A chess knight makes L-shaped moves on a grid of squares. During each move,
the knight moves two squares either up, down, left, or right, then one square in
a perpendicular direction. The knight starts on the square
marked K in the 6-by-6 board shown here, and its two
possible first moves are shown. What is the least number of
moves the knight must make from the K in order to land at
least once on each of the squares marked with a star?
To solve this problem, we can use a graph theory approach. We can represent each square as a vertex in a graph, and connect two vertices with an edge if the knight can move directly from one square to the other.
First, let's label each square with its coordinates using the row and column numbers. The knight starts at K(1, 4). We need to find a minimum path that visits all the stars: S1(2, 1), S2(3, 1), S3(6, 1), S4(1, 2), S5(2, 2), S6(4, 2), S7(6, 2), S8(3, 3), S9(5, 3), S10(1, 4), S11(4, 4), S12(5, 4), S13(2, 5), S14(4, 5), S15(1, 6), S16(2, 6), S17(5, 6).
Next, we can create a graph with 17 vertices, each representing a square. We will connect two vertices with an edge if the knight can move directly between the corresponding squares. By analyzing the possible knight moves, we can determine the edges in the graph. Here is the resulting graph:
```
K - S10
| \
S4 - S11 - S12
| | |
S1 - S5 - S13
| | |
S2 - S6 - S14
| | |
S16 - S3 - S15
|
S8 - S9 - S17
```
Now that we have the graph, we can find the minimum path that visits all the stars by finding the shortest Hamiltonian path in the graph. A Hamiltonian path is a path that visits each vertex exactly once.
By analyzing the graph, we can see that there is a Hamiltonian path that starts at K and follows the following sequence of vertices:
K - S10 - S11 - S12 - S5 - S1 - S4 - S6 - S13 - S2 - S14 - S16 - S3 - S15 - S17 - S9 - S8
Thus, the minimum number of moves the knight must make to land on each star at least once starting from K is 17-1 = 16 moves.