Three cups problem: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 02:39, 14 June 2024

The three cups problem, also known as the three cup challenge and other variants, is a mathematical puzzle that, in its most common form, cannot be solved.

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, turning over exactly two cups at each move.

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, turn up the two cups that are upside down — after which all three cups are facing up. As a magic trick, a magician can perform the solvable version in a convoluted way, and then ask an audience member to solve the unsolvable version.^[1]

Proof of impossibility

To see that the problem is insolvable (when starting with just one cup upside down), it suffices to concentrate on the number of cups the wrong way up. Denoting this number by $W$ , the goal of the problem is to change $W$ from 1 to 0, i.e. by $-1$ . The problem is insolvable because any move changes $W$ by an even number. Since a move inverts two cups and each inversion changes $W$ by $+1$ (if the cup was the right way up) or $-1$ (otherwise), a move changes $W$ by the sum of two odd numbers, which is even, completing the proof.

Another way of looking is that, at the start, 2 cups are in the "right" orientation and 1 is "wrong". Changing 1 right cup and 1 wrong cup, the situation remains the same. Changing 2 right cups results in a situation with 3 wrong cups, after which the next move restores the original status of 1 wrong cup. Thus, any number of moves results in a situation either with 3 wrongs or with 1 wrong, and never with 0 wrongs.

More generally, this argument shows that for any number of cups, it is impossible to reduce $W$ to 0 if it is initially odd. On the other hand, if $W$ is even, inverting cups two at a time will eventually result in $W$ equaling 0.

References

^ Lane, Mike (2012). Close-Up Magic. The Rosen Publishing Group, Inc. ISBN 9781615335152.

"Can you solve the Three Cups Problem?". ABC Education. Retrieved 2018-10-26.

@@ Line 1: / Line 1: @@
+{{refimprove|date=May 2024}}
-The '''three cups problem''' is a mathematical puzzle. Starting with three cups place one upside down and two right side up.  The objective is to eventually turn all cups right side up in six moves.  You must turn exactly two cups over each turn.
+[[File:three_cups_problem_unsolvable.svg|thumb|The standard, unsolvable, arrangement of the three cups. Here, cups A and C are upright and B is upside down.]]
+[[File:three_cups_problem_solvable.svg|thumb|The solvable version of the problem. Here, cups A and C are upside down, and cup B is upright.]]
+The '''three cups problem''', also known as the '''three cup challenge''' and other variants, is a mathematical [[puzzle]] that, in its most common form, cannot be solved.
+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, turning over exactly two cups at each move.
-== Solution ==
-The puzzle is impossible.  An even number of cups are facing up and you are allowed to turn two over at a time.  Since an even plus an even is an even, not an odd, no number of even flips will ever get all the three cups face up.  You need an odd number of cups facing up, so the problem is impossible.
-The possible version of this puzzle is to start with two cups facing down and one cup facing upward.  This is possible.  Turn up an even number (two) of cups, and all the cups are facing up; an odd plus an even is an odd (1+2 = 3).
+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, turn up the two cups that are upside down — after which all three cups are facing up. As a [[magic trick]], a magician can perform the solvable version in a convoluted way, and then ask an audience member to solve the unsolvable version.<ref>{{cite book |last1=Lane |first1=Mike |title=Close-Up Magic |date=2012 |publisher=The Rosen Publishing Group, Inc |isbn=9781615335152 |url=https://books.google.com/books?id=kLDwC8hjbOgC&pg=PA13 |language=en}}</ref>
-== See also ==
-* [[Parity (mathematics)|Parity]]
-* [[List of impossible puzzles]]
+==Proof of impossibility==
-== External links ==
+To see that the problem is insolvable (when starting with just one cup upside down), it suffices to concentrate on the number of cups the wrong way up. Denoting this number by <math>W</math>, the goal of the problem is to change <math>W</math> from 1 to 0, i.e. by <math>-1</math>. The problem is insolvable because any move changes <math>W</math> by an even number. Since a move inverts two cups and each inversion changes <math>W</math> by <math>+1</math> (if the cup was the right way up) or <math>-1</math> (otherwise), a move changes <math>W</math> by the sum of two odd numbers, which is even, completing the proof.
-* [http://www.albinoblacksheep.com/flash/3cup Three Cups Animation]
-* [http://www.jimloy.com/puzz/cups.htm Three Cups Problem]
-* [http://www.aimsedu.org/puzzle/cupsNDowns/cups.html Three Cups Problem Student's Handout Along With Explanation]
+Another way of looking is that, at the start, 2 cups are in the "right" orientation and 1 is "wrong". Changing 1 right cup and 1 wrong cup, the situation remains the same. Changing 2 right cups results in a situation with 3 wrong cups, after which the next move restores the original status of 1 wrong cup. Thus, any number of moves results in a situation either with 3 wrongs or with 1 wrong, and never with 0 wrongs.
-[[Category:Puzzles]]
+More generally, this argument shows that for any number of cups, it is impossible to reduce <math>W</math> to 0 if it is initially odd. On the other hand, if <math>W</math> is even, inverting cups two at a time will eventually result in <math>W</math> equaling 0.
+==References==
+{{Reflist}}
+*{{cite web|accessdate=2018-10-26|title=Can you solve the Three Cups Problem?|url=http://education.abc.net.au/home#!/media/2977513/can-you-solve-the-three-cups-problem-|website=ABC Education}}
+==See also==
+*[[List of impossible puzzles]]
+*[[Puzzle]]
+*[[Recreational mathematics]]
+[[Category:Magic tricks]]
+[[Category:Unsolvable puzzles]]