05 Jan '07 15:18

**Find a non-empty set of positive integers, base 10, no two equal, such that:**

(1) For any number N in the set, none of the numbers in the set has N digits.

(2) For any number N in the set, the sum of the squares of the digits of N is also in the set.

(3) The number of numbers in the set is also in the set.

(4) The largest number in the set is equal to the sum of all of the other numbers in the set.

Among all such sets find one with the fewest numbers, and among those,

find the one with the smallest largest number.

(1) For any number N in the set, none of the numbers in the set has N digits.

(2) For any number N in the set, the sum of the squares of the digits of N is also in the set.

(3) The number of numbers in the set is also in the set.

(4) The largest number in the set is equal to the sum of all of the other numbers in the set.

Among all such sets find one with the fewest numbers, and among those,

find the one with the smallest largest number.