Problem 19 · 1987 AJHSME
Hard
Algebra & Patterns
repeated-squaringgrowth
A calculator has a squaring key x² which replaces the current number displayed with its square. For example, if the display is 000003 and the x² key is depressed, then the display becomes 000009. If the display reads 000002, how many times must you depress the x² key to produce a displayed number greater than 500?
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Answer: A — 4.
Show hints
Hint 1 of 2
Squaring isn't adding — each press multiplies the number by itself, so it grows ferociously. Just list the displays and stop the moment one passes 500.
Still stuck? Show hint 2 →
Hint 2 of 2
Track the exponent instead: each squaring DOUBLES the exponent of 2. Starting at 2¹, the powers go 2 → 4 → 8 → 16…
Show solution
Approach: iterate the squaring and count presses
- Start at 2. Press 1: 2² = 4. Press 2: 4² = 16. Press 3: 16² = 256. Press 4: 256² = 65536.
- 256 is still below 500, so three presses aren't enough; 65536 clears 500, so it takes 4 presses.
- Why this transfers: squaring doubles the exponent, so the display is 2^(2ⁿ) after n presses — that's 2¹, 2², 2⁴, 2⁸, 2¹⁶. Doubling exponents means the size explodes, so 'how many presses' is always small. Don't confuse 4 presses with the wrong-units traps 8 or 250.
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