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Thea has a key on her calculator marked with a blue asterisk*. If an integer is displayed, pressing the asterisk key chops off the first digit and moves it to the end. For example, if 6138 is on the screen, then pressing the asterisk key changes the display to 1386.

Thea enters a positive integer into her calculator, then squares it, then presses the blue asterisk, then squares the result, then presses the blue asterisk key again. After all these steps, the calculator displays 243. What number did Thea originally enter?

*I tried doing LaTeX for the blue asterisks but it didn't work.

Guest Jun 25, 2018

#1**+2 **

Reversing the results, we have :

Before pressing the asterisk key for the last time, the number displayed must have been 324

Taking the square root of this would result in 18

Before pressing the asterisk key for the first time, the display would have shown "81"

Taking the square root of this would have resulted in an original positive integer of "9"

So...the steps - in order - would have resulted in :

9

81

18

324

243

CPhill Jun 25, 2018