physics

Slothouber–Graatsma puzzle

Slothouber–Graatsma puzzle is a physics 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 Slothouber–Graatsma puzzle rather than just read about it. In short: The Slothouber–Graatsma puzzle is a packing problem that calls for packing six 1 × 2 × 2 blocks and three 1 × 1 × 1 blocks into a 3 × 3 × 3 box (all shapes being right angled). The solution to this puzzle is unique (up to mirror reflections and rotations).

Key takeaways

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

Reference excerpt

The Slothouber–Graatsma puzzle is a packing problem that calls for packing six 1 × 2 × 2 blocks and three 1 × 1 × 1 blocks into a 3 × 3 × 3 box (all shapes being right angled). The solution to this puzzle is unique (up to mirror reflections and rotations). It was named after its inventors Jan Slothouber and William Graatsma. The puzzle is essentially the same if the three 1 × 1 × 1 blocks are left out, so that the task is to pack six 1 × 2 × 2 blocks into a cubic box with volume 27.

Solution

The solution of the Slothouber–Graatsma puzzle is straightforward when one realizes that the three 1 × 1 × 1 blocks (or the three holes) need to be placed along a body diagonal of the box, as each of the 3 × 3 layers in the various directions needs to contain such a unit block. This follows from parity considerations, because the larger blocks can only fill an even number of the nine cells in each 3 × 3 layer.

Variations The Slothouber–Graatsma puzzle is an example of a cube-packing puzzle using convex polycubes. More general puzzles involving the packing of convex rectangular blocks exist. The best known example is the Conway puzzle which asks for the packing of eighteen convex rectangular blocks into a 5 × 5 × 5 box. Another 5 × 5 × 5 puzzle attributed to Conway includes six 1 × 2 × 4 and six 2 × 2 × 3 blocks with a solution similar to the Slothouber-Graatsma puzzle. A harder convex rectangular block packing problem is to pack forty-one 1 × 2 × 4 blocks into a 7 × 7 × 7 box (thereby leaving fifteen unit holes); the solution is analogous to the 5 × 5 × 5 case, and has three 1 × 1 × 5 cuboidal holes in mutually perpendicular directions covering all seven slices.

See also Soma cube Bedlam cube Diabolical cube Herzberger Quader (right cuboid)

References

External links The Slothouber-Graatsma puzzle in Stewart Coffin's "The Puzzling World of Polyhedral Dissections" Jan Slothouber and William Graatsma: Cubic constructs William Graatsma and Jan Slothouber: Dutch mathematical art

Illustrations

Worked examples

Example 1 — a first encounter with Slothouber–Graatsma puzzle

Start with the simplest possible case. Write down what Slothouber–Graatsma puzzle claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, 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 Slothouber–Graatsma puzzle 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 Slothouber–Graatsma puzzle 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 Slothouber–Graatsma puzzle

In research: Slothouber–Graatsma puzzle appears in physics 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 Slothouber–Graatsma puzzle 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: Slothouber–Graatsma puzzle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mechanical puzzle cubes, Packing problems, Tiling puzzles, so understanding it makes those chapters shorter.
In everyday life: Look for Slothouber–Graatsma puzzle 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 Slothouber–Graatsma puzzle in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Slothouber–Graatsma puzzle 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 Slothouber–Graatsma puzzle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Slothouber–Graatsma puzzle in simple terms?

Why does Slothouber–Graatsma puzzle matter?

Because it connects several physics 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 Slothouber–Graatsma puzzle?

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 Slothouber–Graatsma puzzle.