Problem of the Week #14: Monday April 10th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
Consider a puzzle where the player is asked to connect dots on the left to dots on the right without letting any connecting lines cross each other.
a). Connect the matching letters without crossing any lines:
b). Is this always possible to do regardless of how the letters on the right are arranged? If yes, explain your strategy. If no, show a case where this is not possible.
c). Connect the matching letters without crossing any lines, consider each pair of letters on the right to be the same point:
d). Is this always possible to do regardless of how the letters on the right (two of each letter, two at each point) are arranged? If yes, prove. If no, show a case where this is not possible.
e). Find one arrangement of letters on the right, three of each, three at each point, such that it’s possible to connect the five letters on the left to all their matching letters without crossing any of the connecting lines.
f). Find an arrangement of letters on the right, three of each, three at each point, such that it’s impossible to connect the five letters on the left to all their matching letters without crossing any of the connecting lines.
g). How many possible arrangements are there for part e)?
h). How many possible arrangements are there for part f)?
i). Share your own problem inspired by this one.
j). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.