John von Neumann, b. Dec. 28, 1903, d. Feb. 8, 1957, was a Hungarian-American mathematician
who made important contributions to the foundations of mathematics, logic, quantum physics,
meteorology, science, computers, and game theory. He was noted for a phenomenal memory and
the speed with which he absorbed ideas and solved problems. In 1925 he received a B.S. diploma
in chemical engineering from Zurich Institute and in 1926 a Ph.D. in mathematics from the
University of Budapest. His Ph.D. dissertation on set theory was an important contribution to the
subject. At the age of 20, von Neumann proposed a new definition of ordinal numbers that was
universally adopted. While still in his twenties, he made many contributions in both pure and
applied mathematics that established him as a mathematician of unusual depth. His Mathematical
Foundations of Quantum Mechanics (1932) built a solid framework for the new scientific
discipline. During this time he also proved the mini-max theorem of GAME THEORY. He
gradually expanded his work in game theory, and with coauthor Oskar Morgenstern he wrote
Theory of Games and Economic Behavior (1944).
There are some numbers which can be expressed by the sum of factorials. For example 9,9=1!+2!+3!. Dr. von Neumann was very interested in such numbers. So, he gives you a number n, and wants you to tell him whether or not the number can be expressed by the sum of some factorials.
Well, it's just a piece of cake. For a given n, you'll check if there are some xi, and let n equal to