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We prove several results addressing the envy-free division problem in the presence of an unpredictable (secretive) player, called the "dragon". There are two basic scenarios. 1. There are $r-1$ players and a dragon. Once the "cake" is…
Ice states, in which frustrated interactions lead to a macroscopic ground-state degeneracy, occur in water ice, in problems of frustrated charge order on the pyrochlore lattice, and in the family of rare-earth magnets collectively known as…
Are dreidels fair? In other words, does the average dreidel have an equal chance of turning up any one of its four sides? To explore this hypothesis, three different dreidels were each spun hundreds of times with the number of occurrences…
Habitable planets are commonly imagined to be temperate planets like Earth, with areas of open ocean and warm land. In contrast, planets in snowball states, where oceans are entirely ice-covered, are believed to be inhospitable. However, we…
Artificial spin ices have transcended their origins in frustrated rare-earth pyrochlores to become a versatile platform for engineering exotic states of matter. Across diverse implementations, from nanomagnets and superconducting vortices…
We examine square and kagome artificial spin ice for colloids confined in arrays of double-well traps. Unlike magnetic artificial spin ices, colloidal and vortex artificial spin ice realizations allow creation of doping sites through double…
Gaia's exquisite parallax measurements allowed for the discovery and characterization of the Q branch in the Hertzsprung-Russell diagram, where massive C/O white dwarfs (WDs) pause their dimming due to energy released during…
We consider the following cake cutting game: Alice chooses a set P of n points in the square (cake) [0,1]^2, where (0,0) is in P; Bob cuts out n axis-parallel rectangles with disjoint interiors, each of them having a point of P as the lower…
The possible existence of stable black holes with entropies larger than the corresponding Schwarzschild black hole has been discussed extensively. The recently proposed "rough" black holes provide a concrete example of this. The fear is…
The third law of thermodynamics dictates that the entropy of a system in thermal equilibrium goes to zero as its temperature approaches absolute zero. In ice, however, a "zero point" or residual entropy can be measured - attributable to a…
A frustrated system is one whose symmetry precludes the possibility that every pairwise interaction (``bond'') in the system can be satisfied at the same time. Such systems are common in all areas of physical and biological science. In the…
The Good is Blondie, a wandering gunman with a strong personal sense of honor. The Bad is Angel Eyes, a sadistic hitman who always hits his mark. The Ugly is Tuco, a Mexican bandit who's always only looking out for himself. Against the…
Artificial particle ices are model systems of constrained, interacting particles. They have been introduced theoretically to study ice-manifolds emergent from frustration, along with domain wall and grain boundary dynamics, doping,…
Ordinary ice has a proton-disordered phase which is kinetically metastable, unable to reach, spontaneously, the ferroelectric (FE) ground state at low temperature where a residual Pauling entropy persists. Upon light doping with KOH at low…
White Dwarfs (WD) capture Dark Matter (DM) as they orbit within their host halos. These captured particles may subsequently annihilate, heating the stellar core and preventing the WD from cooling. The potential wells of WDs are considerably…
The conjecture that dark matter in galaxies is mostly cold fractal gas is developed in the more general context of the thermodynamics of the ideal isothermal gas subject to gravitational instability. This simple gas model already contains…
The low temperature entropy of the the spin ice compounds, such as Ho$_2$Ti$_2$O$_7$ and Dy$_2$Ti$_2$O$_7$, is well described by the nearest-neighbor antiferromagnetic Ising model on the pyrochlore lattice, i.e.\ by the ``ice rules''. This…
Flyby images of (486958) Arrokoth (Ultima Thule, 2014 MU$_{69}$) show a comparatively bright "necklace" in the neck, or cleft between its two lobes, in contrast to its generally low albedo. We suggest that the necklace may be the result of…
Water is the most familiar liquid, and arguably the most complex. Anomalies of supercooled water have been measured during decades, and competing interpretations proposed. Yet, a decisive experiment remains elusive, because of unavoidable…
A remarkable variety of organisms and wet materials are able to endure temperatures far below the freezing point of bulk water. Cryo-tolerance in biology is usually attributed to "anti-freeze" proteins, and yet massive supercooling ($<…