science

Three cups problem

Three cups problem is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Three cups problem rather than just read about it. In short: 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.

Key takeaways

Three cups problem belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Three cups problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Three cups problem from memory before moving on to harder problems.

Reference excerpt

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.

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 {\displaystyle W} , the goal of the problem is to change W {\displaystyle W} from 1 to 0, i.e. by − 1 {\displaystyle -1} . The problem is insolvable because any move changes W {\displaystyle W} by an even number. Since a move inverts two cups and each inversion changes W {\displaystyle W} by + 1 {\displaystyle +1} (if the cup was the right way up) or − 1 {\displaystyle -1} (otherwise), a move changes W {\displaystyle 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 {\displaystyle W} to 0 if it is initially odd. On the other hand, if W {\displaystyle W} is even, inverting cups two at a time will eventually result in W {\displaystyle W} equaling 0.

References

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

See also List of impossible puzzles Puzzle Recreational mathematics

Illustrations

Worked examples

Example 1 — a first encounter with Three cups problem

Start with the simplest possible case. Write down what Three cups problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Three cups problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Three cups problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Three cups problem

In research: Three cups problem appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Three cups problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Three cups problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Magic tricks, Unsolvable puzzles, so understanding it makes those chapters shorter.
In everyday life: Look for Three cups problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Three cups problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Three cups problem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Three cups problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Three cups problem in simple terms?

Why does Three cups problem matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Three cups problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Three cups problem.