Hanoi Game Online' title='Hanoi Game Online' />Tower of Hanoi. The Tower of Hanoi puzzle was invented by the French mathematician Edouard Lucas in 1. We are given a tower of eight disks initially four in the applet below, initially stacked in increasing size on one of three pegs. The objective is to transfer the entire tower to one of the other pegs the rightmost one in the applet below, moving only one disk at a time and never a larger one onto a smaller. The puzzle is well known to students of Computer Science since it appears in virtually any introductory text on data structures or algorithms. Its solution touches on two important topics discussed later on recursive functions and stacks. Hanoi Game RulesOur BoomerInCharge here at BBHQ, Hershel Chicowitz, writes frequently about current events. He is sometimes funny, sometimes. Towers of Hanoi. Play this classic Towers of Hanoi game online. You aim is to move the piles of blocks from one location to the other, remembering that you cant. Add a link to this game to your MySpace, Facebook or web page. Just CopyPaste. Horror Stories of the Hanoi Hilton The Best Vietnam War Movies, Ranked Movies That Fudged the Truth Old School Pics from Vietnam Documentaries About Vietnam Creepy. The applet has several controls that allow one to select the number of disks and observe the solution in a Fast or Slow manner. To solve the puzzle drag disks from one peg to another following the rules. You can drop a disk on to a peg when its center is sufficiently close to the center of the peg. The Tower of Hanoi puzzle was invented by the French mathematician Edouard Lucas in 1883. We are given a tower of eight disks initially four in the applet below. The applet expects you to move disks from the leftmost peg to the rightmost peg. If you are reading this, your browser is not set to run Java applets. Try IE1. 1 or Safari and declare the site https www. Java setup. Recursive solution. Let call the three pegs Src Source, Aux Auxiliary and Dst Destination. To better understand. However one solves the problem, sooner or later the bottom disk will. Src to Dst. At this point in time all the remaining disks will have to be stacked. Aux. After moving the bottom disk from Src to Dst these disks will have to be. Aux to Dst. Therefore, for a given number N of disks, the problem appears to be solved if we. Move the top N 1 disks from Src to Aux using Dst as an intermediary peg. Move the bottom disk from Src to Dst. Move N 1 disks from Aux to Dst using Src as an intermediary peg. Assume there is a function Solve with four arguments number of disks and three pegs source, intermediary and destination in this order. Then the body of the function might look like. SolveN, Src, Aux, Dst. SolveN 1, Src, Dst, Aux. Move from Src to Dst. SolveN 1, Aux, Src, Dst. This actually serves as the definition of the function Solve. The function is recursive in that it calls itself repeatedly with decreasing values of N until a terminating condition in our case N 0 has been met. How To Activate Your Pay As You Go Sim there. To me the sheer simplicity of the solution is breathtaking. For N 3 it translates into. Move from Src to Dst. Move from Src to Aux. Move from Dst to Aux. Move from Src to Dst. Move from Aux to Src. Move from Aux to Dst. Move from Src to Dst. Of course Move means moving the topmost disk. For N 4 we get the following sequence. Move from Src to Aux. Move from Src to Dst. Move from Aux to Dst. Move from Src to Aux. Move from Dst to Src. Move from Dst to Aux. Move from Src to Aux. Move from Src to Dst. Move from Aux to Dst. Move from Aux to Src. Move from Dst to Src. Move from Aux to Dst. Move from Src to Aux. Move from Src to Dst. Move from Aux to Dst. Recurrence relations. Let TN be the minimum number of moves needed to solve the puzzle with N disks. From the previous section T3 7 and T4 1. One can easily convince oneself that T2 3 and T1 1. A trained mathematician would also note that T0 0. Now let us try to derive a general formula. The recursive solution above involves moving twice N 1 disks from one peg to another and making. It then follows that. TN TN 1 1 TN 1 2. TN 1 1. The inequality suggests that it might be possible to move N disks with fewer than 2. TN 1 1 moves. Which is actually not the case. Indeed, when the time comes to move the bottom disk, N 1 smaller disks will have been moved from Src to Aux in at least TN 1 moves. Since we are trying to use as few steps as possible, we may assume that that portion of the task took exactly TN 1 moves. It takes just one move to move the