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Seek simple and succinct solutions in these systems by sussing-out the significant symmetries.

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When we write the integers from 1 to 100, how many times does the digit 9 appear?

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Which of the following expresses the correct formula for the number of unit squares with exactly 1 red side in an \( n \times n \) square with \( n \geq 2 \) ?

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Find the number of \(2\) digit numbers whose digit sum is a perfect square.

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If you take a \(3 \times 3 \times 3\) Rubik cube and break it up into the \( 3^3 \) cubes, how many cubes would have stickers on exactly \(2\) faces?

**Bonus question**: Can you generalize this for a \(N \times N \times N\) Rubik cube?

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