxes, illusions & games

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The largest disk is 1, so it is on the middle (final) peg.

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The largest disk is 1, so it is on the middle (final) peg.

1334:

The largest disk is 0, so it is on the left (initial) peg.

1057:, etc. for odd height of the tower and traverses the pegs 228: 220:

Animation of an iterative algorithm-solving 6-disk problem

97:

consisting of three rods and a number of disks of various

4771:"A Recursive Solution to Bicolor Towers of Hanoi Problem" 3216: 3196: 1276: 495:− 1 disk that we have just placed on the spare, from the 1267:

Disk positions may be determined more directly from the

5325: 4854:"Tower Of Hanoy Patience (AKA Tower Of Hanoi Patience)" 3106:. Thus intuitively, we could interpret the fraction of 2380:

disks to the destination peg, using only the remaining

1710:

The game graph of level 7 shows the relatedness to the

1389:= 1), it is one peg to the left, i.e. on the right peg. 1095:

For the very first move, the smallest disk goes to peg

389:

that same general solving procedure that we are seeking

5177: 5175: 5173: 4768: 4466:

Klavzar, Sandi; Milutinovi, Uro; Petrb, Ciril (2002).

4465: 3953:(2). National Council of Teachers of Mathematics: 84. 3200:

Initial configuration of bicolor Towers of Hanoi (n=4)

1378:= 1), it is one peg to the left, i.e. on the left peg. 1317:

is even, the disk is located one peg to the right, if

1191:

Call the moves detailed above a disk's "natural" move.

391:

applies, and the total solution is then found in some

3150: 3112: 3086: 3016: 2748: 2685: 2632: 2612: 2592: 2521: 2482: 2462: 2412: 2386: 2360: 2340: 2334:

Without disturbing the peg that now contains the top

2302: 2282: 2250: 2230: 2187: 2164: 2141: 2087:

The smallest disk is the first and last disk to move.

909: 876: 843: 5154:. Framingham, Mass.: NESFA Press. pp. 399–417. 4620:"UPenn CIS 194 Introduction to Haskell Assignment 1" 3418:

A 15-disk version of the puzzle appears in the game

3220:

Final configuration of bicolor Towers of Hanoi (n=4)

58:

