Tiling of a Möbius strip can be performed immediately by carrying over a tiling of a rectangle with the same two-sided surface area. However, additional tilings are possible by cutting tiles across glued edges. An example of such a tiling is the strip constructed from a rectangle consisting of two halves of a width 2 square (which are rejoined when edges are connected) separated by a square (Stewart 1997). Unfortunately, since the long top and bottom edges must be glued together, this example is not constructible out of paper. It also suffers from having the unit square share a boundary with itself. In 1993, S. J. Chapman found a tiling free of the latter defect (although still suffering from the former) which can be constructed using five squares. No similar tiling is possible using fewer tiles (Stewart 1997)...
The Curry triangle, also sometimes called the missing square puzzle, is a dissection fallacy created by American neuropsychiatrist L. Vosburgh Lions as an example of a phenomenon discovered by Paul Curry. The figure apparently shows that a triangle of area 60, a triangle of area 58 containing a rectangular hole, and a broken rectangle of area 59 can all be formed out of the same set of 6 polygonal pieces. The explanation for this lies in the inaccuracy of the initial subdivision. In the diagrams, the small and large right triangles are similar, hence they cannot have perpendicular sides of lengths and , respectively, as apparently shown in the drawing.
The stomachion is a 14-piece dissection puzzle similar to tangrams. It is described in fragmentary manuscripts attributed to Archimedes as noted by Magnus Ausonius (310-395 A.D.). The puzzle is also referred to as the "loculus of Archimedes" (Archimedes' box) or "syntemachion" in Latin texts. The word stomachion has as its root the Greek word , meaning "stomach." Note that Ausonius refers to the figure as the "ostomachion," an apparent corruption of the original Greek.The puzzle consists of 14 flat pieces of various shapes arranged in the shape of a square, with the vertices of pieces occurring on a grid. Two pairs of pieces are duplicated. Like tangrams, the object is to rearrange the pieces to form interesting shapes such as the elephant illustrated above (Andrea).Taking the square as having edge lengths 12, the pieces have areas 3, 3, 6, 6, 6, 6, 9, 12, 12, 12, 12, 12, 21, and 24, giving them relative..
A solid dissection puzzle invented by Piet Hein during a lecture on Quantum Mechanics by Werner Heisenberg. There are seven soma pieces composed of all the irregular face-joined cubes (polycubes) with cubes. The object is to assemble the pieces into a cube. There are 240 essentially distinct ways of doing so (Beeler 1972, Berlekamp et al. 1982), as first enumerated one rainy afternoon in 1961 by J. H. Conway and Mike Guy.A commercial version of the cube colors the pieces black, green, orange, white, red, and blue. When the 48 symmetries of the cube, three ways of assembling the black piece, and ways of assembling the green, orange, white, red, and blue pieces are counted, the total number of solutions rises to .
With three cuts, dissect an equilateral triangle into a square. The problem was first proposed by Dudeney in 1902, and subsequently discussed in Dudeney (1958), and Gardner (1961, p. 34), Stewart (1987, p. 169), and Wells (1991, pp. 61-62). The solution can be hinged so that the four pieces collapse into either the triangle or the square. Two of the hinges bisect sides of the triangle, while the third hinge and the corner of the large piece on the base cut the base in the approximate ratio 0.982:2:1.018.