129. Digital Squares. A very good puzzle is to find a number which, together with its square, contains each of the nine digits, excluding zero, exactly once. Thus, if the square of the number 378 were 152694, it would fit our criteria. But in fact, its square is 142884, which gives us two fours and three eights, while 6, 5, and 9 are missing.

There are only two solutions; they can be found in a quarter of an hour if you proceed correctly.
129. The problem has only two solutions: the numbers 567 (\(567^{2}=\) \(321489)\) and 854 (\(854^{2}=729316\)). When searching for a solution, only three-digit numbers should be considered, the sum of whose digits is 9, 18, and 27 or 8, 17, and 26. The smallest three-digit number whose square is a six-digit number is 317.