biggest disk from Src to Dst. One then needs exactly TN 1 more steps to finish the task. Therefore the minimum number of moves needed to solve the puzzle with N disks equals TN 1 1 TN 1 2. TN 1 1 moves. In other words. Thus we can define the quantity TN as. T0 0. TN 2. TN 1 1 for N 0. We may compute T1 2. T0 1 1, T2 2. T1 1 3,T3 2. T2 1 7 and so on sequentially. The above expression is known as a recurrence relation which, as you might have noticed, is but a recursive function. TN is defined in terms of only one of its preceding values. Other recurrence relations may be more complicated, for example, fN 2fN 1 3fN 2. Recurrence relations appear under various guises in numerous branches of Mathematics and applications. Returning to the definition of TN, define SN TN 1. Then S0 1 and SN TN 1 2. TN 1 1 1 2. TN 1 2 2TN 1 1 2. SN 1. Which is to say that SN could be defined as. S0 1. SN 2. SN 1 for N 0. The latter is solved easily in the closed non recurrent form SN2. N. TN 2. N 1 for N 0. Remark 1. The latest implementation of the applet sports a Suggest move button that exploits an algorithm devised by Romek Zylla who graciously put up on the Web an explanation of his algorithm. The algorithm actually provides another, a non recursive solution to the puzzle. Remark 2. There are quite a few variations on the puzzle. For example, you may want to experiment with its bicolor or 3 colors versions. Reference. A. Beck, M. N. Bleicher, D. W. Crowe, Excursions into Mathematics, A K Peters, 2. Tower of Hanoi. Tower of Hanoi, the Hard Way. Bicolor Towers of Hanoi. Colors Tower of Hanoi. Colors Tower of Hanoi AlgorithmHanoing. Sierpinski Gasket and Tower of HanoiContactFront pageContentsGamesCopyright 1. Alexander Bogomolny. Operation Linebacker II Wikipedia. Free Download Ocean Express Games Full Version. Operation Linebacker II was a US Seventh Air Force and US Navy. Task Force 7. 7aerial bombing campaign, conducted against targets in the Democratic Republic of Vietnam North Vietnam during the final period of US involvement in the Vietnam War. The operation was conducted from 1. December 1. 97. 2, leading to several informal names such as The December Raids and The Christmas Bombings. Unlike the Operation Rolling Thunder and Operation Linebacker interdiction operations, Linebacker II was to be a maximum effort bombing campaign to destroy major target complexes in the Hanoi and Haiphong areas, which could only be accomplished by B 5. It saw the largest heavy bomber strikes launched by the US Air Force since the end of World War II. Linebacker II was a modified extension of the Operation Linebacker bombings conducted from May to October, when the emphasis of the new campaign shifted to attacks by B 5. BackgroundeditPeace is at handeditOn 8 October 1. U. S. National Security Advisor Dr. Henry Kissinger and North Vietnamese Politburo member Le Duc Tho met in Paris to discuss new proposals by both nations, hoping to reach mutually agreeable terms for a peace settlement for the decade old Vietnam war. Tho presented a new North Vietnamese plan which included proposals for a cease fire, the withdrawal of American forces, and an exchange of prisoners of war. All three Vietnamese combatant governmentsNorth Vietnam, the Republic of Vietnam South Vietnam, and the Provisional Revolutionary Government of South Vietnam PRGwould remain intact, as would their separate armies. Hanoi no longer demanded that South Vietnamese president Nguyen Van Thieu be removed from office, the U. S. did not have to cease its aid to the southern government, and both Washington and Hanoi could continue to resupply their allies or forces on a parity basis. No new North Vietnamese forces were to be infiltrated from the north, and the U. S. agreed to extend post war reconstruction assistance to North Vietnam. The new terms on the table also included the establishment of a National Council of National Reconciliation and Concord, a loosely defined administrative structure which was to work toward general and local elections within South Vietnam. Political power would be shared by three groups, the Saigon government, the PRG, and a third force group to be mutually agreed upon by the other two parties. Since it was to work by consensus, nothing could be accomplished by the new council without the agreement of President Thieu. When the two sides convened again on 1. October, there were two main areas of disagreement the periodic replacement of South Vietnams American weaponry, and the release of political prisoners held by the Saigon government. The North Vietnamese