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| The '''three cups problem''' is a mathematical [[puzzle]] that, in its most common form, cannot be solved. | | The title of the author is Nestor. One of my preferred hobbies is tenting and now I'm attempting to make money with it. I presently reside in Alabama. His working day job is a cashier and his wage has been really satisfying.<br><br>Visit my blog [http://Calvaryhill.net/xe/board_DsjO50/23803 calvaryhill.net] |
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| In the beginning position of the problem, one cup is upside-down and the other two are right-side up. The objective is to '''turn all cups right-side up''' in no more than six moves. You must turn over exactly two cups at each move.
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| ==Solvable version==
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| [[File:Threecupsproblem.jpg|thumb|left|The solvable version of the Three Cups Problem is shown here. In the impossible version, cups A and C are upright, and cup B is turned down.]]
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| The solvable (but trivial) version of this puzzle begins with one cup right-side up and two cups upside-down. To solve the puzzle in a single move, you need only turn up the two cups that are upside down — after which all three cups are facing up.
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| {{Clear left}}
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| ==Proof of impossibility==
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| The proof that the problem is impossible to solve is done by means of exhausting cases.
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| As a starting position, we place cup A up, cup B down, and cup C up — the reverse of the figure above. Using a well known formula for [[combination]]s,<ref><math>\frac{n!}{k!(n-k)!}</math> where <math>k\leq n</math>. ''n'' is the number of items, and ''k'' is the number of items to be selected from them.</ref> we find that two items can be selected, without regard to their order, from three items in three ways. In this instance, the three ways are:
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| ;1. Select cups A<sup>U</sup> and B<sup>D</sup>
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| ;2. Select cups A<sup>U</sup> and C<sup>U</sup>
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| ;3. Select cups B<sup>D</sup> and C<sup>U</sup>
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| Where ''U'' means Up, and ''D'' means Down.
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| If we flip selection '''1''' we get A<sup>D</sup> and B<sup>U</sup>.
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| If we flip selection '''2''' we get A<sup>D</sup> and C<sup>D</sup>. Cup B remains down, so now all three cups are down.
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| If we flip selection '''3''' we get B<sup>U</sup> and C<sup>D</sup>.
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| Thus we see that in all three cases at least one cup remains down after flipping. Reversing any one of these flips restores the starting position of the cups.
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| This exhausts the possible combinations of two cups selected (without order) from among three; no other combination of two cups is possible. Therefore the problem cannot be solved.
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| ==See also==
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| * [[List of impossible puzzles]]
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| * [[Puzzle]]
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| * [[Recreational mathematics]]
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| ==Notes==
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| {{Reflist}}
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| [[Category:Puzzles]]
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The title of the author is Nestor. One of my preferred hobbies is tenting and now I'm attempting to make money with it. I presently reside in Alabama. His working day job is a cashier and his wage has been really satisfying.
Visit my blog calvaryhill.net