Tower of Hanoi interactive display at Mexico City's

5170: 4997:"TURF: Toward a unified framework of EHR usability" 4494:"Variations on the Four-Post Tower of Hanoi Puzzle" 4468:"Variations on the Four-Post Tower of Hanoi Puzzle" 755:The following is a procedure for moving a tower of 429:peg, and all the rest of the disks are larger than 5082: 3913: 3261:. There also exists a variant of this task called 3169: 3135: 3098: 3072: 2991: 2699: 2671: 2618: 2598: 2578: 2503: 2468: 2445: 2398: 2372: 2346: 2323: 2288: 2268: 2236: 2208: 2170: 2147: 950: 895: 862: 813:> 1, then again use this procedure to move the 779:> 1, then first use this procedure to move the 713:− 1 disks from one peg to another one, first from 693:. Then move the largest disk and finally move the 3326:nanosheets with a Tower of Hanoi-like structure. 5346: 5181: 4963:"Representations in distributed cognitive tasks" 4590: 4511: 4140:"Tower of Hanoi instructions in English, page 1" 3233:A variation of the puzzle has been adapted as a 2700:{\displaystyle \left\lfloor \cdot \right\rceil } 2131:The Frame–Stewart algorithm is described below: 1370:Disk three is 0, so it is on another peg. Since 34:A model set of the Tower of Hanoi (with 8 disks) 1381:Disk four is 1, so it is on another peg. Since 409:number the disks from 1 (smallest, topmost) to 5251:"The Fourth Dimension: The Celestial Toymaker" 4879: 4131: 3334:In the science fiction story "Now Inhale", by 1395:Disk six is 0, so it is on another peg. Since 3825: 3322:In 2014, scientists synthesized multilayered 2726:General shortest paths and the number 466/885 1883:to be 2, 12, 1872, 6563711232, ... (sequence 748:. Hence disk 0 is the smallest one, and disk 4430: 3438:2011 episode of the American version of the 2220:The algorithm can be described recursively: 2097: 522:The full Tower of Hanoi solution then moves 379:click a grey button to expand or collapse it 5144:. Vol. 63, no. 2. pp. 31–77. 4518:Herter, Felix; Rote, Günter (2018-11-14) . 4235:(1st ed.). Boston MA: Addison-Wesley. 4065: 4035: 3073:{\displaystyle 466/885\cdot 2^{n}-1/3+o(1)} 1649: 1303:Assume that the initial peg is on the left. 623:being the remaining third peg and assuming 5214: 4995:Zhang, Jiajie; Walji, Muhammad F. (2011). 4994: 4517: 4491: 3911: 3832: 3818: 3204: 2126: 641:). If a solution is known moving from peg 537:Logical analysis of the recursive solution 5096: 5012: 4737: 4540: 4158: 4059: 1829:disks from one peg to another one. Then: 1520:Learn how and when to remove this message 1255:Learn how and when to remove this message 1012:Learn how and when to remove this message 961: 903:moves, and step 2 takes one move, giving 587:Learn how and when to remove this message 433:, so they can be safely ignored; to move 359:Learn how and when to remove this message 279:Learn how and when to remove this message 4186: 4010: 4008: 3248: 3215: 3195: 1705: 1412:The source and destination pegs for the 507:, so they are placed on top of the disk 370: 215: 150:, and claiming that according to legend 53: 37: 29: 5135: 4875: 4873: 4696: 4643:Hinz, A. (1989). "The Tower of Hanoi". 3371:Professor Layton and the Diabolical Box 3237:with nine playing cards under the name 229:Simpler statement of iterative solution 5347: 4825: 4798: 4699:Another Fine Math You've Got Me Into.. 4381: 4354: 4262:(9 ed.). Addison Wesley Longman. 4255: 4230: 4189:Famous Puzzles of Great Mathematicians 4137: 4041: 3431:This was first used as a challenge in 3402:Star Wars: Knights of the Old Republic 135:The puzzle was invented by the French 5326: 5277:"Tower of Hanoi (video game concept)" 4960: 4721: 4117: 4090: 4005: 3986: 3329: 3284:The Tower of Hanoi is also used as a 3136:{\displaystyle 466/885\approx 52.6\%} 2476:disks to the destination peg, taking 1469: 308: 211: 109:Only one disk may be moved at a time. 23:. For the Vietnamese skyscraper, see 4870: 4725:SIAM Journal on Discrete Mathematics 4669: 4642: 4328:The Tower of Hanoi – Myths and Maths 4286: 4211: 4044:"The Tower of Hanoi: A Bibliography" 4018:The Tower of Hanoi – Myths and Maths 3940: 3391:The puzzle is featured regularly in 1502:adding citations to reliable sources 1473: 1237:adding citations to reliable sources 1208: 994:adding citations to reliable sources 965: 569:adding citations to reliable sources 540: 341:adding citations to reliable sources 312: 261:adding citations to reliable sources 232: 4598:"University of Toronto CSC148 Slog" 4391:Bull. Belg. Math. Soc. Simon Stevin 4100:(in French). Paris: Chambon et Baye 3881: 1646:gives moves for the 8-disk puzzle. 603:(height) disks from a starting peg 13: 5136:Russell, Eric Frank (April 1959). 