Pattern Lock System

This is equivalent to finding the number of Hamiltonian paths in a $3 \times 3$ grid graph. We will first find the number of directed paths; we'll simply divide by 2 to obtain the number of undirected paths (each path can be traversed in two directions).

First, color the points in a checker pattern, so the corners and the center are white, while the edge points are black. Then every edge in a path must connect two points of different colors, so the path alternates between colors. Since there are one more white point than black, the path must begin and end on a white point.

Now we simply count everything carefully. We'll divide into two cases:

• If the path starts on the center point, there are 4 ways the second point can be (any of the four directions; they all give the same thing), then 2 ways the third point can be (either turn left or turn right), and now the path is completely determined. We have $4 \times 2 = 8$ paths here.
• If the path starts on a corner point, without loss of generality, say that the corner is bottom-left. Now the path can go either up or right; either case is symmetrical (flip the grid), so let's say it goes up, and we remember to multiply the final answer by 2. Now, there must be an edge that goes into the center point; among all four possible edges, turns out each one can be extended into a path uniquely. So we have $4 \times 2 = 8$ paths as well.

Since there are 4 corner points, the corners contribute 32 paths in total. Adding the center point gives 40 directed paths, or $\boxed{20}$ undirected paths.

Well by a bit of trial and error, one notices that only the corner point and the centre points can be used as start or end of a pattern. We cannot use the edge centre points. Now we can also conjecture that for every pair of such points, there exists exactly one path (This is simple enough to verify on paper, accounting for symmetry). So we have 5 points, of which we need to select two non-same points in order. Hence 5*4=20

For a purely undirected approach, consider the central point as our way to classify patterns. We can have a pattern where the center only has one edge connected to it. Said edge could be in any of four directions. Then outer boundary would missing exactly one edge at the point connected to the center, and there are two possibilities there. So there are eight patterns where the center is connected to the outer edge exactly once. Next, consider patterns where the center point is connected to two points on opposite sides of the square. This path could be horizontal or vertical, and in either case we could have the pattern oriented to an S shape to the curve or a Z shape (rotated appropriately). This gives four more patterns. Finally, the center point could connect with edges to two adjacent sides of the square. In this case, the corner meeting those two sides has to connect to one of the two edges, and either is a valid choice which forces the rest of the outside to be connected to the other edge. (See for example #1 in the question.) There are four pairs of adjacent sides with two orientations possible for each, making another 8 patterns. We have exhausted the possibilities and thus there are 8 + 4 + 8 = 20 patterns total.