Problem 18 · 1989 AJHSME
Hard
Algebra & Patterns
involution
Many calculators have a reciprocal key 1/x that replaces the current number displayed with its reciprocal. For example, if the display is 00004 and the 1/x key is pressed, then the display becomes 000.25. If 00032 is currently displayed, what is the fewest positive number of times you must depress the 1/x key so the display again reads 00032?
Show answer
Answer: B — 2.
Show hints
Hint 1 of 3
Don't assume it takes many presses — just do it once and see what's on the screen, then ask whether one more press undoes it.
Still stuck? Show hint 2 →
Hint 2 of 3
Flipping a fraction upside down, then flipping again, lands you exactly where you started: the reciprocal of the reciprocal is the original number.
Still stuck? Show hint 3 →
Hint 3 of 3
32 = 32⁄1; one press flips it to 1⁄32; the next press flips it back.
Show solution
Approach: the reciprocal undoes itself
- Press once: 32 (which is 32⁄1) flips to 1⁄32. Press again: 1⁄32 flips back to 32. So the display returns after exactly 2 presses.
- Why this works: taking a reciprocal twice cancels itself — like flipping a card over and over, every even number of presses returns the original. An operation that is its own undo is called self-inverse, and it always cycles with period 2 (unless the number is 1, which never changes).
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