5042:The American Journal of Psychiatry 4924:"Specific impairments of planning" 3130: 3093: 2579:{\displaystyle 2T(k,r)+T(n-k,r-1)} 1792: 1767: 1698: 1682: 1666: 1660:The game can be represented by an 1204: 669:to another peg, preferably to peg 505:the same general solving procedure 463:the same general solving procedure 445:peg, without violating the rules: 14: 5396: 5319: 5300:. Sega Amusements. Archived from 5001:Journal of Biomedical Informatics 4778:Recreational Mathematics Magazine 4066:de Parville, Henri (1883-12-27). 3228: 3181: 1327:For example, in an 8-disk Hanoi: 653:= 1, then move the disk from peg 406:be the total number of disks, and 188:, and may be associated with any 3488: 2042:A(n) = A(n−1) n C(n−1) n A(n−1). 1991:to the neighbouring target peg: 1931:to the neighbouring target peg: 1634:), a sequence also known as the 1478: 1213: 970: 951:{\displaystyle T_{h}=2T_{h-1}+1} 794:Now the largest disk, i.e. disk 545: 469:as a top disk on the source peg. 317: 237: 5290: 5269: 5243: 5208: 5129: 5076: 5029: 4988: 4954: 4916: 4846: 4819: 4805:. Sterling Publishing Company. 4792: 4769:Prasad Vithal Chaugule (2015). 4762: 4715: 4690: 4663: 4636: 4612: 4576:from the original on 2020-12-16 4485: 4459: 4424: 4375: 4348: 4332:Springer Science+Business Media 4319: 4295: 4280: 4249: 4224: 4205: 4180: 4152: 3912:Hofstadter, Douglas R. (1985). 3244: 1914: 1901: 1489:needs additional citations for 1284:most significant (leftmost) bit 1224:needs additional citations for 981:needs additional citations for 771:being the remaining third peg: 685:− 1 smaller disks must go from 556:needs additional citations for 328:needs additional citations for 248:needs additional citations for 4191:. AMS Bookstore. p. 197. 4111: 4084: 3980: 3934: 3905: 3381:Rise of the Planet of the Apes 3090: 3067: 3061: 2573: 2549: 2540: 2528: 2498: 2486: 2440: 2416: 2354:disks, transfer the remaining 2318: 2306: 2203: 2191: 1800:The graphs clearly show that: 1730:In general, for a puzzle with 1696:For three disks the graph is: 611:(from) onto a destination peg 1: 5385:Divide-and-conquer algorithms 4934:(1089): 199–209. 1982-06-25. 4826:Hedges, Sid G. (2018-03-06). 4433:American Mathematical Monthly 3868:The Nine Billion Names of God 1896: 1532:The binary numeral system of 437:disks from a source peg to a 383:The key to solving a problem 71:The problem of Benares Temple 4982:10.1016/0364-0213(94)90021-3 4799:Arnold, Peter (2003-05-28). 4528:Theoretical Computer Science 4384:"La quatrieme tour de Hanoi" 3862:Recursion (computer science) 3099:{\displaystyle n\to \infty } 2586:moves. Therefore, the count 2269:{\displaystyle 1\leq k<n} 511:without violating the rules. 81:and sometimes pluralized as 42:An animated solution of the 7: 5298:"Tower of Hanoi / Andamiro" 5192:10.1007/978-3-030-01978-5_3 4829:Everybody's Book of Hobbies 4645:L'Enseignement Mathématique 3468: 1446:(m + (m & -m)) % 3 1442:(m - (m & -m)) % 3 1424:. To use the syntax of the 1275:There is one binary digit ( 817:− 1 smaller disks from peg 783:− 1 smaller disks from peg 697:− 1 smaller disks from peg 195: 10: 5401: 5142:Astounding Science Fiction 4535:. Berlin, Germany: 40–54. 4187:Petković, Miodrag (2009). 3279:human–computer interaction 3185: 2456:Finally, transfer the top 2446:{\displaystyle T(n-k,r-1)} 1653: 1438:((m | m - 1) + 1) % 3 130: 18: 5014:10.1016/j.jbi.2011.08.005 4684:10.1080/00207168908803728 4672:Internat. J. Comput. Math 4551:10.1016/j.tcs.2017.11.017 4492:Stockmeyer, Paul (1994). 4291:. Lyon: Aimé Vingtrinier. 4124:Récréations mathématiques 3920:. New York: Basic Books. 2515:The entire process takes 2098:With four pegs and beyond 1719:call the pegs a, b, and c 1569:the destination peg, and 1538:positional numeral system 514:The base case is to move 144:Récréations mathématiques 4403:10.36045/bbms/1420071861 3874: 2709:nearest integer function 1650:Graphical representation 1434:(m & m - 1) % 3 127:is the number of disks. 16:Mathematical puzzle game 5365:19th-century inventions 4369:10.1093/comjnl/39.4.353 4330:(1st ed.). Basel: 3947:The Mathematics Teacher 3941:Cohn, Ernst M. (1963). 3274:representational effect 3205:Bicolor Towers of Hanoi 3188:Magnetic Tower of Hanoi 3170:{\displaystyle 2^{n}-1} 2155:be the number of disks. 2127:Frame–Stewart algorithm 1644:IntegerExponent, 2] + 1 1552:is the number of times 1029:is the number of times 896:{\displaystyle T_{h-1}} 863:{\displaystyle 2^{h}-1} 413:(largest, bottom-most). 399:label the pegs A, B, C, 4940:10.1098/rstb.1982.0082 4501:Congressus Numerantium 4475:Congressus Numerantium 4289:Théorie du Baguenodier 3851:Backup rotation scheme 3350:The Celestial Toymaker 3286:backup rotation scheme 3254: 3221: 3201: 3171: 3137: 3100: 3074: 2993: 2701: 2673: 2620: 2600: 2580: 2505: 2504:{\displaystyle T(k,r)} 2470: 2447: 2400: 2374: 2348: 2325: 2324:{\displaystyle T(k,r)} 2290: 2270: 2238: 2210: 2209:{\displaystyle T(n,r)} 2178:be the number of pegs. 2172: 2149: 2074:A(2) = 1 1 2 1 2 1 1. 