Problem 23 · 2024 Math Kangaroo
Stretch
Number Theory
divisibilitycasework
Cristina has 12 cards numbered 1 to 12. She places eight of them in a circle so that the sum of any two adjacent numbers is a multiple of 3. Which numbers does Cristina not use?
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Answer: E — 3, 6, 9, 12
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Hint 1 of 2
Sort 1-12 by remainder when divided by 3: three groups of four.
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Hint 2 of 2
Two adjacent numbers sum to a multiple of 3 only if both are multiples of 3, or one is a 'remainder 1' and the other a 'remainder 2'.
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Approach: classify by residue mod 3 and build the circle
- Mod 3 the numbers split into {3,6,9,12}, {1,4,7,10}, {2,5,8,11}.
- Adjacent sums divisible by 3 force the circle to alternate the remainder-1 and remainder-2 groups, using all eight of those numbers.
- That leaves the multiples of 3 unused, so Christina does not use 3, 6, 9, 12.
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