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Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to be possible by Miklós Laczkovich in 1990; the decomposition makes heavy use of the axiom of choice and is therefore non-constructive. Laczkovich estimated the number of pieces in his decomposition at roughly 1050. More recently, Andrew Marks and Spencer Unger (2017) gave a completely constructive solution using Borel pieces.

In particular, it is impossible to dissect a circle and make a square using pieces that could be cut with an idealized pair of scissors (that is, having Jordan curve boundary). The pieces used in Laczkovich's proof are non-measurable subsets.

Laczkovich actually proved the reassembly can be done using translations only; rotations are not required. Along the way, he also proved that any simple polygon in the plane can be decomposed into finitely many pieces and reassembled using translations only to form a square of equal area. The Bolyai–Gerwien theorem is a related but much simpler result: it states that one can accomplish such a decomposition of a simple polygon with finitely many polygonal pieces if both translations and rotations are allowed for the reassembly.

It follows from a result of Wilson (2005) that it is possible to choose the pieces in such a way that they can be moved continuously while remaining disjoint to yield the square. Moreover, this stronger statement can be proved as well to be accomplished by means of translations only.

These results should be compared with the much more paradoxical decompositions in three dimensions provided by the Banach–Tarski paradox; those decompositions can even change the volume of a set. However, in the plane, a decomposition into finitely many pieces must preserve the sum of the Banach measures of the pieces, and therefore cannot change the total area of a set (Wagon 1993).
See also

Squaring the circle, a different problem: the task (which has been proven to be impossible) of constructing, for a given circle, a square of equal area with straightedge and compass alone.

References

Hertel, Eike; Richter, Christian (2003), "Squaring the circle by dissection" (PDF), Beiträge zur Algebra und Geometrie, 44 (1): 47–55, MR 1990983.
Laczkovich, Miklos (1990), "Equidecomposability and discrepancy: a solution to Tarski's circle squaring problem", Journal für die Reine und Angewandte Mathematik, 404: 77–117, doi:10.1515/crll.1990.404.77, MR 1037431.
Laczkovich, Miklos (1994), "Paradoxical decompositions: a survey of recent results", Proc. First European Congress of Mathematics, Vol. II (Paris, 1992), Progress in Mathematics, 120, Basel: Birkhäuser, pp. 159–184, MR 1341843.
Marks, Andrew; Unger, Spencer (2017), "Borel circle squaring", Annals of Mathematics, 186 (2): 581–605, arXiv:1612.05833, doi:10.4007/annals.2017.186.2.4.
Tarski, Alfred (1925), "Probléme 38", Fundamenta Mathematicae, 7: 381.
Wilson, Trevor M. (2005), "A continuous movement version of the Banach–Tarski paradox: A solution to De Groot's problem" (PDF), Journal of Symbolic Logic, 70 (3): 946–952, doi:10.2178/jsl/1122038921, MR 2155273.
Wagon, Stan (1993), The Banach–Tarski Paradox, Encyclopedia of Mathematics and its Applications, 24, Cambridge University Press, p. 169, ISBN 9780521457040.

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