2034:C(n) = A(n−1) n A(n−1) 1797: 1772: 1715: 1703: 1687: 1671: 1426:C programming language 962:Non-recursive solution 952: 897: 864: 831:mathematical induction 798:can be moved from peg 721:and subsequently from 531:mathematical induction 380: 221: 180:, and the priests are 62: 51: 35: 5229:10.1145/988284.988286 5054:10.1176/ajp.156.5.777 4697:Stewart, Ian (2004). 3993:mathworld.wolfram.com 3450:Benjamin "Coach" Wade 3361:to play a ten-piece, 3252: 3219: 3199: 3172: 3138: 3101: 3075: 3008:asymptotic expression 2994: 2702: 2674: 2621: 2601: 2581: 2506: 2471: 2448: 2401: 2375: 2349: 2326: 2291: 2271: 2239: 2211: 2173: 2150: 1796: 1771: 1709: 1702: 1686: 1670: 1565:be the starting peg, 953: 898: 865: 752:− 1 the largest one. 677:− 1 disks are on peg 374: 219: 57: 41: 33: 5380:Mathematical puzzles 4357:The Computer Journal 4334:. pp. 241–259. 4138:Stockmeyer, Paul K. 4068:"Revue des Sciences" 4042:Stockmeyer, Paul K. 3959:10.5951/MT.56.2.0084 3386:intelligence of apes 3293:recursive algorithms 3148: 3110: 3084: 3014: 2746: 2683: 2630: 2610: 2590: 2519: 2480: 2460: 2410: 2384: 2358: 2338: 2300: 2280: 2248: 2228: 2185: 2162: 2139: 1790:for three disks is: 1556:can be divided by 2. 1498:improve this article 1464:count trailing zeros 1233:improve this article 1084:In alternate moves: 990:improve this article 907: 874: 841: 835:recurrence relations 565:improve this article 337:improve this article 257:improve this article 5217:ACM SIGPLAN Notices 4894:2014NanoL..14.7188Y 4657:10.5169/seals-57378 4587:(15/18/19/24 pages) 4382:Bousch, T. (2014). 3987:Weisstein, Eric W. 3853:, a TOH application 3794:Nikoli puzzle types 3476:Part of a series on 3213:held in July 1988. 2399:{\displaystyle r-1} 2373:{\displaystyle n-k} 2276:, transfer the top 1760:Sierpiński triangle 1712:Sierpiński triangle 1331:Move 0 = 00000000. 1114:Also observe that: 681:. Hence, first all 453:− 1 disks from the 171:age of the universe 5375:Mechanical puzzles 5355:1883 introductions 5328:Weisstein, Eric W. 5107:10.1242/jeb.048173 4832:. Read Books Ltd. 4802:Card Games for One 4259:Mathematical Ideas 4072:Journal des débats 3799:Puzzle video games 3784:Impossible puzzles 3680:Puzzle video games 3336:Eric Frank Russell 3330:In popular culture 3316:Linepithema humile 3267:executive function 3255: 3222: 3202: 3167: 3133: 3096: 3070: 2989: 2697: 2669: 2616: 2596: 2576: 2501: 2466: 2443: 2396: 2370: 2344: 2321: 2286: 2266: 2234: 2206: 2168: 2145: 2010:one step clockwise 1968:one step clockwise 1950:one step clockwise 1798: 1773: 1716: 1704: 1688: 1672: 1470:Gray-code solution 1422:bitwise operations 1103:is odd and to peg 948: 893: 860: 381: 309:Recursive solution 222: 212:Iterative solution 63: 52: 36: 5370:French inventions 5201:978-3-030-01976-1 5161:978-1-886778-10-8 4970:Cognitive Science 4961:Zhang, J (1994). 4902:10.1021/nl503879a 4839:978-1-5287-8344-6 4812:978-0-600-60727-4 4748:10.1137/050628660 4708:978-0-7167-2342-4 4701:. Courier Dover. 4287:Gros, L. (1872). 4269:978-0-321-07607-6 4242:978-0-201-91465-8 4198:978-0-8218-4814-2 4173:978-0-7611-1826-8 3927:978-0-465-04540-2 3842: 3841: 3703: 3702: 3378:In the 2011 film 3002:For large enough 2974: 2968: 2942: 2935: 2922: 2894: 2888: 2862: 2855: 2842: 2814: 2796: 2783: 2757: 2657: 2619:{\displaystyle k} 2599:{\displaystyle k} 2469:{\displaystyle k} 2347:{\displaystyle k} 2289:{\displaystyle k} 2237:{\displaystyle k} 2171:{\displaystyle r} 2148:{\displaystyle n} 2078: 2077: 2021:to the target peg 1979:to the target peg 1943:to the target peg 1788:Hamiltonian cycle 1734:disks, there are 1530: 1529: 1522: 1265: 1264: 1257: 1022: 1021: 1014: 759:disks from a peg 597: 596: 589: 369: 368: 361: 289: 288: 281: 91:mathematical game 5392: 5341: 5340: 5331:"Tower of Hanoi" 5313: 5312: 5310: 5309: 5294: 5288: 5287: 5285: 5284: 5273: 5267: 5266: 5264: 5262: 5247: 5241: 5240: 5212: 5206: 5205: 5179: 5168: 5165: 5145: 5133: 5127: 5126: 5100: 5080: 5074: 5073: 5033: 5027: 5026: 5016: 4992: 4986: 4985: 4967: 4958: 4952: 4951: 4920: 4914: 4913: 4877: 4868: 4867: 4865: 4864: 4850: 4844: 4843: 4823: 4817: 4816: 4796: 4790: 4789: 4775: 4766: 4760: 4759: 4741: 4719: 4713: 4712: 4694: 4688: 4687: 4667: 4661: 4660: 4640: 4634: 4633: 4631: 4629: 4624: 4616: 4610: 4609: 4607: 4605: 4594: 4588: 4584: 4582: 4581: 4575: 4544: 4524: 4515: 4509: 4508: 4498: 4489: 4483: 4482: 4472: 4463: 4457: 4456: 4428: 4422: 4421: 4419: 4413:. Archived from 4388: 4379: 4373: 4372: 4352: 4346: 4345: 4323: 4317: 4316: 4314: 4313: 4307:math.toronto.edu 4299: 4293: 4292: 4284: 4278: 4277: 4272:. Archived from 4253: 4247: 4246: 4233:Hacker's delight 4228: 4222: 4221: 4209: 4203: 4202: 4184: 4178: 4177: 4156: 4150: 4149: 4147: 4146: 