Problem 14 · 2023 Math Kangaroo
Medium
Counting & Probability
Number Theory
careful-countingcasework
The digits 0 to 9 can be formed using matchsticks (see diagram). How many different positive whole numbers can be formed this way with exactly 6 matchsticks?
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Answer: C — 6
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Hint 1 of 2
First note how many matchsticks each digit 0..9 needs.
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Hint 2 of 2
List all numbers (one or more digits) whose matchstick counts add to exactly 6, ignoring leading zeros.
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Approach: count matchsticks per digit, then list totals equal to 6
- Stick counts: 1 uses 2; 7 uses 3; 4 uses 4; 2, 3, 5 use 5; 0, 6, 9 use 6; 8 uses 7.
- One-digit (6 sticks, no leading zero): 6 and 9.
- Two-digit (sticks add to 6): 2 + 4 gives 14 and 41; 3 + 3 gives 77.
- Three-digit (sticks add to 6): each digit needs at least 2, so all three are 1 — the number 111.
- That is 6, 9, 14, 41, 77, 111 — 6 numbers, so the answer is C.
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