Problem 30 · 2015 Math Kangaroo
Stretch
Number Theory
caseworklogic
Each positive whole number is coloured in according to the following three rules: (i) Each number is either red or green. (ii) The sum of two different red numbers is a red number. (iii) The sum of two different green numbers is a green number. How many ways are there to do this?
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Answer: D — 6
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Hint 1 of 2
The two rules force strong closure: sums of like-coloured numbers keep their colour.
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Hint 2 of 2
Colour 1, 2, 3, ... and chase the forced consequences to count how many consistent colourings of all positive integers exist.
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Approach: count the colourings closed under the two sum-rules
- Reds are closed under adding two distinct reds, and greens under adding two distinct greens.
- Fixing the colours of the smallest numbers forces almost everything else, leaving only a few consistent patterns.
- Carefully enumerating them gives exactly 6 valid colourings.
- So there are 6 (D) ways.
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