4135: 4129: 4128: 4115: 4109: 4108: 4106: 4105: 4088: 4082: 4081: 4079: 4078: 4063: 4057: 4056: 4054: 4053: 4048: 4039: 4033: 4032: 4012: 4003: 4002: 4000: 3999: 3989:"Tower of Hanoi" 3984: 3978: 3977: 3975: 3973: 3938: 3932: 3931: 3919: 3909: 3903: 3902: 3900: 3899: 3889:"A000225 - OEIS" 3885: 3834: 3827: 3820: 3789:Maze video games 3778: 3743:Packing problems 3738:Optical illusion 3716: 3505: 3504: 3501: 3492: 3473: 3472: 3444:. Both players ( 3176: 3174: 3173: 3168: 3160: 3159: 3142: 3140: 3139: 3134: 3120: 3105: 3103: 3102: 3097: 3079: 3077: 3076: 3071: 3051: 3040: 3039: 3024: 2998: 2996: 2995: 2990: 2985: 2984: 2979: 2975: 2970: 2969: 2964: 2955: 2948: 2944: 2943: 2938: 2936: 2928: 2923: 2915: 2905: 2904: 2899: 2895: 2890: 2889: 2884: 2875: 2868: 2864: 2863: 2858: 2856: 2848: 2843: 2835: 2825: 2824: 2819: 2815: 2807: 2797: 2789: 2784: 2776: 2771: 2770: 2758: 2750: 2706: 2704: 2703: 2698: 2696: 2678: 2676: 2675: 2670: 2662: 2658: 2644: 2625: 2623: 2622: 2617: 2605: 2603: 2602: 2597: 2585: 2583: 2582: 2577: 2510: 2508: 2507: 2502: 2475: 2473: 2472: 2467: 2452: 2450: 2449: 2444: 2405: 2403: 2402: 2397: 2379: 2377: 2376: 2371: 2353: 2351: 2350: 2345: 2330: 2328: 2327: 2322: 2295: 2293: 2292: 2287: 2275: 2273: 2272: 2267: 2243: 2241: 2240: 2235: 2215: 2213: 2212: 2207: 2177: 2175: 2174: 2169: 2154: 2152: 2151: 2146: 2068:C(2) = 1 1 2 1 1 2043: 2035: 2028: 2027: 2019:counterclockwise 2001:counterclockwise 1977:counterclockwise 1961:to the start peg 1941:counterclockwise 1929:counterclockwise 1888: 1662:undirected graph 1645: 1640:Wolfram Language 1629: 1525: 1518: 1514: 1511: 1505: 1482: 1474: 1447: 1443: 1439: 1435: 1345: 1279:) for each disk. 1260: 1253: 1249: 1246: 1240: 1217: 1209: 1017: 1010: 1006: 1003: 997: 974: 966: 957: 955: 954: 949: 941: 940: 919: 918: 902: 900: 899: 894: 892: 891: 869: 867: 866: 861: 853: 852: 633:symmetric group 592: 585: 581: 578: 572: 549: 541: 364: 357: 353: 350: 344: 321: 313: 284: 277: 273: 270: 264: 241: 233: 207: 122: 60:Universum Museum 5400: 5399: 5395: 5394: 5393: 5391: 5390: 5389: 5345: 5344: 5322: 5317: 5316: 5307: 5305: 5296: 5295: 5291: 5282: 5280: 5279:. Giantbomb.com 5275: 5274: 5270: 5260: 5258: 5249: 5248: 5244: 5213: 5209: 5202: 5180: 5171: 5162: 5149: 5134: 5130: 5098:10.1.1.231.9201 5081: 5077: 5034: 5030: 4993: 4989: 4965: 4959: 4955: 4922: 4921: 4917: 4888:(12): 7188–94. 4878: 4871: 4862: 4860: 4852: 4851: 4847: 4840: 4824: 4820: 4813: 4797: 4793: 4773: 4767: 4763: 4720: 4716: 4709: 4695: 4691: 4668: 4664: 4641: 4637: 4627: 4625: 4622: 4618: 4617: 4613: 4603: 4601: 4600:. April 5, 2014 4596: 4595: 4591: 4579: 4577: 4573: 4522: 4516: 4512: 4496: 4490: 4486: 4470: 4464: 4460: 4445:10.2307/2304268 4429: 4425: 4417: 4386: 4380: 4376: 4353: 4349: 4342: 4324: 4320: 4311: 4309: 4301: 4300: 4296: 4285: 4281: 4270: 4254: 4250: 4243: 4229: 4225: 4210: 4206: 4199: 4185: 4181: 4174: 4160:Moscovich, Ivan 4157: 4153: 4144: 4142: 4136: 4132: 4116: 4112: 4103: 4101: 4089: 4085: 4076: 4074: 4064: 4060: 4051: 4049: 4046: 4040: 4036: 4029: 4013: 4006: 3997: 3995: 3985: 3981: 3971: 3969: 3939: 3935: 3928: 3910: 3906: 3897: 3895: 3887: 3886: 3882: 3877: 3846:ABACABA pattern 3838: 3809: 3808: 3779: 3776: 3769: 3768: 3767: 3753:Problem solving 3717: 3712: 3705: 3704: 3637: 3584:Disentanglement 3502: 3499: 3471: 3357:villain forces 3332: 3263:Tower of London 3259:problem-solving 3247: 3231: 3207: 3190: 3184: 3155: 3151: 3149: 3146: 3145: 3116: 3111: 3108: 3107: 3085: 3082: 3081: 3047: 3035: 3031: 3020: 3015: 3012: 3011: 2980: 2963: 2956: 2954: 2950: 2949: 2937: 2927: 2914: 2913: 2909: 2900: 2883: 2876: 2874: 2870: 2869: 2857: 2847: 2834: 2833: 2829: 2820: 2806: 2802: 2801: 2788: 2775: 2766: 2762: 2749: 2747: 2744: 2743: 2728: 2686: 2684: 2681: 2680: 2643: 2639: 2631: 2628: 2627: 2611: 2608: 2607: 2591: 2588: 2587: 2520: 2517: 2516: 2481: 2478: 2477: 2461: 2458: 2457: 2411: 2408: 2407: 2385: 2382: 2381: 2359: 2356: 2355: 2339: 2336: 2335: 2301: 2298: 2297: 2281: 2278: 2277: 2249: 2246: 2245: 2229: 2226: 2225: 2186: 2183: 2182: 2163: 2160: 2159: 2140: 2137: 2136: 2129: 2100: 2041: 2033: 1917: 1904: 1899: 1884: 1882: 1867: 1858: 1849: 1837: 1824: 1658: 1652: 1643: 1625: 1526: 1515: 1509: 1506: 1495: 1483: 1472: 1445: 1441: 1437: 1433: 1360: 1343: 1261: 1250: 1244: 1241: 1230: 1218: 1207: 1205:Binary solution 1018: 1007: 1001: 998: 987: 975: 964: 930: 926: 914: 910: 908: 905: 904: 881: 877: 875: 872: 871: 848: 844: 842: 839: 838: 733:− 1 problem to 639: 593: 582: 576: 573: 562: 550: 539: 425:top disks on a 365: 354: 348: 345: 334: 322: 311: 285: 274: 268: 265: 254: 242: 231: 214: 205: 198: 154:at a temple in 133: 120: 75:Tower of Brahma 28: 17: 12: 11: 5: 5398: 5388: 5387: 5382: 5377: 5372: 5367: 5362: 5360:1889 documents 5357: 5343: 5342: 5321: 5320:External links 5318: 5315: 5314: 5289: 5268: 5242: 5207: 5200: 5169: 5167: 5166: 5160: 5140:. Novelettes. 