Thea has a key on her calculator marked *. If an integer is displayed, pressing the * key chops off the first digit and moves it to the end. For example, if 6138 is on the screen, then pressing the * key changes the display to 138.
Thea enters a positive integer into her calculator, then squares it, then presses the * key, then squares the result, then presses the * key again. After all these steps, the calculator displays 243. What number did Thea originally enter?
Also, I know this question was previously asked, but no one ever answered it 
+33652 The example in your first paragraph is not consistent with the words! Assuming the example should be 6138 goes to 1386, then:
Prior to the last * the number was 324
The square root of this is 18
Prior to the first * the number was 81.
The original number was therefore the square root of this, namely, 9.
