Find the two-digit number, with its unit digit greater than its tens digit,…

For any two-digit number with tens digit x and units digit y, the number equals 10x + y and its digit-reversal equals 10y + x. Their sum, 11(x + y), is a perfect square only for specific values of the digit sum (x + y); for a two-digit number this sum lies between 1 and 18, and here, with the added constraint y > x, it lies between 3 and 17. Checking each multiple of 11 in that range shows that only x + y = 11 gives a perfect square (11 x 11 = 121), since 11 being prime means 11k is a perfect square only when k itself is 11 times a perfect square. Their difference, 9(y - x), is a perfect square only when the digit difference (y - x) is itself a perfect square; since the tens digit x is at least 1 and the units digit y is at most 9, (y - x) can be at most 8, so the perfect-square values actually achievable here are 1 and 4 (not 9).

1. Let the required number be 10x + y, where x is the tens digit and y is the units digit, with y > x (unit digit greater than tens digit) as given.

2. Reversing the digits gives 10y + x.

3. Sum condition: (10x + y) + (10y + x) = 11(x + y) must be a perfect square, so x + y = 11 (giving 11 x 11 = 121 = 11 squared).

4. Difference condition: (10y + x) - (10x + y) = 9(y - x) must be a perfect square, so (y - x) must itself be a perfect square achievable within the digit range, i.e. 1 or 4 (a difference of 9 is not possible here since the tens digit x is at least 1, which caps y - x at 8).

5. List the digit pairs (x, y) with x + y = 11 and y > x: (2, 9), (3, 8), (4, 7), (5, 6), with respective differences y - x = 7, 5, 3, 1.

6. Only the pair (5, 6) has a difference of 1, which is a perfect square (1 squared); the others (7, 5, 3) are not perfect squares.

7. So x = 5 and y = 6, giving the number 10(5) + 6 = 56.

Verification: the reverse of 56 is 65. Their sum, 56 + 65 = 121 = 11 squared, is a perfect square, and their difference, 65 - 56 = 9 = 3 squared, is also a perfect square -- both conditions hold simultaneously, confirming the number.

Hence, the required two-digit number is 56.