5128: 5091:(Pt 1): 50–8. 5075: 5028: 5007:(6): 1056–67. 4987: 4953: 4915: 4869: 4858:bbcmicro.co.uk 4845: 4838: 4818: 4811: 4791: 4761: 4732:(3): 610–622. 4714: 4707: 4689: 4678:(1–4): 57–65. 4662: 4635: 4611: 4589: 4510: 4484: 4458: 4423: 4420:on 2017-09-21. 4397:(5): 895–912. 4374: 4363:(4): 353–356. 4347: 4340: 4318: 4294: 4279: 4276:on 2004-08-21. 4268: 4248: 4241: 4223: 4216:(in Russian). 4204: 4197: 4179: 4172: 4151: 4130: 4119:Lucas, Édouard 4110: 4092:Lucas, Édouard 4083: 4058: 4034: 4028:978-3034802369 4027: 4004: 3979: 3933: 3926: 3904: 3879: 3878: 3876: 3873: 3872: 3871: 3864: 3859: 3854: 3848: 3840: 3839: 3837: 3836: 3829: 3822: 3814: 3811: 3810: 3807: 3806: 3801: 3796: 3791: 3786: 3780: 3775: 3774: 3771: 3770: 3766: 3765: 3760: 3755: 3750: 3745: 3740: 3735: 3730: 3725: 3719: 3718: 3711: 3710: 3707: 3706: 3701: 3700: 3694: 3693: 3692: 3691: 3683: 3682: 3676: 3675: 3674: 3673: 3668: 3660: 3659: 3649: 3648: 3647: 3646: 3635: 3630: 3625: 3617: 3616: 3615: 3614: 3609: 3604: 3599: 3591: 3586: 3581: 3576: 3568: 3567: 3561: 3560: 3559: 3558: 3556:Self-reference 3553: 3548: 3543: 3535: 3534: 3528: 3527: 3526: 3525: 3520: 3512: 3511: 3503: 3498: 3497: 3494: 3493: 3485: 3484: 3478: 3477: 3470: 3467: 3458:Genshin Impact 3331: 3328: 3246: 3243: 3239:Tower of Hanoy 3235:solitaire game 3230: 3229:Tower of Hanoy 3227: 3206: 3203: 3186:Main article: 3183: 3182:Magnetic Hanoi 3180: 3166: 3163: 3158: 3154: 3132: 3129: 3126: 3123: 3119: 3115: 3095: 3092: 3089: 3069: 3066: 3063: 3060: 3057: 3054: 3050: 3046: 3043: 3038: 3034: 3030: 3027: 3023: 3019: 3000: 2999: 2988: 2983: 2978: 2973: 2967: 2962: 2959: 2953: 2947: 2941: 2934: 2931: 2926: 2921: 2918: 2912: 2908: 2903: 2898: 2893: 2887: 2882: 2879: 2873: 2867: 2861: 2854: 2851: 2846: 2841: 2838: 2832: 2828: 2823: 2818: 2813: 2810: 2805: 2800: 2795: 2792: 2787: 2782: 2779: 2774: 2769: 2765: 2761: 2756: 2753: 2727: 2724: 2695: 2692: 2689: 2668: 2665: 2661: 2656: 2653: 2650: 2647: 2642: 2638: 2635: 2615: 2595: 2575: 2572: 2569: 2566: 2563: 2560: 2557: 2554: 2551: 2548: 2545: 2542: 2539: 2536: 2533: 2530: 2527: 2524: 2513: 2512: 2500: 2497: 2494: 2491: 2488: 2485: 2465: 2454: 2442: 2439: 2436: 2433: 2430: 2427: 2424: 2421: 2418: 2415: 2395: 2392: 2389: 2369: 2366: 2363: 2343: 2332: 2320: 2317: 2314: 2311: 2308: 2305: 2285: 2265: 2262: 2259: 2256: 2253: 2233: 2218: 2217: 2205: 2202: 2199: 2196: 2193: 2190: 2179: 2167: 2156: 2144: 2128: 2125: 2099: 2096: 2095: 2094: 2091: 2088: 2085: 2076: 2075: 2072: 2069: 2066: 2063: 2062: 2059: 2056: 2053: 2049: 2048: 2045: 2044: 2039: 2036: 2031: 2023: 2022: 2011: 2004: 2003:to a spare peg 1981: 1980: 1969: 1962: 1951: 1944: 1916: 1913: 1903: 1900: 1898: 1895: 1878: 1872: 1871: 1870: 1869: 1863: 1854: 1844: 1839: 1835: 1820: 1814: 1811: 1808: 1805: 1724: 1723: 1720: 1654:Main article: 1651: 1648: 1636:ruler function 1558: 1557: 1528: 1527: 1486: 1484: 1477: 1471: 1468: 1462:, disk number 1410: 1409: 1408: 1407: 1404: 1393: 1390: 1379: 1368: 1365: 1358: 1355: 1354: 1353: 1350: 1340: 1339: 1338: 1335: 1325: 1324: 1323: 1322: 1307: 1304: 1298: 1297: 1296: 1290: 1287: 1280: 1263: 1262: 1221: 1219: 1212: 1206: 1203: 1202: 1201: 1198: 1195: 1192: 1185: 1184: 1153: 1122: 1119: 1093: 1092: 1089: 1020: 1019: 978: 976: 969: 963: 960: 947: 944: 939: 936: 933: 929: 925: 922: 917: 913: 890: 887: 884: 880: 859: 856: 851: 847: 827: 826: 807: 792: 637: 595: 594: 553: 551: 544: 538: 535: 520: 519: 512: 489: 472:Move the disk 470: 415: 414: 407: 400: 367: 366: 325: 323: 316: 310: 307: 303: 302: 299: 296: 287: 286: 245: 243: 236: 230: 227: 213: 210: 197: 194: 132: 129: 117: 116: 113: 110: 87:pyramid puzzle 67:Tower of Hanoi 44:Tower of Hanoi 21:Tower of Hanoy 15: 9: 6: 4: 3: 2: 5397: 5386: 5383: 5381: 5378: 5376: 5373: 5371: 5368: 5366: 5363: 5361: 5358: 5356: 5353: 5352: 5350: 5338: 5337: 5332: 5329: 5324: 5323: 5304:on 2012-03-01 5303: 5299: 5293: 5278: 5272: 5256: 5252: 5246: 5238: 5234: 5230: 5226: 5222: 5218: 5211: 5203: 5197: 5193: 5189: 5185: 5178: 5176: 5174: 5163: 5157: 5153: 5147: 5146: 5143: 5139: 5132: 5124: 5120: 5116: 5112: 5108: 5104: 5099: 5094: 5090: 5086: 5079: 5071: 5067: 5063: 5059: 5055: 5051: 5047: 5043: 5039: 5032: 5024: 5020: 5015: 5010: 5006: 5002: 4998: 4991: 4983: 4979: 4975: 4971: 4964: 4957: 4949: 4945: 4941: 4937: 4933: 4929: 4925: 4919: 4911: 4907: 4903: 4899: 4895: 4891: 4887: 4883: 4876: 4874: 4859: 4855: 4849: 4841: 4835: 4831: 4830: 4822: 4814: 4808: 4804: 4803: 4795: 4787: 4783: 4779: 4772: 4765: 4757: 4753: 4749: 4745: 4740: 4735: 4731: 4727: 4726: 4718: 4710: 4704: 4700: 4693: 4685: 4681: 4677: 4673: 4666: 4658: 4654: 4650: 4646: 4639: 4621: 4615: 4599: 4593: 4586: 4572: 4568: 