The vertices of a 20-gon are numbered from 1 to 20 in such a way that the numbers of adjacent vertices differ by either 1 or 2. The sides of the 20-gon whose ends differ by only 1 are colored red. How many red sides are there? So first let's draw out this 20-gon with all of its vertices. We know that we need to use the numbers 1 to 20 somehow, and so that means the number 1 has to be somewhere, so let's just put it at the top. So now what can the neighbors of the number 1 be? Well, they have to differ from 1 by either 1 or 2, and there are only two numbers that do that, you know, 2 and 3. So 2 and 3 have to be the neighbors of 1, so let's just put them down like this. Well now, what are the neighbors of 2? Well, they have to differ from 2 by 1 or 2, so we have 1, 3, and 4 as options, but we've already used up 1 and 3, so we're forced to use 4. Well now let's look at the neighbors of 3. Well, the only possibilities are 1, 2, 4, and 5, but we've already used 1, 2, and 4, so we have to use 5. Now let's look at the neighbors of 4. Well, the only options we have are 2, 3, 5, and 6, but we've already used 2, 3, and 5, so we have to use 6. And now for the neighbors of 5, we can do a similar procedure to deduce that the only thing can be 7, and we can just keep doing this around the circle until we're done, and we have to put 20 over here. And so this numbering for the vertices was uniquely determined, and so now we just count how many red sides have to be there. The only things that differ by 1 are these two and these two, so there are two red sides.

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