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A number which is simultaneously a pentagonal number P_n and a square number S_m. Such numbers exist when 1/2n(3n-1)=m^2. (1) Completing the square gives ...

Combine the two above squares on the left into the single large square on the right.

There are two types of squares inscribing reference triangle DeltaABC in the sense that all vertices lie on the sidelines of ABC. The first type has two adjacent vertices of ...

A centered polygonal number consisting of a central dot with four dots around it, and then additional dots in the gaps between adjacent dots. The general term is n^2+(n+1)^2, ...

A number which is simultaneously a heptagonal number H_n and square number S_m. Such numbers exist when 1/2n(5n-3)=m^2. (1) Completing the square and rearranging gives ...

A number which is simultaneously a nonagonal number N_m and a square number S_n and therefore satisfies the Diophantine equation 1/2m(7m-5)=n^2. (1) Completing the square and ...

A magic square-type arrangement of the words in the Latin sentence "Sator Arepo tenet opera rotas" ("the farmer Arepo keeps the world rolling"). This square has been found in ...

Johnson solid J_(10).

A number which is simultaneously square and triangular. Let T_n denote the nth triangular number and S_m the mth square number, then a number which is both triangular and ...

Square line picking is the selection of pairs of points (corresponding to endpoints of a line segment) randomly placed inside a square. n random line segments can be picked ...