PALO ALTO, CALIF. — A century-old math puzzle so complicated that its handwritten solution would cover the island of Manhattan was finally cracked by an international research team that had been working on it for four years.

The 18-member group of mathematicians and computer scientists was convened by the American Institute of Mathematics in Palo Alto to map a theoretical object known as the "Lie group E8."

Lie (pronounced Lee) groups were invented by 19th-century Norwegian mathematician Sophus Lie in his study of symmetrical objects, especially spheres, and differential calculus.

The E8 group, discovered in 1887, is the most complicated Lie group, with 248 dimensions, and was long considered impossible to solve.