The last house cannot be green. By fact 5, remove green as an option for the center house. This house also cannot be white, due to fact 4, leaving the colour red. This also leads us to conclude that the second and last houses are green and white, respectively.

Continuing with fact 3, we remove the Dane as an option for the green house. Also remove tea from the first house. Fact 2 says the Swede keeps dogs, so remove d for dogs in all houses which doesn't have S wede as an option. Fact 7 says the Norwegian smokes Dunhill, and fact 10 says the next house keeps horses. Fact 12 says the one smoking Blue Master drinks beer, so remove b from all houses that don't match — therefore the Norwegian must drink water. We will also remove b for Blue Master in houses that doesn't have a beer option.

As you can see, this puzzle is quite trivial to solve once we have a good approach. Just keep chipping away at the problem, removing possible values for each cell. Fact 15 means that the blend l is smoked in the second house, so update the table according to this.

### See a Problem?

Fact 6 says the first house cannot have a bird option, as Dunhill is smoked there. Fact 13 forces us to remove the German option for the second house, since blend is smoked there. Fact 3 again gives us tea for this house, and this leaves beer for the last house.

Now everything falls into place. We will now skip all the intermediate steps and just show you the final table. Einstein's problem and a solution by elimination By Christian Stigen Larsen 01 Jan The Zebra Puzzle is a famous puzzle that has been said to have been invented by Einstein. The Puzzle There are five houses in unique colors: Blue, green, red, white and yellow.

## Spherically symmetric static solutions in the Einstein-Cartan theory

Each person drinks a unique beverage: Beer, coffee, milk, tea and water. The Schwarzschild metric is the unique external solution for a spherically symmetric body in a surrounding empty space. This suggests that General Relativity shares with Newtonian gravity the property that the external field of any spherical body depends only on its total mass and not on the radial distribution of the matter. For historical information see Eisenstaedt and for a general discussion of global properties of spacetimes, including those discussed here, see Hawking and Ellis The Schwarzschild solution provided a pattern for later investigations of singularities and of black holes.

This solution's uniqueness shows that General Relativity does not admit monopolar gravitational waves. It is also the lowest order approximation to the field of real astronomical bodies such as the Earth and the Sun. Calculating geodesics in this field has enabled accurate predictions of light-bending by the Sun and the advance of the perihelion of Mercury, two of the "classical tests" of general relativity theory.

The Schwarzschild solution is a special case of the Kerr solution found in which represents the external field of a rotating black hole. This can be written as an instance of Eq. The Schwarzschild and Kerr solutions provide the background for studies of the physics in the field of black holes, which are used in modelling X-ray binary sources and active galactic nuclei in astronomy.

The Schwarzschild and Kerr black holes can be readily generalized to include non-zero electromagnetic charges and using Eq. There are uniqueness theorems showing with some technical caveats that these families are the unique stationary black holes with spherical topology of a non-singular event horizon.

These solutions give the geometry of the "standard model" in modern cosmology , and thus provide the background for an enormous number of papers studying cosmological physics, including perturbations of the solutions. These spacetimes are spherically symmetric and conformally flat, hence exemplifying both the major types of simplification of the Einstein equations, and must contain a energy-momentum tensor of perfect fluid type.

The first of Eq. Many explicit solutions are known. These spherically symmetric solutions are the solutions for Eq. They generalize the FLRW solutions for dust to inhomogeneous solutions. Since dust is believed to be an appropriate representation of the universe's matter content on the large scale at the present time, LTB solutions have been much used to provide exact models of structures in the universe see Bolejko et al The Einstein-Maxwell plane wave solutions were first found by Baldwin and Jeffrey in Plane waves admit a 5-dimensional isometry group containing an Abelian 3-dimensional subgroup acting on null hypersurfaces.

They are spacetime homogeneous i. These spacetimes provide an important example of unexpected global structure. The sandwich wave structure resolved the issue of whether the gravitational waves first found, using approximations, by Einstein could be merely coordinate effects: Bondi, Pirani and Robinson showed that free test particles are relatively accelerated by passage through the wave region, implying that the wave must carry energy.

Plane waves are the first approximation for gravitational radiation far from a source in an otherwise empty space.

They are a special case of the more general pp -waves, solutions with a covariantly constant null Killing vector reresenting plane-fronted gravitational waves with parallel rays and found in by Brinkman. This vacuum solution can be generalized to non-vacuum cases. Taub-NUT spacetime has very unexpected global properties.

The NUT region contains closed timelike lines and no sensible Cauchy surfaces, there are two inequivalent maximal analytic extensions of the Taub region or one non-Hausdorff manifold with both extensions , the spacetime is nonsingular in the sense of a curvature singularity, and there are geodesics of finite affine parameter length. These properties gave rise to the title of Misner's paper some of these properties are shared by the other Taub-NUT metrics.

The solution had a great influence on studies of exact solutions and cosmological models which are spatially-homogeneous, and more generally on those which are hypersurface-homogeneous and self-similar, on cosmology in general, and on our understanding of global analysis and singularities in space-times. Belinski, V A and Verdaguer, E Gravitational solitons. Cambridge University Press, Cambridge. Selected solutions of Einstein's field equations: their role in general relativity and astrophysics. In Einstein's field equations and their physical implications.

Notes Phys. Springer, Heidelberg. B G Schmidt. Structures in the Universe by exact methods: formation, evolution, interactions. Gravitational waves in general relativity. Exact plane waves. A Eisenstaedt, J Histoire et singularities de la solution de Schwarzschild: Exact Sci.

## Quote by Albert Einstein: “If I had an hour to solve a problem I'd spend ”

Exact scalar field cosmologies. Springer, Heidelberg. B G Schmidt. Structures in the Universe by exact methods: formation, evolution, interactions.

Gravitational waves in general relativity. Exact plane waves. A Eisenstaedt, J Histoire et singularities de la solution de Schwarzschild: Exact Sci. Exact scalar field cosmologies. Griffiths, J B Colliding plane waves in general relativity. Oxford mathematical monographs. Oxford University Press, Oxford. Exact space-times in Einstein's general relativity. The large scale structure of space-time. Inhomogeneous cosmological models.

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Misner, C W Taub-NUT space as a counterexample to almost anything. In Relativity theory and astrophysics, vol. Ehlers, pp. Olver, P J Applications of Lie groups to differential equations. Springer-Verlag, Heidelberg. Penrose, R A remarkable property of plane waves in general relativity. Generally covariant integral formulation of Einstein's field equations.

### The Puzzle

Physical Review A Stephani, H Differential equations — Their solutions using symmetries. Exact solutions of Einstein's field equations, 2nd edition. Synge, J L Relativity: the general theory. North-Holland, Dordrecht. MacCallum , Scholarpedia, 8 12 Jump to: navigation , search. Post-publication activity Curator: Malcolm A. MacCallum Contributors:. Jerry Griffiths. Olivier Minazzoli. Riccardo Guida. Sponsored by: Dr.

Categories : Scholarpedia Physics Space-time and gravitation General relativity. Namespaces Page Discussion. Views Read View source View history. Contents 1 Summary 2 Einstein's equations 3 Making the equations tractable 3. Reviewed by : Dr.