, Volume 77, Issue 2, pp 247–256 | Cite as

Cuboidal dice and Gibbs distributions

  • Wolfgang Riemer
  • Dietrich Stoyan
  • Danail Obreschkow


What are the face-probabilities of a cuboidal die, i.e. a die with different side-lengths? This paper introduces a model for these probabilities based on a Gibbs distribution. Experimental data produced in this work and drawn from the literature support the Gibbs model. The experiments also reveal that the physical conditions, such as the quality of the surface onto which the dice are dropped, can affect the face-probabilities. In the Gibbs model, those variations are condensed in a single parameter, adjustable to the physical conditions.


Cuboidal dice Asymmetric dice Tossing Gibbs distribution Boltzmann distribution 



The authors thank Robert Allin for valuable discussions about an earlier version of this paper. D.O. acknowledges the discussions with Nick Jones.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Wolfgang Riemer
    • 1
  • Dietrich Stoyan
    • 2
  • Danail Obreschkow
    • 3
  1. 1.Zentrum für Lehrerbildung KölnPulheimGermany
  2. 2.Institut für StochastikFreibergGermany
  3. 3.International Centre for Radio Astronomy Research, M468University of Western AustraliaCrawleyAustralia

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