Did you catch the part about 14? Not a square number, but the save is to add an extra digit to make it 144. Let's call that a square-save. Here's the section of the song, if you want to enjoy the 14 square-save in song.
With that as the inspiration, can you always square-save any positive integer? In other words, can you always add some extra digits to make a perfect square?
What do you make of the following sequence:
1, 5, 6, 2, 23, 8, 27, 9, 3, 10, 34, 11, 37, 12, 39, 4, 42, 43, 14, 45 . . .
What number comes next?
Do you notice any patterns?
Extra Extra Credit
One square-save of 10 is to add a 0, making 100. One hundred, of course, is 10 * 10. Are there any other numbers that can do this, e.g., their own square is a square-save?