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Let N_{0}denote a three digit number with not all digits identical. Arrange the digits in descending order and subtract from this number the number that is obtained by arranging the digits in ascending order. Let N_{1}denote the result, written as a three digit number (e.g., 42 is written as 042). Now perform the same operation on N_{1}that you performed on N_{0}and let N_{2}denote the result. Repeat to construct the sequence N_{3}, N_{4},....

(a) Show that there exists a number N_{x}such that if N_{0}=N_{x}then N_{1}=N_{x}

(b) Show that N_{6=Nx for any initial number N0}

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# Junior Question

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