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2. Missing square puzzle - Wikipedia

   en.wikipedia.org/wiki/Missing_square_puzzle

   The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If θ is the angle between two opposing sides in each quadrilateral, then the ratio of the two areas is given by sec 2 θ. For θ = 5°, this is approximately 1.00765, which corresponds to a difference of about 0.8%.

3. Fair cake-cutting - Wikipedia

   en.wikipedia.org/wiki/Fair_cake-cutting

   For two cakes, they prove that an EF allocation may not exist when there are 2 agents and each cake is cut into 2 pieces. However, an EF allocation exists when there are 2 agents and one cake is cut into 3 pieces (the least-wanted piece is discarded), or when there are 3 agents and each cake is cut into 2 pieces (one agent is ignored; the ...

4. Tarski's axioms - Wikipedia

   en.wikipedia.org/wiki/Tarski's_axioms

   Continuity: φ and ψ divide the ray into two halves and the axiom asserts the existence of a point b dividing those two halves Axiom schema of Continuity. Let φ(x) and ψ(y) be first-order formulae containing no free instances of either a or b. Let there also be no free instances of x in ψ(y) or of y in φ(x). Then all instances of the ...

5. Divide and choose - Wikipedia

   en.wikipedia.org/wiki/Divide_and_choose

   Divide and choose (also Cut and choose or I cut, you choose) is a procedure for fair division of a continuous resource, such as a cake, between two parties. It involves a heterogeneous good or resource ("the cake") and two partners who have different preferences over parts of the cake (both want as much of it as possible). The procedure ...

6. Hilbert's third problem - Wikipedia

   en.wikipedia.org/wiki/Hilbert's_third_problem

   Therefore, these two shapes cannot be scissors-congruent. A polyhedron's invariant is defined based on the lengths of its edges and the angles between its faces. If a polyhedron is cut into two, some edges are cut into two, and the corresponding contributions to the Dehn invariants should therefore be additive in the edge lengths.

7. Banach–Tarski paradox - Wikipedia

   en.wikipedia.org/wiki/Banach–Tarski_paradox

   "Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original?" The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different ...