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Theory of algorithms (other) 11-13

A cat tries to catch a mouse in labyrinths A, B, and C. The cat walks first, beginning with the node marked with the letter “K”. Then the mouse $($ from the node “M”$)$ moves, then again the cat moves, etc. From any node the cat and mouse go to any adjacent node. If at some point the cat and mouse are in the same node, then the cat eats the mouse.
Can the cat catch the mouse in each of the cases A, B, C?

A                                                       B                                                          C




Boundedness, monotonicity , Quadratic inequaities and systems of inequalities 14-17

For which natural K does the number reach its maximum value?

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