142. Sum of Cubes. The numbers 407 and 370 match the sum of the cubes of their digits. For example, the cube of 4 is 64, the cube of 0 is 0, and the cube of 7 is 343. Adding 64, 0, and 343 gives 407. Similarly, the cube of 3 (27) added to the cube of 7 (343) results in 370.

Could you find a number, not containing zero and having the same property? Of course, we exclude the trivial case of the number being 1.
142. The desired number is 153. The cubes of the numbers 1, 5, and 3 are 1, 125, and 27, respectively, and their sum is 153.

[The author did not notice the fourth number: 371. If 1 is not counted, then 407, 370, 153, and 371 are the only four numbers that equal the sum of the cubes of their digits. For a more general problem of finding numbers that equal the sum of the \(n\)-th powers of their digits, see Joseph S. Madachy’s book “Mathematics on Vacation” (N. Y., 1966, pp. 163-165). - M. G.]