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May 7th, 2018, 02:31 AM   #1
lua
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Painted numbers

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Is it possible to paint each point on the number line corresponding to the natural number with one of three colors - blue, red or green - so that not all points are painted with the same color, that the sum of the blue and red number is always green, sum of the blue and green number is red, and the sum of red and green number is always a blue number?
Can anybody halp. I had one solution, but I saw error in it.

That try of the solution assumed that equations in the problem are true for 3 fixed numbers, and then the answer is no.
But, colored numbers in the equations of the problem can be different (but only painted with the same color).

Last edited by lua; May 7th, 2018 at 02:52 AM.
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May 7th, 2018, 03:20 AM   #2
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Think "mod 3". That is, paint the numbers 1= biue, 2= red, 3= green, 4= blue, 5= red, 6= green, etc. Every number equal to 0 mod 3 is green, every number equal to 1 mod 3 is blue, and every number equal to 2 mod 3 is red.
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May 7th, 2018, 03:34 AM   #3
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OK. In that case:

$\displaystyle G\equiv 0 (mod 3)$,
$\displaystyle B\equiv 1 (mod 3)$ and
$\displaystyle R\equiv 2 (mod 3)$.

Equations from problem become:
$\displaystyle B+R\equiv 1+2\equiv 0 (mod 3) = G$,
$\displaystyle B+G\equiv 1+0\equiv 1 (mod 3) = B$ (not $\displaystyle R$), and
$\displaystyle R+G\equiv 2+0\equiv 2 (mod 3) = R$ (not $\displaystyle B$).
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May 7th, 2018, 04:14 AM   #4
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I'll try another way.

It is obvious that $\displaystyle 1$, $\displaystyle 2$ and $\displaystyle 3$ must be painted with three different colors (because $\displaystyle 1+2=3$). Then, $\displaystyle 4$ must be painted with the same color as $\displaystyle 2$ ($\displaystyle 1+3=4$). $\displaystyle 5$ must be painted differently from $\displaystyle 2$ and from $\displaystyle 3$ ($\displaystyle 2+3=5$), but differently from $\displaystyle 1$ and $\displaystyle 4$ ($\displaystyle 1+4=5$), as well. To obtain that, we would have to have 4 different colors:

$\displaystyle 1$ is painted with the first color, $\displaystyle 2$ (and $\displaystyle 4$) with the second, $\displaystyle 3$ with the third. $\displaystyle 5$ must be painted differently from all of these, but it is impossible because we have only three different colors.

So, answer to the question in the problem is no.
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May 7th, 2018, 05:03 AM   #5
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In my opinion, it is easier to think as follows.

Let r, b, and g be respectively the smallest red, blue, and green numbers and, without loss of generality, r < b < g. Thus 1 is a red number.

Thus 1 + b = g. How about g + 1. Well it must be blue and equals b + 2. And b + 2 + 1 must be green and so on. Thus every number not less than b is either blue or green. And that means that the sum of any two numbers not less than b must be blue or green, including the sum of a blue and a green number. But the sum any pair of blue and green numbers must be red.

Contradiction.
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May 7th, 2018, 05:25 AM   #6
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I understand. Thanks!
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