Driving Around In Circles

Apparently, if you actually try this out, you find out that the farmer goes in an infinite loop between plots 2,4 and 7.

Here's a mathematical solution to this problem: Starting from the first point the farmer clears, label the plots of land as 1, 2, 3, ... n clockwise around the circle. Now, the farmer moves 1, 3, 6, 10, 15 ... times around the circle. We see that every (3k+1)th move he makes will land at plots 1 (mod 3) and all the other moves will take him to plots 0 (mod 3). Therefore, any plots that were assigned numbers 2 (mod 3) will never be visited so he will not be able to clear those lands.

At the 18th move, all 9 plots have flowers once again. So, it doesn't matter how many times he keeps doing it, it will cycle endlessly. The only difference between subsequent cycles and the first one is the starting point, i.e., the offset increases with each subsequent cycle.

Let's label the plots as 0, 1, 2, ..., 8. Starting from plot 0, after going to plot $a_n$ the next plot will be the plot $a_n + n$ , but since this is a circular configuration it will go back once it reaches plot 8 so we can say that after being in plot $a_n$ , the plot $a_{n+1} \equiv a_n + n \pmod 9$ .

The plot on the $n_{th}$ move can then be described as $1 + 2 + ... + n \pmod 9 \equiv \frac{n(n-1)}{2} \pmod 9$ which turns out that it cicles thru the values 1, 3 and 6, so the other plots can never be reached.

If you number the plots 1 through 9, start at 1 so that the next plot hit is 2, then 4, then 7, and continue to follow the rule, the plots the farmer gets to are 1, 2, 4, 7, 2, 7, 4, 2, and 1. Then the cycle repeats: 1, 2, 4, 7, 2, 7, 4, 2, and 1. Making a move of 1, then 2, then 3, and so on up to 9 is repeated as the moves become 10, 11, 12, and so on up to 18. The cycles of 9 are endlessly repeated. Great problem!

As the set is of (2,4,7) if we enhance it, it will never clear all of our plots. i.e from 1 to 9