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<details><summary>Key insights</summary>

1. The number of digits in the square must be exactly one more than the number of digits in X.  A one-digit integer can have a two-digit square; those are easy to check by inspection, and the example given in the question is the only case here.  A two-digit integer can
have a three-digit square, but that maxes out at 31 (and nothing in that range qualifies, by inspection).  32^2 is 4 digits, so from there up, it is not possible to assemble N^2 by inserting a 2 into N.

2. ... If we're limited to integers.  But X is a *number*, not specifically an *integer*.

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<details><summary>Two more solutions</summary>

I actually worked this out from first principles before noticing the similarity to the examples in the question.

1.1 squared is 1.21, and 0.5 squared is 0.25, both of which satisfy the requirement.  These are, it turns out, just the original examples divided by 10.  Dividing instead by 100 doesn't work though unless you're willing to be inconsistent about zeros -- 0.11 squared is .0121, which technically works, but for consistency it's really 0.0121, which doesn't.

Because of what I said about the number of digits in the previous section, I *think* 11, 5, 1.1, and 0.5 are the only solutions involving rational numbers.
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