had made significant modifications to their past negotiating position and were hurrying to get the agreement signed before November, believing that President Richard Nixon would be more willing to make concessions before, rather than after, the upcoming presidential election. Simple .Obj File Example. Although there were still some issues to be finalized, Kissinger was generally satisfied with the new terms and so notified Nixon, who gave his approval to the settlement. The finalized agreement was to be signed in Hanoi on 3. October. citation neededKissinger then flew on to Saigon on the 1. Thieu. The South Vietnamese president was not happy with either the new agreement or with Kissinger, who he felt had betrayed him. Although Kissinger knew Thieus negotiating position, he had not informed him of the changes made in Paris nor had his approval been sought. Kissinger had negotiated on behalf of the South Vietnamese government provisions that he, Thieu, had already rejected. Thieu completely castigated the agreement and proposed 1. He went further, demanding that the Demilitarized Zone separating the two Vietnams be recognized as a true international border and not as a provisional military demarcation line as had been stipulated in the Geneva Accords and that South Vietnam be recognized as a sovereign state. The supreme irony, in the words of Stanley Karnow, had now arrived having fought a war to defend South Vietnams independence, the United States was now denying its legitimacy. Thieu then went one step further on 2. October, and publicly released an altered version of the text that made the South Vietnamese provisions look even worse than they actually were. The North Vietnamese leadership, believing that they had been hoodwinked by Kissinger, responded by broadcasting portions of the agreement that gave the impression that the agreement conformed to Washington and Saigons objectives. Kissinger, hoping to both reassure the Communists of Americas sincerity, and convince Thieu of the administrations dedication to a compromise, held a televised press conference at the White House during which he announced, We believe that peace is at hand. On 2. 0 November, the South Vietnamese revisions, and 4. Nixon, were presented to the North Vietnamese delegation by Kissinger. These new demands included That the DMZ be accepted as a true international boundary that a token withdrawal of North Vietnamese troops take place that the North Vietnamese guarantee an Indochina wide cease fire and that a strong international peace keeping force the ICCS be created for supervising and enforcing the cease fire. Once the North Vietnamese read the new demands, they began to retract their own concessions and wanted to bargain anew, leading Kissinger to proclaim that they were stalling. The talks, scheduled to last ten days, ended on 1. December, with both parties agreeing to resume negotiations. Teams of experts from each side met to discuss technicalities and protocols on 1. December, during which time the North Vietnamese representatives submitted a Vietnamese language text of the protocol on prisoners containing several important changes that Hanoi had failed to gain in the main negotiating sessions. At a subsequent meeting of experts on 1. December, the North Vietnamese side stone walled from beginning to end. The talks broke down that day, and the Hanoi negotiators refused to set a date for the resumption of negotiations. PreludeeditDecisionseditNixon was now working against a January deadline. Kissingers peace is at hand statement had raised expectations of a settlement among the US population. Even weightier on the Presidents mind was the fact that the new Ninety third Congress would go into session on 3 January, and the President feared that the heavily Democratic legislative branch would preempt his pledge of peace with honor by legislating an end to the war. Also prompting the President toward some form of rapid offensive action was the cost of the force mobilization that had accompanied Operation Linebacker. The additional aircraft and personnel assigned to Southeast Asia for the operation was straining the Pentagons budget. The cost of maintaining this augmentation force totaled over 4 billion by mid autumn and Secretary of Defense. Melvin Laird insisted that the President request a supplementary defense appropriation from Congress to pay for it. Nixon and Kissinger were convinced that the legislative branch would seize the opportunity to simply write the United States out of the war. After returning from Paris on 1.