The biggest problem with a 120-sided die is not its size, or its weight, or even its price. The biggest problem with a 120-sided die is no one knows what to do with it, a fact not lost on the people who created it. “We were a little concerned to make this because it’s so expensive and there’s no real use for it,” says Robert Fathauer.

Fathauer is one half of Dice Lab, a small company in Phoenix that explores the wonder of polyhedra in dice form. The D120 is its most ambitious project yet, one that, frankly, makes absolutely zero sense but is awesome just the same.

Most specialty dice, of which Dice Lab offers six varieties, run three to five bucks apiece. The D120 costs $12, making it the Rolls-Royce of dice. More notable than its price is its mathematical improbability. All dice are polyhedra (Greek for many-sided), but the D120 is a special variety called disdyakis triacontahedron. It features 120 scalene triangular faces and 62 vertices. That creates the largest number of symmetrical faces possible for an icosahedron and the biggest, most complex fair dice possible. To be considered fair, a dice must be equally likely to land on any of its sides when you roll it.

This is not an original idea. We were just the people crazy enough to actually do it. Henry Segerman, mathematician and co-founder of Dice Lab

Creating the world’s most complex dice presents more than a few technical challenges, which helps explain its enduring appeal to mathematicians. “This is not an original idea,” says Henry Segerman, a mathematician at Oklahoma State University and co-founder of Dice Lab. “We were just the people crazy enough to actually do it.”

Size presents the first challenge. Anyone could make a disdyakis triacontahedron big enough to easily engrave all those numbers. But try using it. "It would be heavier and bigger and more expensive," Segerman says. At about 2 inches in diameter and 3 ounces in weight, the D120 is hefty, but still small enough to cause a few design headaches. Look at the numbers and you’ll notice a slight distortion of the triple-digit numerals where they squeeze into the sharp end of each facet. "You need to be aware of how close the digits are to the edge of the triangle," Segerman says. "You don’t want the numbers getting cut into when its rounding off the edges of the dice."

Positioning the numbers poses a bigger challenge. Most dice place the largest number opposite the smallest. On a six-sided dice, for example, you find the six opposite the one, the five opposite the two, and so on. This mitigates any chance of the dice rolling too high or too low should there be a distortion during manufacturing. The D120 follows suit, placing the number 120 opposite the number one. But knowing those positions does little to inform the placement of everything else, leaving the designers with what they call "tons and tons of choices."

"It’s about 1098, even with that restriction," Fathauer says. “That’s about one percent of a googol1, about 10 18 times the number of known atoms in the universe. It’s, like, a crazy big number.”

To ensure they created a numerically balanced dice, Fathauer and Segerman sought help from Bob Bosch, a mathematician at Oberlin University. Bosch specializes in operations research, a field that combines mathematics, computer science, and economics. More specifically, he focuses on optimization, or trying to perform a task as well as it can be performed. So he wrote a program to cycle through every potential number placement. "Now of course some tasks are easy and don't require us to undertake any sophisticated analysis, but others seem to be extremely difficult," he says. "I found the task of assigning numbers to the faces of Henry and Robert's 120-sided polyhedron to be extremely difficult but also enormously fun."

The die designers wanted every vertex sum—the sum of the numbers of each triangle that meets a common point—to equal certain ... well, it gets tricky, so I'll let Bosch explain it:

Their 120-sided polyhedron has 12 vertices where 10 triangles meet. Henry and Robert wanted the numbers on the 10 faces that surround a vertex of this type to add up to 605, which is 10 times 60.5 (the average of all the numbers from 1 to 120). In addition, the polyhedron has 20 vertices where 6 triangles meet. Henry and Robert wanted the numbers on the 6 faces that surround a vertex of this type to add up to 363, which is 6 times 60.5. Finally, the polyhedron has 30 vertices where 4 triangles meet. Here, they wanted the vertex sums to be 242, which is 4 times 60.5. Henry and Robert did not know (nor did I) if it was possible to construct a numbering satisfying all of these conditions.

Bosch started by feeding the data into a program that he hoped would yield equal numbering at all 62 vertices. It worked, but it wasn't perfect. Some of the vertices' sums were still off. After spending more than a month on the problem, he wrote a script that repeatedly selected a collection of contiguous faces at random while keeping all of the numbers on those faces as they were in the near-perfect numbering. The script focused on numbers in the remaining two vertices that remained off. "I set the script running, took my son to the movies, and when we returned, my computer had stopped," he says. "It had either crashed or had found a perfect numbering. Fortunately it was the latter."

While the dice undoubtedly is a mathematical feat, Segerman says it isn't much to look at. Unlike a dodecahedron, with its handsome polygonal faces, spikes punctuate the surface of the D120, making it look uneven despite its symmetry. Still, there is a beauty to it. "It’s growing on me," he says. The D120 lands with a thump when thrown and rattles along until wobbling to a stop. Pointless, yes, but Segerman says it’s essentially the most versatile dice on the market. "This is the dice you want to take with you to a desert island,” he says. Even if you'd have no idea what to do with it.

1. Correction 9:10 05/10/16: This story was updated to accurately reflect a quote