Problem 7 · 2022 Math Kangaroo
Medium
Number Theory
place-value
Bodil lays these seven cards — 4, 69, 113, 9, 51, 5, 67 — next to each other, so that the smallest possible 12-digit number that can be made from these cards is formed. What are the last three digits of this number?
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Answer: A — 699
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Hint 1 of 3
A number is smallest when its very first digit is as small as possible, then the next, and so on.
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Hint 2 of 3
When two cards could go either way, try both joins and keep the smaller one (for example 51 then 5 makes 515, which beats 551).
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Hint 3 of 3
Once you have the whole order, the last card you place gives the final three digits.
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Approach: build the number digit by digit, smallest first
- Want the smallest start, so a card beginning with 1 leads: the card 113 goes first.
- Next pick the card that keeps the number smallest at each step, comparing tricky pairs by trying both joins (51 before 5 since 515 < 551).
- The order 113, 4, 51, 5, 67, 69, 9 builds 113451567699.
- Its last three digits are 699, so the answer is A.
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