Can the Dice be Fair by Dynamics?

We consider the dynamics of the three-dimensional model of the die which can bounce with dissipation on the table. It is shown that for the realistic values of the initial energy the probabilities of the die landing on the face which is the lowest one at the beginning is larger than the probabilities of landing on any other face.

[1]  I. Neĭmark,et al.  Dynamics of Nonholonomic Systems , 1972 .

[2]  Zhang Kechen Uniform distribution of initial states: The physical basis of probability. , 1990 .

[3]  Yue Zeng-yuan,et al.  On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process , 1985 .

[4]  Kapitaniak Uncertainty in coupled chaotic systems: Locally intermingled basins of attraction. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  Murray,et al.  Probability of a tossed coin landing on edge. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  Feldberg,et al.  Iterated-map approach to die tossing. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[7]  Tomasz Kapitaniak,et al.  Bifurcations from locally to globally riddled basins , 1998 .

[8]  D. Frail,et al.  Identification of PSR1758 – 23 as a runaway pulsar from the supernova remnant W28 , 1993, Nature.

[9]  Vulovic,et al.  Randomness of a true coin toss. , 1986, Physical review. A, General physics.

[10]  杨骁,et al.  On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process , 1985 .

[11]  P. Richter,et al.  How random is dice tossing? , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Joseph Ford,et al.  How random is a coin toss , 1983 .

[13]  Tsuyoshi Mizuguchi,et al.  Dynamics of Coin Tossing , 2006 .

[14]  T. Kapitaniak,et al.  Dynamics of coin tossing is predictable , 2008 .

[15]  J. Keller The Probability of Heads , 1986 .

[16]  J. W. Humberston Classical mechanics , 1980, Nature.

[17]  Coin tossing as a billiard problem. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  Brian Mirtich,et al.  Fast and Accurate Computation of Polyhedral Mass Properties , 1996, J. Graphics, GPU, & Game Tools.

[19]  T. Mckeown Mechanics , 1970, The Mathematics of Fluid Flow Through Porous Media.

[20]  J. Yorke,et al.  Final state sensitivity: An obstruction to predictability , 1983 .

[21]  J. Yorke,et al.  Fractal basin boundaries , 1985 .

[22]  Joseph B. Keller,et al.  Fair dice , 1989 .

[23]  Persi Diaconis,et al.  c ○ 2007 Society for Industrial and Applied Mathematics Dynamical Bias in the Coin Toss ∗ , 2022 .

[24]  J. Yorke,et al.  The transition to chaotic attractors with riddled basins , 1994 .