The Square Root of 4444222225
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Let us find a positive x, such that:

 x² = 4444222225

without a calculator or the pen-and-paper algorithm.
The firs step will be to look at its digits:

Four fours, then four twos and then 25.
Or
 4444222225 = 4444000000 + 222200 + 25 =
            = 4444⋅10⁶ + 2222⋅100 + 25 =
            = 1111 ⋅ 100(40000 + 2) + 25 =
            = 1111 ⋅ 100(44440 - 4438) + 25 =
            = 1111 ⋅ 100(44440 - 4444 + 6) + 25 =
            = 1111 ⋅ 100(40 ⋅ 1111 - 4⋅1111 + 6) + 25 =
            = 1111 ⋅ 100(36 ⋅ 1111 + 6) + 25 =
            = 36 ⋅ 100 ⋅ 1111² + 6 ⋅ 100 ⋅ 1111 + 25 =
            = 6² ⋅ 10² ⋅ 1111²+6 ⋅ 100 ⋅ 1111 + 25 =
            = 66660² + 666600 + 5² =
            = 66660² + 10 ⋅ 66660 + 5² =
            = 66660² + 2 ⋅ 5 ⋅ 66660 + 5² =
            = (66660 + 5)² =
            = 66665²

Thus,

 +-----------+
 | x = 66665 |
 +-----------+

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