Geometry problem solved after more than three centuries that began as a royal bet

A centuries-old gambling story about cubes and curiosity has sparked a modern mathematical upset: two Austrian researchers have built a shape that refuses the famous “Rupert” trick. What began as a bet in a royal court has evolved into a deep geometric puzzle tackled with computers, theorems and a surprising new object called the Noperthedron.

The idea is delightfully simple to picture: could one solid, turned just right and bored through, allow another identical solid to pass through it? Over the last 300 years, mathematicians have found surprisingly many shapes that admit such a passage. Now, for the first time, researchers have proven a shape that does not—ending a streak of shapes that seemed, until recently, to always find a way through themselves.

How a royal dare turned into a long-running math question

In the 17th century a wager between Prince Rupert of the Rhine and the noted English mathematician John Wallis launched what would become a celebrated geometric curiosity. Prince Rupert, an accomplished metallurgist and glassmaker, reportedly demonstrated that a cube could be tilted and hollowed so another cube of the same size could slide through it. Wallis later expressed the result in mathematical terms, and the phenomenon became known as the Rupert property or the Rupert tunnel.

That playful exchange—perhaps originating from the sort of gambling parlor experiments common in the age—left mathematicians with a concrete question: which solids allow a congruent copy to pass through a hole carved inside them, and which do not?

What mathematicians mean by the Rupert property

At its core, the Rupert property asks whether a solid S admits a cut and a reorientation such that another copy of S can be moved through the cut without intersecting the interior of the first. In practice, researchers reduce the three-dimensional question to a two-dimensional test using shadows.

• Cast the solid’s silhouette in the best possible orientation—the shadow that covers the largest area.
• Check whether the silhouette of a congruent copy can be placed entirely inside that shadow.
• If such an orientation and cut exist, the solid is said to possess the Rupert property.

This method allows computer programs and human researchers alike to search configurations quickly. Over time, the list of Rupert solids grew: besides cubes, mathematicians and hobbyists found Rupert tunnels in tetrahedra, octahedra, and many other polyhedra. By 2017 researchers had extended the result to both the dodecahedron and the icosahedron—shapes familiar to soccer fans and tabletop gamers alike.

Progress through history: from Prince Rupert to modern proofs

Discoveries have come in fits and starts across centuries:

1. 17th century: Prince Rupert’s cube demonstration and John Wallis’s formal statement of the property.
2. 1968 onwards: mathematicians and enthusiasts generalize Rupert tunnels to other regular polyhedra.
3. 2017: proofs appear showing both the dodecahedron and icosahedron admit Rupert tunnels.
4. Recent years: computational searches produce millions of candidate shapes and tunnels, revealing extremely tight fits in some cases—sometimes only tiny fractions of a percent larger than the space through which they pass.

Even with high-powered search algorithms, finding shapes that lack the Rupert property seemed elusive. Computer-generated families of shapes nearly always revealed a narrow passage—prompting speculation that the Rupert property might be ubiquitous among “reasonable” convex solids.

The Noperthedron: a deliberately unaccommodating shape

Jakob Steininger and Sergey Yurkevich—two friends from Austria—set out to test whether a true counterexample could exist. Using a mix of geometric insight and computational rigor, they designed a polyhedron they dubbed the Noperthedron—a playful name that fuses “Rupert” with a clear refusal.

The Noperthedron is not a simple block or platonic solid. Its surface is built from many polygonal facets arranged with careful asymmetry so that no orientation and cut allow a congruent copy to pass through. In technical terms, the object was constructed from a complex assembly of triangular faces paired with a couple of larger polygonal facets, designed to eliminate every possible Rupert tunnel.

• Designed structure: a dense tiling of triangular faces plus two large regular polygonal faces.
• Intended effect: remove any orientation where the silhouette could contain a congruent silhouette.
• Result: the shape does not admit a Rupert tunnel—making it a genuine counterexample.

The geometric ingenuity lies in arranging faces so that the cross-sectional shadows never line up in a way that would allow passage—no matter how one rotates or shifts a second copy.

How they proved it: theorems, shadows and massive computation

Proving that a shape cannot have a Rupert tunnel is more demanding than finding one. Steininger and Yurkevich developed two complementary theorems—one global and one local—that break the problem into manageable pieces. The global theorem partitions the possible orientations into regions; the local theorem analyzes each region’s behavior down to its boundary points. Together they let a computer program exclude entire classes of placements without checking every conceivable micro-adjustment.

Their computational pipeline then performed an exhaustive search across these regions. The program subdivided the configuration space into millions of small “blocks” and verified that none of them admitted a valid passage. According to the researchers, the final run examined on the order of tens of millions of such blocks to rule out any potential tunnel.

Tom Murphy, a software engineer who has previously explored thousands of shapes in search of Rupert tunnels, noted that shapes that truly deny the Rupert property are exceptionally rare. Many candidate solids that look unpromising still succumb to a narrow tunnel when a program searches thoroughly. That rarity makes the Noperthedron both surprising and significant.

Key steps in their method

• Translate the 3D passage problem into a family of 2D silhouette containment problems.
• Prove theorems that isolate critical points and reduce the search domain.
• Run a high-resolution computational sweep, subdividing orientation space into millions of test cells.
• Confirm there is no valid placement for a congruent copy to pass through.

Because the proof couples rigorous theorems with an exhaustive computational verification, it meets modern standards for certifying geometric nonexistence statements.

What this result changes—and what the researchers say next

By producing a verified solid without the Rupert property, Steininger and Yurkevich have shown that the centuries-old puzzle has a negative answer in some cases: not every convex polyhedron will admit a same-sized copy to pass through it. Their accomplishment turns a long sequence of positive examples into a more nuanced picture where both behaviors occur.

The researchers describe themselves as amateurs in the sense that their work began as fascination in spare time, yet their methods were rigorous and their result definitive. They intend to keep exploring geometric curiosities—searching for new counterexamples, tighter bounds, and fresh questions that mix classical geometry with modern computation.

Animations and demonstrations of many Rupert shapes exist online; for readers who enjoy visual puzzles, watching these tunnels in action helps make the idea tangible.

You might also like:

• Lemon-shaped exoplanet challenges planet formation models
• Uneaten mystery: every fig hides this surprising insect inside its sweet flesh
• Diamond rain on lemon-shaped planet: unusual weather turns precipitation into diamonds
• Archimedes manuscript lost page found in museum reveals hidden text under illustration
• Venus lava tubes may exist: what it means for geology and exploration

Michael Thompson

Michael Thompson is an experienced journalist covering U.S. and global news. With ten years on the front lines, he breaks down political and economic stories that matter. His precise writing and keen attention to detail help you grasp the real‑world impact of every event.