You want to color the numbers $\{1,2,3,4,5\}$ with two colors (red and blue).
The prohibited rule is:
- there cannot be three numbers of the same color $x,y,z$ that satisfy $x+y=z$ (allowing $x=y$).
Question: is there any color that meets the rule?
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The impossible coloring from 1 to 5
Hints
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- Without loss of generality, color 1 red.
- The 3 can be neither red nor blue: contradiction.
- Since 1+1=2, 2 cannot be red. Then 2 is blue.
Solution
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Answer: No, it is impossible.
Without loss of generality, color 1 red.
- Like $1+1=2$, 2 cannot be red. Then 2 is blue.
- Like $2+2=4$, 4 cannot be blue. Therefore 4 is red.
- Like $1+4=5$, 5 cannot be red. Therefore 5 is blue.
Now look at number 3:
- If 3 is red, then $1+3=4$ gives a forbidden red triple.
- If 3 is blue, then $2+3=5$ gives a forbidden blue triple.
The 3 can be neither red nor blue: contradiction.
Therefore there is no valid coloring for $\{1,2,3,4,5\}$.
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