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Q: I am a square number. I am an odd two-digit number. The sum of my digits is a two-digit number. What number am I?

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There are infinitely many of them. The square of every odd number will be an odd square number.

The square of any odd number will be an odd square number. eg 12 = 1 32 = 9 52 = 25 etc

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Assuming you know that your number is a perfect square, the square root of an even number is even, and the square root of an odd number is odd.

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Related questions

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I cannot see how odd or even will determine the number of digits in the square root. For example:100 is an even number, and its square root is 10 (2 digits)121 is an odd number, and its square root is 11 (2 digits)Maybe the question is not phrased properly for what the questioner is asking.

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9

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23?

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16.

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There can't be a prime number that has a square root because the square root would be a factor of the number.

There is no such number. If the four digits are even, the entire number is even, not odd.