The skwedge has been independently discovered by mathematicians, puzzle designers, linguists, and computer scientists across four centuries. Here are three of those stories.
In the early 1660s, a shipwright brought a shape to the attention of Sir Robert Moray, one of the founding members of England's new Royal Society. The shape was practical: it showed up in how ship hulls curved from a round cross-section at the middle of the boat to a straight edge at the bow. The shipwright wanted to know its volume. Moray forwarded the problem to the best geometer he knew.
That geometer was John Wallis. Wallis solved it. The result was a short paper he sent back to Moray as a letter, published in 1684. He titled it Cono-cuneus, or, The Shipwright's Circular Wedge. The volume he proved, (π/2)R²h, has not changed in the 340 years since. It is Skwedge curvum, Species I.
Wallis came to mathematics late. He was trained as a theologian and spent the English Civil War deciphering intercepted enemy letters for Parliament. He was so good at it that he became indispensable to whoever was in power, surviving the transition from Cromwell to Charles II without losing his Oxford chair.
He took up mathematics seriously at 31, after stumbling across a textbook. Within a few years he was producing work that Newton later credited as foundational. He introduced the symbol ∞ for infinity. He developed methods for computing areas under curves that became part of the toolkit of calculus. He helped found the Royal Society, the organization that would become the model for every national academy of science that followed.
The conocuneus paper was a small piece of a large life. But it is the one that connects directly to what you are teaching. When Wallis proved (π/2)R²h, he was answering a practical question from a craftsman. The math served the work. That connection is worth naming in a classroom.
In the 1660s there was no formula for the volume of a curved solid unless someone had worked it out specifically. You could measure a box. You could calculate a cylinder. But a shape that curved from a circle to a line? That required methods that were new at the time, and Wallis was one of a handful of people in the world who had them.
The key was Cavalieri's principle, developed by the Italian mathematician Bonaventura Cavalieri a few decades earlier: if you can describe the cross-sectional area of a solid at every height, you can compute its volume by adding all those slices up. This is the same idea that eventually became the integral in calculus. Wallis knew how to apply it. The shipwright did not. That is why the problem traveled from a dockyard to Oxford.
Here is a question that sounds simple: given the same circle at the bottom and the same line at the top, how do you fill in the space between them?
The surprising answer is: there is more than one correct way. Not approximately different. Exactly different. The same boundary, three geometrically distinct interiors, volumes that differ by up to 13.13%. This is not a mistake. It is a feature of the problem, and it has a name: the problem is ill-posed.
Hadamard spent his career trying to understand when a mathematical problem had a trustworthy answer. He noticed that some problems were solid: give them slightly different inputs and they produce slightly different outputs, in a predictable way. Other problems were treacherous: tiny changes in the inputs caused wildly unpredictable outputs, or produced no answer at all, or produced too many answers to choose between.
In 1902, he gave these ideas formal names. A problem is well-posed if it has exactly one solution and that solution behaves predictably. A problem is ill-posed if it has no solution, more than one solution, or a solution that is wildly sensitive to small changes in the inputs.
Hadamard originally thought ill-posed problems were physically meaningless: if a math problem has more than one answer, it probably doesn't correspond to anything real. History proved him wrong. Ill-posed problems turn out to be everywhere: in medical imaging, geophysics, archaeology, and, as it turns out, in the geometry of shapes that have been in human hands for two million years.
Imagine someone asks you: "I have a rectangle with a perimeter of 20. What is its area?"
You cannot answer. A 1×9 rectangle has perimeter 20 and area 9. A 4×6 has perimeter 20 and area 24. A 5×5 has perimeter 20 and area 25. Same perimeter, completely different areas. The problem is ill-posed because the answer is not unique.
Now imagine someone asks: "I have a square with a perimeter of 20. What is its area?" That has exactly one answer: 25. That problem is well-posed.
The skwedge boundary problem is like the first question. Given a circle at the bottom and a line at the top, the interior is not determined. At least three geometrically distinct, mathematically valid solids share the same boundary. The problem is ill-posed, and the gap between the answers is 13.13%.
Why does this matter in a classroom? Most geometry problems students encounter are well-posed by design. The skwedge is one of the rare cases where a visually simple question genuinely has multiple correct answers, each provably different from the others. It demonstrates that "not enough information" is a real mathematical condition, not just a test-taking excuse.
In August 1958, a puzzle appeared in Scientific American under the heading "The Cork Plug." The challenge: can you carve a single solid that fits snugly through a circular hole, a square hole, and a triangular hole? All three. One object.
Gardner gave two versions. The 2026 papers establish that the larger of his two versions is exactly Skwedge projectivum: not just a similar shape, the same shape, with the same cross-sections, the same shadows, and the same volume formula. Many 3D printer makers have been printing the "cork plug" or ptsi·nɬ for years without any need for the species name projectivum. Wallis found a sister shape, the curvum, while trying to figure out the volume of a ship's hull. The 2026 taxonomy unified these constructions under one family for the first time.
Gardner had no graduate training in mathematics. He studied philosophy, worked as a journalist, and edited a children's magazine before pitching an article on paper-folding puzzles to Scientific American in 1956. The editors liked it enough to offer him a monthly column. He wrote it for 25 years.
His method was to find ideas at the intersection of play and rigor: problems that looked like puzzles but pointed toward genuine mathematics. The cork plug is a perfect example. It appears as a craftwork challenge: what shape would you need to carve? The answer is a solid with three distinct cross-sections, each matching one of the holes. What Gardner described as a puzzle, a geometer would call the projective maximum of the circle-to-line loft family.
Gardner never took a calculus course. As mathematician Keith Devlin noted, Gardner had no PhD and never wrote a mathematical paper, yet the world's best mathematicians corresponded with him regularly and regarded him as a colleague. His columns introduced Conway's Game of Life, Penrose tiles, RSA cryptography, and the art of M.C. Escher to readers who would not otherwise have found them. The cork plug sits quietly in the same list.
Gardner was not the first to pose the three-hole puzzle. The earliest known printed version appears in Wilhelm Catel's 1790 German puzzle book, 168 years before Gardner gave it wide attention in Scientific American.
Gardner was not the first to pose the three-hole puzzle. The earliest known printed version appears in Peter Friedrich Catel's 1790 German toy catalog, Mathematisches und physikalisches Kunst-Cabinet (p. 16, Fig. 42).
Catel sold a plum-wood puzzle board with three holes, square, round, and triangular, and challenged buyers to find the single solid that passes through all three and fills each one completely. He suggested cutting it from "Brod, Käse, Kork oder Holz" (bread, cheese, cork, or wood).
"Die mathematischen Löcher (Fig. 42.) bestehet aus einem von Pflaumbaumholz verfertigten Brette, 9 Zoll lang und 2½ Zoll breit, worin ein viereckiges, ein rundes und ein dreieckiges Loch sind. Die Aufgabe davon ist: daß man die Figur angeben solle, welche durch alle 3 Löcher gehen könne, und doch solche vollkommen verstopfe oder ausfülle. Man kann solche von Brod, Käse, Kork oder Holz schneiden lassen."
The Mathematical Holes: a board with a square, round, and triangular hole. The puzzle: find the figure that passes through all three and completely fills each one. One may cut it from bread, cheese, cork, or wood.
Gardner's 1958 "cork plug" echoed Catel's own material list, 168 years later.