I have this problem: Q4 of this
- Prove by induction that no matter how the dots are coloured red and blue, it is possible to have a successful trip around the circle if you start at the correct point.
But I have another point that I don't know how to do
2.Use the above proof to give a recursive denition for the "correct point".
Can you help me please ?
Well, you did it already, but let me try to answer what they are asking. Let CorrectPoint($x_1,x_2,\ldots,x_{2n}$) denote the point that causes a win for $2n$ dots.
First, CorrectPoint($x_1,x_2$) is equal to whichever of $x_1$ and $x_2$ is the red dot.
Second, for $n\geq 2$ and given $(x_1,x_2,\ldots,x_{2n})$, first identify a pair $(x_i,x_{i+1})$, such that $x_i$ is red, and $x_{i+1}$ is blue, and then define
CorrectPoint($x_1,x_2,\ldots,x_{2n}$) = CorrectPoint($x_1,x_2,\ldots,x_{i-1},x_{i+2},\ldots,x_{2n}$).
I hope this was what you were aiming for.