science

Levmore–Cook moving-knives procedure

Levmore–Cook moving-knives procedure 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 Levmore–Cook moving-knives procedure rather than just read about it. In short: The Levmore–Cook moving-knives procedure is a procedure for envy-free cake-cutting among three partners. It is named after Saul X.

Key takeaways

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

Reference excerpt

The Levmore–Cook moving-knives procedure is a procedure for envy-free cake-cutting among three partners. It is named after Saul X. Levmore and Elizabeth Early Cook who presented it in 1981. It assumes that the cake is two-dimensional. It requires a referee, two knives and four cuts, so some partners may receive disconnected pieces.

Procedure We name the partners Alice, Bob and Carl. Initially, Alice cuts the cake to three pieces equal in her eyes. Bob and Carl each point to their favorite piece. Easy case: Bob and Carl point to different pieces. Each receives his favorite piece and Alice the remaining piece. Hard case: Bob and Carl point to the same piece. Say this is piece X and the other pieces are Y and Z. Now the referee (ideally it could be Alice) takes one knife and moves it horizontally over piece X, and Alice takes the other knife and moves it vertically over piece X:

Knife #1 is moved horizontally and continuously by the referee from the left of piece X to its right. It divides piece X to two pieces: the left piece XL and the right piece XR. Knife #2 is moved vertically and simultaneously by Alice, to the left of Knife #1, such that XL is divided to two equal pieces in her eyes: the left-top XLT and the left-bottom XLB. Initially XR=X, so for Bob and Carl it is bigger than Y and Z. Moreover, Initially XLT and XLB are empty so XR is bigger than the two pairs: Y+XLT and Z+XLB. As Knife #1 moves rightwards, XR shrinks while XLT and XLB grows. At some point, either Bob or Carl thinks that XR equals one of the two pairs. The first one that thinks there is equality, shouts "stop!" and receives his chosen pair, either Y+XLT or Z+XLB. Alice receives the other pair, and the non-shouter receives XR.

Analysis We analyze the case when Bob shouted "stop!" and picked the pair Y+XLT. Alice gets Z+XLB and Carl gets XR. The division is envy-free because:

For Alice, Z>X>XR so Alice does not envy Carl. Moreover, Z=Y and XLB=XLT so Alice does not envy Bob. For Bob, Y+XLT=XR>Z+XLB, so Bob does not envy. For Carl, XR is larger than both pairs (since he did not shout) so he does not envy. The other cases are analogous.

Variants It is possible to let the shouter choose one of the four pairs: Y+XLT, Y+XLB, Z+XLT, Z+XLB. This modification favors the non-shouter, since the shouter will typically shout "stop" sooner. Levmore and Cook presented a generalization of their procedure for 4 partners. However, it was later shown that this generalization does not work in all cases.

See also The Stromquist moving-knives procedure uses four knives, but only two of them should cut, so each partner receives a connected piece.

References

Worked examples

Example 1 — a first encounter with Levmore–Cook moving-knives procedure

Start with the simplest possible case. Write down what Levmore–Cook moving-knives procedure 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 Levmore–Cook moving-knives procedure 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 Levmore–Cook moving-knives procedure 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 Levmore–Cook moving-knives procedure

In research: Levmore–Cook moving-knives procedure 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 Levmore–Cook moving-knives procedure 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: Levmore–Cook moving-knives procedure is common in secondary-school and first-year university syllabi. It links to neighbouring topics Cake-cutting, Fair division protocols, so understanding it makes those chapters shorter.
In everyday life: Look for Levmore–Cook moving-knives procedure 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Levmore–Cook moving-knives procedure” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Levmore–Cook moving-knives procedure in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Levmore–Cook moving-knives procedure 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 Levmore–Cook moving-knives procedure out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Levmore–Cook moving-knives procedure in simple terms?

The Levmore–Cook moving-knives procedure is a procedure for envy-free cake-cutting among three partners. It is named after Saul X.

Why does Levmore–Cook moving-knives procedure 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 Levmore–Cook moving-knives procedure?

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 Levmore–Cook moving-knives procedure.