Double check 2.d4 d5 3.Nc3 dxe4 4.Nxe4 Nf6 5.Qd3 e5 6.dxe5 Qa5+ 7.Bd2 Qxe5 8.0-0-0 Nxe4 9.Qd8+ (sacrificing a queen in order to set up a double check) Kxd8 10.Bg5++ and White mates after 10...Ke8 11.Rd8# or 10...Kc7 11.Bd8#. A double check was also seen in the celebrated Evergreen Game, Anderssen–Dufresne, 1852. Anderssen won with 20.Rxe7+! Nxe7 21.Qxd7+!! (a queen sacrifice to set up a deadly double check) Kxd7 22.Bf5++ Ke8 (or 22...Kc6 23.Bd7#) 23.Bd7+ Kf8 24.Bxe7#. In chess with variant rules or fairy pieces, other ways of delivering a double check may be possible. Triple, quadruple and even quintuple checks may
Quadruple product Quadruple product In mathematics, the quadruple product is a product of four vectors in three-dimensional Euclidean space. The name "quadruple product" is used for two different products, the scalar-valued scalar quadruple product and the vector-valued vector quadruple product or vector product of four vectors . The scalar quadruple product is defined as the dot product of two cross products: where a, b, c, d are vectors in three-dimensional Euclidean space. It can be evaluated using the identity: or using the determinant: The vector quadruple product is defined as the cross product of two cross products: where a, b, c, d