The gnomes puzzle: hint

Just in case you missed it, here's the original puzzle.

Suppose we have n=2 gnomes, and the possible hat colors are red and blue. Gnome 1 (G1) can always guess that his hat is the same color as gnome 2's hat; while gnome 2 (G2) can always guess that her hat is the opposite color as gnome 1's.

We can check each possible placement of hat colors to see that, in each case, one of the gnomes correctly guesses his/her own hat color. A green highlight indicates a correct guess of a gnome's own hat color.

G1 hat   G2 hat   G1 guess   G2 guess  
blue blue blue red
blue red red red
red blue blue blue
red red red blue

This is a solution for n=2.

Once you've thought about this for long enough (don't just give up!) you should check out this solution to the puzzle.

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