Problem 28 · 2015 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
I have noted down six digits of Erich’s seven-digit phone number in the correct order. I don’t know which digit I have missed out and where I have missed it out. What is the maximum number of tries that I have to make to be sure that I have used the correct phone number? (Note: the first digit could also be 0!)
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Answer: C — 64
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Hint 1 of 2
You know 6 digits in order with one digit deleted somewhere; rebuild the candidates by inserting one unknown digit.
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Hint 2 of 2
Count insertions of a digit (0-9) into the 8 gaps, then remove the duplicates that arise next to matching digits.
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Approach: count insertions of one digit into the known string
- Insert one missing digit into one of the 8 positions (including before the first digit).
- Each position allows 10 digits, giving 8 * 10 = 80 raw candidates.
- Inserting a digit equal to an adjacent known digit repeats an earlier candidate; removing those duplicates leaves 64.
- So the maximum number of tries is 64 (C).
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