{"trustable":true,"sections":[{"title":"","value":{"format":"HTML","content":"\u003cb\u003eBackground\u003c/b\u003e\r\u003cbr\u003eCharlie Darkbrown sits in another one of those boring Computer Science lessons: At the moment the teacher just explains the standard Tower of Hanoi problem, which bores Charlie to death!\r\u003cbr\u003e\u003ccenter\u003e\u003cimg src\u003d\"CDN_BASE_URL/b49f7faee4f57434cc9faca3dc837dd6?v\u003d1714657851\"\u003e\u003c/center\u003e\r\u003cbr\u003eThe teacher points to the blackboard (Fig. 4) and says: \"So here is the problem:\r\u003cbr\u003e\u003cul\u003e\u003cli\u003eThere are three towers: A, B and C.\r\u003cbr\u003e\u003c/li\u003e\u003cli\u003eThere are n disks. The number n is constant while working the puzzle.\r\u003cbr\u003e\u003c/li\u003e\u003cli\u003eAll disks are different in size.\r\u003cbr\u003e\u003c/li\u003e\u003cli\u003eThe disks are initially stacked on tower A increasing in size from the top to the bottom.\r\u003cbr\u003e\u003c/li\u003e\u003cli\u003eThe goal of the puzzle is to transfer all of the disks from tower A to tower C.\r\u003cbr\u003e\u003c/li\u003e\u003cli\u003eOne disk at a time can be moved from the top of a tower either to an empty tower or to a tower with a larger disk on the top.\u003c/li\u003e\u003c/ul\u003e\r\u003cbr\u003eSo your task is to write a program that calculates the smallest number of disk moves necessary to move all the disks from tower A to C.\"\r\u003cbr\u003eCharlie: \"This is incredibly boring—everybody knows that this can be solved using a simple recursion.I deny to code something as simple as this!\"\r\u003cbr\u003eThe teacher sighs: \"Well, Charlie, let\u0027s think about something for you to do: For you there is a fourth tower D. Calculate the smallest number of disk moves to move all the disks from tower A to tower D using all four towers.\"\r\u003cbr\u003eCharlie looks irritated: \"Urgh. . . Well, I don\u0027t know an optimal algorithm for four towers. . . \"\r\u003cbr\u003e\u003cb\u003eProblem\u003c/b\u003e\r\u003cbr\u003eSo the real problem is that problem solving does not belong to the things Charlie is good at. Actually, the only thing Charlie is really good at is \"sitting next to someone who can do the job\". And now guess what — exactly! It is you who is sitting next to Charlie, and he is already glaring at you.\r\u003cbr\u003eLuckily, you know that the following algorithm works for n \u0026lt;\u003d 12: At first k \u0026gt;\u003d 1 disks on tower A are fixed and the remaining n-k disks are moved from tower A to tower B using the algorithm for four towers.Then the remaining k disks from tower A are moved to tower D using the algorithm for three towers. At last the n - k disks from tower B are moved to tower D again using the algorithm for four towers (and thereby not moving any of the k disks already on tower D). Do this for all k 2 ∈{1, .... , n} and find the k with the minimal number of moves.\r\u003cbr\u003eSo for n \u003d 3 and k \u003d 2 you would first move 1 (3-2) disk from tower A to tower B using the algorithm for four towers (one move). Then you would move the remaining two disks from tower A to tower D using the algorithm for three towers (three moves). And the last step would be to move the disk from tower B to tower D using again the algorithm for four towers (another move). Thus the solution for n \u003d 3 and k \u003d 2 is 5 moves. To be sure that this really is the best solution for n \u003d 3 you need to check the other possible values 1 and 3 for k. (But, by the way, 5 is optimal. . . )"}},{"title":"Input","value":{"format":"HTML","content":"There is no input."}},{"title":"Output","value":{"format":"HTML","content":"For each n (1 \u0026lt;\u003d n \u0026lt;\u003d 12) print a single line containing the minimum number of moves to solve the problem for four towers and n disks."}},{"title":"Sample","value":{"format":"HTML","content":"\u003ctable class\u003d\u0027vjudge_sample\u0027\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n \u003cth\u003eInput\u003c/th\u003e\n \u003cth\u003eOutput\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\u003cpre\u003eNo input.\u003c/pre\u003e\u003c/td\u003e\n \u003ctd\u003e\u003cpre\u003eREFER TO OUTPUT.\u003c/pre\u003e\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n"}}]}