I promise I won't look

Android's lock screen feature allows people to protect their phones by joining points on a grid to make a pattern. Let $L$ be the size of the longest possible length of the pattern that can be drawn on an android phone with a $3 \times 3$ pattern lock. If the grid is spaced equally by one unit what is $L \sqrt{31}$ rounded to the nearest integer?

The rules for drawing a pattern are:

• You can only use a point once

• You cannot "jump" a point if it is in between the two points and is part of the line that satisfies the first two points. Ie You cannot go from 1 to 9 without also hitting 5. But you can go directly from 2 to 9 without including any other point.

• ONCE a point is "taken", it may THEN be skipped. A point cannot be skipped "in transit" only if it hasn't already been taken.

To fully understand the rules, it is best to try them out on a real phone.

Details and Assumptions

• The points are equally spaced vertically and horizontally by one unit.

• If it had been a $2 \times 2$ grid, $L$ would be $1 + 2 \sqrt{2}$.

×