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The problem of finding in how many ways a plane convex polygon of sides can be divided into triangles by diagonals. Euler first proposed it to Christian Goldbach in 1751, and the solution is the Catalan number .

The volumes of any -dimensional solids can always be simultaneously bisected by a -dimensional hyperplane. Proving the theorem for (where it is known as the pancake theorem) is simple and can be found in Courant and Robbins (1978).The proof is more involved for (Hunter and Madachy 1975, p. 69), but an intuitive proof can be obtained by the following argument due to G. Beck (pers. comm., Feb. 18, 2005). Note that given any direction , the volume of a solid can be bisected by a plane with normal . To see this, start with a plane that has all of the solid on one side and move it parallel to itself until the solid is completely on its other side. There must have been an intermediate position where the plane bisected the solid.Now take a sphere centered at the origin large enough to contain the three solids. Each point on the surface of the sphere indicates a direction. For any direction and each solid, find a plane that bisects the solid with that..

Given three jugs with pints in the first, in the second, and in the third, obtain a desired amount in one of the vessels by completely filling up and/or emptying vessels into others. This problem can be solved with the aid of trilinear coordinates (Tweedie 1939).A variant of this problem asks to obtain a fixed quantity of liquid using only two initially empty buckets of capacities and and a well containing an inexhaustible supply of water.This two bucket variant is used in the film Die Hard: With a Vengeance (1995). The characters John McClane and Zeus Carver (played by Bruce Willis and Samuel L. Jackson) solve the two bucket variant with two jugs and water from a public fountain in order to try to prevent a bomb from exploding by obtaining 4 gallons of water using only 5-gallon and 3-gallon jugs.General problems of this type are sometimes collectively known as "decanting problems."..

Consider intersecting ellipses. The maximal number of regions into which these divide the plane aregiving values for , 2, ... of 2, 6, 14, 26, 42, 62, 86, 114, ... (OEIS A051890).

The maximum number of pieces into which a cylinder can be divided by oblique cuts is given by(1)(2)(3)where is a binomial coefficient.This problem is sometimes also called cake cutting or pie cutting, and has the same solution as space division by planes. For , 2, ... cuts, the maximum number of pieces is 2, 4, 8, 15, 26, 42, ... (OEIS A000125). Unsurprisingly, the numbers of this sequence are called cake numbers.

Consider intersecting circles. The maximal number of regions into which these divide the plane aregiving values for , 2, ... of 2, 4, 8, 14, 22, 32, 44, 58, ... (OEIS A014206).

The average number of regions into which randomly chosen planes divide a cube is(Finch 2003, p. 482).The maximum number of regions is presumably the same as for spacedivision by planes, namely(Yaglom and Yaglom 1987, pp. 102-106). For , 2, ... planes, this gives the values 2, 4, 8, 15, 26, 42, ... (OEIS A000125), a sequence whose values are sometimes called the "cake numbers" due to their relation to the cake cutting problem.

The number of regions into which space can be divided by mutually intersecting spheres isgiving 2, 4, 8, 16, 30, 52, 84, ... (OEIS A046127) for , 2, ....

Determining the maximum number of pieces in which it is possible to divide a circle for a given number of cuts is called the circle cutting or pancake cutting problem. The minimum number is always , where is the number of cuts, and it is always possible to obtain any number of pieces between the minimum and maximum. The first cut creates 2 regions, and the th cut creates new regions, so(1)(2)(3)Therefore,(4)(5)(6)(7)(8)Evaluating for , 2, ... gives 2, 4, 7, 11, 16, 22, ... (OEIS A000124). This is equivalent to the maximal number of regions into which a plane can be cut by lines.

The maximal number of regions into which space can be divided by planes is(Yaglom and Yaglom 1987, pp. 102-106). For , 2, ..., these give the values 2, 4, 8, 15, 26, 42, ... (OEIS A000125), a sequence whose values are sometimes called the "cake numbers" due to their relation to the cake cutting problem. This is the same solution as for cylinder cutting.

If a circular pizza is divided into 8, 12, 16, ... slices by making cuts at equal angles from an arbitrary point, then the sums of the areas of alternate slices are equal.There is also a second pizza theorem. This one gives the volume of a pizza of thickness and radius :

The maximum number of regions that can be created by cuts using space division by planes, cube division by planes, cylinder cutting, etc., is given by(Yaglom and Yaglom 1987, pp. 102-106). For , 2, ... planes, this gives the values 2, 4, 8, 15, 26, 42, ... (OEIS A000125), a sequence whose values are sometimes called the cake numbers.

It is always possible to "fairly" divide a cake among people using only vertical cuts. Furthermore, it is possible to cut and divide a cake such that each person believes that everyone has received of the cake according to his own measure (Steinhaus 1999, pp. 65-71). Finally, if there is some piece on which two people disagree, then there is a way of partitioning and dividing a cake such that each participant believes that he has obtained more than of the cake according to his own measure.There are also similar methods of dividing collections of individually indivisible objects among two or more people when cash payments are used to even up the final division (Steinhaus 1999, pp. 67-68).Ignoring the height of the cake, the cake-cutting problem is really a question of fairly dividing a circle into equal area pieces using cuts in its plane. One method of proving fair cake cutting to always be possible relies on the Frobenius-König..

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