In 1998, Marlow Anderson and Todd Feil used linear algebra to prove that not all configurations are solvable and also to prove that there are exactly four winning scenarios, not including redundant moves, for any solvable 5×5 problem. The 5×5 grid of Lights Out can be represented as a 25x1 column vector with a 1 and 0 signifying a light in its on and off state respectively. Each entry is an element of Z2, the field of integers modulo 2. Anderson and Feil found that in order for a configuration to be solvable (deriving the null vector from the original configuration) it must be orthogonal to the two vectors N1 and N2 below (pictured as a 5×5 array but not to be confused with matrices).