Let's determine that four-digit number in which the sum of the digits, the sum and difference of the third and fourth digits, and the fourth digit itself are all non-zero square numbers. The two-digit number formed by the first two digits in the decimal system is also a square number.
The fourth digit can only be 1, 4, or 9, but we cannot find a third digit that, when added to or subtracted from 1 or 9, results in a square number. Therefore, the fourth digit can only be 4, and the third digit can only be 5 (since \(5 + 4 = 3^2\) and \(5 - 4 = 1^2\)). The two-digit number formed by the first two digits must be a square number, so the sum of the first two digits, \(x\), must be at least 7 (for 16 and 25) and at most 13 (for 49). The sum of the four digits, \(4 + 5 + x\), must also be a square number, with the smallest possible value being 16 (\(=9 + 7\)) and the largest possible value being 22 (\(=9 + 13\)). The only square number between 16 and 22 is 16, so \(x\) must be 7. Therefore, the two-digit number formed by the first two digits can only be 16 or 25. Thus, the two four-digit numbers that satisfy our conditions are 1654 and 2554.

Kornélia Jójárt (Esztergom, Dobó K. g. I. o. t.)

Note: (Jakob BERNOULLI (pronounced: Bernullyi) was born in 1654 and died in 1705. He was a renowned professor at the University of Basel. His 300th birthday is celebrated on December 27th of this year. This explains the origin of the problem.