4564: 4560: 4556: 4552: 4548: 4543: 4538: 4534: 4530: 4529: 4521: 4514: 4506: 4502: 4495: 4488: 4480: 4476: 4469: 4462: 4454: 4450: 4446: 4442: 4438: 4434: 4427: 4416: 4412: 4408: 4404: 4400: 4396: 4392: 4385: 4378: 4370: 4366: 4362: 4358: 4351: 4343: 4341:9783034802369 4337: 4333: 4329: 4322: 4308: 4304: 4298: 4290: 4283: 4275: 4271: 4265: 4261: 4260: 4252: 4244: 4238: 4234: 4227: 4219: 4215: 4208: 4200: 4194: 4190: 4183: 4175: 4169: 4165: 4161: 4155: 4141: 4134: 4126: 4125: 4120: 4114: 4099: 4098: 4093: 4087: 4073: 4069: 4062: 4045: 4038: 4030: 4024: 4020: 4019: 4011: 4009: 3994: 3990: 3983: 3968: 3964: 3960: 3956: 3952: 3948: 3944: 3937: 3929: 3923: 3918: 3917: 3908: 3894: 3890: 3884: 3880: 3869: 3865: 3863: 3860: 3858: 3855: 3852: 3849: 3847: 3844: 3843: 3835: 3830: 3828: 3823: 3821: 3816: 3815: 3813: 3812: 3805: 3804:Puzzle topics 3802: 3800: 3797: 3795: 3792: 3790: 3787: 3785: 3782: 3781: 3773: 3772: 3764: 3761: 3759: 3756: 3754: 3751: 3749: 3746: 3744: 3741: 3739: 3736: 3734: 3731: 3729: 3726: 3724: 3721: 3720: 3715: 3709: 3708: 3699: 3696: 3695: 3690: 3687: 3686: 3685: 3684: 3681: 3678: 3677: 3672: 3669: 3667: 3664: 3663: 3662: 3661: 3658: 3654: 3651: 3650: 3644: 3640: 3636: 3634: 3631: 3629: 3626: 3624: 3621: 3620: 3619: 3618: 3613: 3610: 3608: 3605: 3603: 3600: 3598: 3596: 3592: 3590: 3587: 3585: 3582: 3580: 3577: 3575: 3572: 3571: 3570: 3569: 3566: 3563: 3562: 3557: 3554: 3552: 3549: 3547: 3544: 3542: 3539: 3538: 3537: 3536: 3533: 3530: 3529: 3524: 3521: 3519: 3516: 3515: 3514: 3513: 3510: 3507: 3506: 3496: 3495: 3491: 3487: 3486: 3483: 3480: 3479: 3475: 3474: 3466: 3464: 3460: 3459: 3453: 3451: 3447: 3443: 3441: 3434: 3429: 3427: 3423: 3422: 3416: 3414: 3410: 3409: 3404: 3403: 3398: 3394: 3389: 3387: 3383: 3382: 3376: 3373: 3372: 3366: 3364: 3360: 3356: 3352: 3351: 3346: 3345: 3339: 3337: 3327: 3325: 3320: 3318: 3317: 3311: 3309: 3304: 3302: 3298: 3294: 3289: 3287: 3282: 3280: 3275: 3270: 3268: 3264: 3260: 3251: 3242: 3240: 3236: 3226: 3218: 3214: 3212: 3198: 3194: 3189: 3179: 3164: 3161: 3156: 3152: 3127: 3124: 3121: 3117: 3113: 3087: 3064: 3058: 3055: 3052: 3048: 3044: 3041: 3036: 3032: 3028: 3025: 3021: 3017: 3009: 3005: 2986: 2981: 2976: 2971: 2965: 2960: 2957: 2951: 2945: 2939: 2932: 2929: 2924: 2919: 2916: 2910: 2906: 2901: 2896: 2891: 2885: 2880: 2877: 2871: 2865: 2859: 2852: 2849: 2844: 2839: 2836: 2830: 2826: 2821: 2816: 2811: 2808: 2803: 2798: 2793: 2790: 2785: 2780: 2777: 2772: 2767: 2763: 2759: 2754: 2751: 2742: 2741: 2740: 2737: 2732: 2723: 2721: 2716: 2714: 2710: 2693: 2690: 2687: 2666: 2663: 2659: 2654: 2651: 2648: 2645: 2640: 2636: 2633: 2613: 2593: 2570: 2567: 2564: 2561: 2558: 2555: 2552: 2546: 2543: 2537: 2534: 2531: 2525: 2522: 2495: 2492: 2489: 2483: 2463: 2455: 2437: 2434: 2431: 2428: 2425: 2422: 2419: 2413: 2406:pegs, taking 2393: 2390: 2387: 2367: 2364: 2361: 2341: 2333: 2315: 2312: 2309: 2303: 2283: 2263: 2260: 2257: 2254: 2251: 2231: 2223: 2222: 2221: 2200: 2197: 2194: 2188: 2180: 2165: 2157: 2142: 2134: 2133: 2132: 2124: 2122: 2118: 2113: 2110: 2107: 2104: 2092: 2089: 2086: 2083: 2082: 2081: 2073: 2070: 2067: 2065: 2064: 2060: 2057: 2054: 2051: 2050: 2047: 2046: 2040: 2037: 2032: 2030: 2029: 2026: 2020: 2016: 2012: 2009: 2005: 2002: 1998: 1994: 1993: 1992: 1990: 1986: 1978: 1974: 1970: 1967: 1963: 1960: 1956: 1952: 1949: 1945: 1942: 1938: 1934: 1933: 1932: 1930: 1926: 1921: 1912: 1909: 1894: 1892: 1887: 1881: 1877: 1866: 1862: 1857: 1853: 1847: 1843: 1840: 1834: 1831: 1830: 1828: 1823: 1819: 1815: 1812: 1809: 1806: 1803: 1802: 1801: 1795: 1791: 1789: 1784: 1782: 1778: 1770: 1766: 1763: 1761: 1757: 1752: 1750: 1746: 1742: 1737: 1733: 1728: 1721: 1718: 1717: 1713: 1708: 1701: 1697: 1694: 1691: 1685: 1681: 1678: 1675: 1669: 1665: 1663: 1657: 1647: 1641: 1637: 1633: 1628: 1622: 1620: 1616: 1612: 1608: 1604: 1600: 1596: 1592: 1588: 1584: 1580: 1576: 1572: 1568: 1564: 1555: 1551: 1547: 1546: 1545: 1541: 1539: 1535: 1524: 1521: 1513: 1503: 1499: 1493: 1492: 1487:This section 1485: 1481: 1476: 1475: 1467: 1465: 1461: 1456: 1454: 1449: 1431: 1427: 1423: 1419: 1415: 1405: 1402: 1398: 1394: 1391: 1388: 1384: 1380: 1377: 1373: 1369: 1366: 1363: 1362: 1356: 1351: 1348: 1347: 1341: 1336: 1333: 1332: 1330: 1329: 1328: 1320: 1316: 1312: 1308: 1305: 1302: 1301: 1299: 1294: 1293: 1291: 1288: 1285: 1281: 1278: 1274: 1273: 1272: 1270: 1259: 1256: 1248: 1238: 1234: 1228: 1227: 1222:This section 1220: 1216: 1211: 1210: 1199: 1196: 1193: 1190: 1189: 1188: 1182: 1178: 1174: 1170: 1166: 1162: 1158: 1154: 1151: 1147: 1143: 1139: 1135: 1131: 1127: 1123: 1120: 1117: 1116: 1115: 1112: 1110: 1106: 1102: 1098: 1090: 1087: 1086: 1085: 1082: 1080: 1076: 1072: 1068: 1064: 1060: 1056: 1052: 1048: 1044: 1040: 1036: 1032: 1028: 1016: 1013: 1005: 995: 991: 985: 984: 979:This