Problem 16 · 2020 Math Kangaroo
Hard
Counting & Probability
Number Theory
careful-countingcasework
Mary numbered the faces of three cards with the numbers 1 to 6. Using the three cards she can make three-digit numbers, for example 135 or 234, but some numbers cannot be made, such as 126. Which of the following numbers CANNOT be made?
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Answer: D — 245
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Hint 1 of 2
Each card shows two numbers (opposite faces), and you read one number from each of the three cards.
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Hint 2 of 2
Find the secret pairing of {1..6} into three cards; a number is impossible if two of its digits live on the same card.
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Approach: recover the card pairings, then test each number
- Three cards carry digits 1..6, two per card; a three-digit number takes one digit from each card.
- Examples 135 and 234 are possible while 126 is not, which pins down which digits share a card.
- Checking the options, 245 needs two digits on the same card, so it cannot be made.
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