Following a 15-year collaborative effort, mathematician Eric Harshbarger and a loose network of collaborators have designed a set of five 60-sided dice containing numbers 1 through 300 with no repeats. Unveiled to establish absolute permutation fairness, the configuration ensures any subset of players rolling equal dice sizes achieves perfectly equal odds.
Over a dinner conversation at a gaming convention around 2010, board game designer James Ernest posed a deceptively straightforward query to his friend Eric Harshbarger, a mathematician at Auburn University in Alabama. Ernest wanted to know if a set of dice could be engineered so that any group of players—whether two participants or any larger number—could each grab a die, roll, and share an entirely equal statistical probability of securing the highest value to go first. Crucially, the system required an absence of ties or mandatory rerolls, a challenge that required untangling complex probability distributions across varying group sizes.
Harshbarger lacked an immediate solution, but the conceptual puzzle initiated a decade-and-a-half investigative trajectory alongside a loose network of collaborators. The preliminary hurdle involved ensuring distinct numerical assignments across every die to prevent deadlocks. However, the true mathematical hurdle lay in distributing those digits so that probability parity held true not just for a complete set of rolling participants, but for any arbitrary subset drawn from the bag. If three, four, or five individuals stepped up to play, each rolling a distinct die from the same collection, their mathematical likelihood of winning had to remain identical.
Early breakthroughs arrived when Harshbarger looped in his childhood friend Robert Ford, a mathematician at Dalton State College. Within weeks, the duo engineered a functional three-player solution relying on numbers 1 through 18 spread across three standard six-sided dice. Ford subsequently calculated a four-player framework utilizing four 12-sided dice entirely by hand. By 2012, Harshbarger began presenting these four-player sets at mathematics conferences, prompting extensive public interest and retail distribution through specialty outlets like Maths Gear in the U.K. and Math Art Fun in the U.S. During peak production phases, Harshbarger manually laser-etched blank 12-sided dice and inked every individual face by hand before mailing custom sets globally.
As the investigative framework expanded, the team realized the mechanical configurations governed far more than initial turn selection. The randomized permutations dictated the entire sequential turn order for all participants. According to Harshbarger, the probability of players finishing in any specific sequence—such as order A, B, C, D versus C, B, D, A—mirrored exact mathematical equilibrium. This phenomenon, designated by the researchers as permutation fairness, established a rigorous baseline for every subsequent iteration.
Overcoming Astronomical Combinatorial Limits via Software Engineering
Scaling the mathematical model from four players to five presented an exponential computational barrier. While the team understood a five-player same-shape configuration was theoretically possible, exploring the total solution space required evaluating more configurations than the total number of atoms in the observable universe, roughly estimated at 10 to the 128th power combinations. Harshbarger noted that brute-force computation remained entirely unfeasible, stating that even with a billion billion years and advanced artificial intelligence tools, a manual processing sweep was impossible.
Progress stalled for years as researchers hit recurring walls involving unwieldy die dimensions and manufacturing constraints. The primary operational objective required keeping individual dice small enough to comfortably fit in a human hand and remain viable for tabletop gaming. That limitation ruled out massive configurations such as theoretical 180-sided objects.
The breakthrough materialized in mid-2023 when Canadian software engineer Paul Meyer contacted Harshbarger via email. Meyer had independently analyzed patterns embedded within Harshbarger’s published four-player data and developed a specialized algorithm designed to exploit those underlying symmetries. To the surprise of the collaborators, Meyer’s computational script successfully identified a specific configuration of five 60-sided dice—known geometrically as hexecontahedrons—that satisfied every structural requirement without mathematical compromise.
In Plain English: The Clinical Takeaway
- Equitable Probability: The newly developed five-die set ensures that any combination of players rolling together maintains absolute statistical equality, eliminating bias based on player count.
- Beyond First Turns: The numerical distribution guarantees permutation fairness, meaning the complete order of gameplay sequences follows a strictly randomized distribution.
- Tangible Scale: Despite evaluating astronomical combinations exceeding universal atomic counts, the resulting physical dice are restricted to 60 sides to ensure practical usability and hand-held manufacturing.
Translating Abstract Mathematics into Permanent Architectural Display
Following verification of Meyer’s algorithm, the five-player set transitioned from theoretical data architecture into physical production. The timing coincided with Auburn University’s construction of a new facility to house its mathematics department, which sought sculptural installations to engage students and visitors with visual representations of advanced quantitative concepts.
To fulfill this objective, Harshbarger constructed five monumental wooden replicas of the 60-sided hexecontahedrons inside his personal workshop. Each oversized geometric sculpture was carved from a distinct timber variety, encompassing pine, poplar, oak, walnut, and mahogany. These large-scale artistic installations were permanently installed in Auburn’s new mathematics building, serving as a physical touchpoint intended to demonstrate that complex mathematical principles frequently manifest through tangible, accessible design. Harshbarger emphasized that the underlying motivation centers on engaging public curiosity, observing that compelling mathematical puzzles often emerge from simple foundational questions that resist easy resolution.
| Player Capacity | Die Geometry | Numerical Range | Primary Collaborators |
|---|---|---|---|
| 3 Players | Standard 6-sided dice | 1 through 18 | Eric Harshbarger, Robert Ford |
| 4 Players | 12-sided dice (dodecahedrons) | Custom Distribution | Eric Harshbarger, Robert Ford |
| 5 Players | 60-sided dice (hexecontahedrons) | 1 through 300 (no repeats) | Eric Harshbarger, Robert Ford, Paul Meyer |