section 977: 973: 968: 967: 959: 945: 942: 937: 934: 931: 927: 923: 920: 915: 911: 888: 885: 882: 878: 857: 854: 849: 845: 836: 832: 824: 820: 816: 812: 808: 805: 801: 797: 793: 790: 786: 782: 778: 774: 773: 772: 770: 766: 762: 758: 753: 751: 747: 742: 740: 736: 732: 728: 724: 720: 716: 712: 708: 704: 700: 696: 692: 688: 684: 680: 676: 672: 668: 664: 660: 656: 652: 648: 644: 640: 636: 630: 626: 622: 618: 614: 610: 606: 602: 591: 588: 580: 570: 566: 560: 559: 554:This section 552: 548: 543: 542: 534: 532: 527: 525: 517: 513: 510: 506: 502: 498: 494: 490: 487: 486:a simple step 483: 479: 475: 471: 468: 464: 460: 456: 452: 448: 447: 446: 444: 440: 436: 432: 428: 424: 420: 417:Assuming all 412: 408: 405: 401: 398: 397: 396: 394: 390: 386: 378: 377:the SVG file, 373: 363: 360: 352: 342: 338: 332: 331: 326:This section 324: 320: 315: 314: 306: 300: 297: 294: 293: 292: 283: 280: 272: 262: 258: 252: 251: 246:This section 244: 240: 235: 234: 226: 218: 209: 203: 193: 191: 187: 183: 179: 174: 172: 168: 163: 161: 157: 153: 149: 145: 141: 140:Édouard Lucas 138: 137:mathematician 128: 126: 114: 111: 108: 107: 106: 104: 100: 96: 92: 88: 84: 80: 76: 72: 69:(also called 68: 61: 56: 49: 45: 40: 32: 26: 22: 5334: 5306:. Retrieved 5302:the original 5292: 5281:. Retrieved 5271: 5259:. Retrieved 5254: 5245: 5220: 5216: 5210: 5183: 5151: 5141: 5138:"Now Inhale" 5131: 5088: 5085:J. Exp. Biol 5084: 5078: 5048:(5): 777–9. 5045: 5041: 5031: 5004: 5000: 4990: 4973: 4969: 4956: 4931: 4927: 4918: 4885: 4882:Nano Letters 4881: 4861:. Retrieved 4857: 4848: 4828: 4821: 4801: 4794: 4780:(4): 37–48. 4777: 4764: 4739:math/0310109 4729: 4723: 4717: 4698: 4692: 4675: 4671: 4665: 4648: 4644: 4638: 4626:. Retrieved 4614: 4602:. Retrieved 4592: 4578:. Retrieved 4532: 4526: 4513: 4504: 4500: 4497:(Postscript) 4487: 4478: 4474: 4471:(Postscript) 4461: 4439:(3): 216–9. 4436: 4432: 4426: 4415:the original 4394: 4390: 4377: 4360: 4356: 4350: 4327: 4321: 4310:. Retrieved 4306: 4297: 4288: 4282: 4274:the original 4258: 4251: 4232: 4226: 4217: 4213: 4207: 4188: 4182: 4163: 4154: 4143:. Retrieved 4133: 4123: 4113: 4102:. Retrieved 4096: 4086: 4075:. Retrieved 4071: 4061: 4050:. Retrieved 4037: 4021:. Springer. 4017: 3996:. Retrieved 3992: 3982: 3970:. Retrieved 3950: 3946: 3936: 3915: 3907: 3896:. Retrieved 3892: 3883: 3857:Baguenaudier 3723:Brain teaser 3594: 3579:Construction 3456: 3454: 3439: 3432: 3430: 3419: 3417: 3406: 3400: 3390: 3379: 3377: 3369: 3367: 3348: 3342: 3341:In the 1966 3340: 3333: 3321: 3314: 3312: 3308:frontal lobe 3305: 3290: 3283: 3271: 3256: 3245:Applications 3232: 3223: 3210: 3208: 3191: 3003: 3001: 2735: 2733: 2729: 2719: 2717: 2712: 2514: 2219: 2130: 2123:Gray codes. 2114: 2111: 2108: 2105: 2101: 2079: 2061:A(1) = 1 1, 2024: 2018: 2014: 2007: 2000: 1996: 1988: 1984: 1982: 1976: 1972: 1965: 1958: 1954: 1947: 1940: 1936: 1928: 1924: 1922: 1918: 1915:Cyclic Hanoi 1907: 1905: 1902:Linear Hanoi 1879: 1875: 1873: 1864: 1860: 1855: 1851: 1845: 1841: 1832: 1826: 1821: 1817: 1799: 1785: 1780: 1776: 1774: 1764: 1758:figure, the 1753: 1748: 1744: 1740: 1731: 1729: 1725: 1695: 1692: 1689: 1679: 1676: 1673: 1659: 1623: 1618: 1614: 1610: 1606: 1602: 1598: 1594: 1590: 1586: 1582: 1578: 1574: 1570: 1566: 1562: 1559: 1553: 1549: 1542: 1531: 1516: 1510:January 2024 1507: 1496:Please help 1491:verification 1488: 1459: 1457: 1452: 1450: 1432:is from peg 1429: 1417: 1413: 1411: 1400: 1396: 1386: 1382: 1375: 1371: 1361:= 11011000. 1346:= 11111111. 1326: 1318: 1314: 1310: 1266: 1251: 1245:January 2024 1242: 1231:Please help 1226:verification 1223: 1186: 1180: 1176: 1172: 1168: 1164: 1160: 1156: 1149: 1145: 1141: 1137: 1133: 1129: 1125: 1113: 1108: 1104: 1100: 1096: 1094: 1083: 1078: 1074: 1070: 1066: 1062: 1058: 1054: 1050: 1046: 1042: 1038: 1034: 1030: 1026: 1023: 1008: 1002:January 2024 999: 988:Please help 983:verification 980: 828: 822: 818: 814: 810: 803: 799: 795: 788: 784: 780: 776: 768: 764: 760: 756: 754: 749: 745: 743: 738: 734: 730: 726: 722: 718: 714: 710: 706: 702: 698: 694: 690: 686: 682: 678: 674: 670: 666: 662: 658: 654: 650: 646: 642: 634: 628: 624: 620: 616: 612: 608: 604: 600: 598: 583: 577:January 2024 574: 563:Please help 558:verification 555: 528: 523: 521: 515: 508: 504: 500: 496: 492: 485: 481: 477: 473: 466: 462: 458: 454: 450: 442: 441:peg using a 438: 434: 430: 426: 422: 418: 416: 410: 403: 392: 388: 382: 355: 349:January 2024 346: 335:Please help 330:verification 327: 304: 290: 275: 269:January 2024 266: 255:Please help 250:verification 247: 223: 201: 199: 175: 164: 143: 134: 124: 118: 86: 85:, or simply 82: 79:Lucas' Tower 78: 74: 70: 66: 64: 47: 43: 5223:(1): 9–10. 5148:Reprinted: 4651:: 289–321. 4628:January 31, 4220:(2): 10–14. 4166:. 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