I come across maths problems of a variety of difficulties that I find interesting. Here, I will attempt to note them down as I see them. Of particular note are the problems that I dub "Wesley Problems", which can be found lower on the page.
During my undergraduate years, my good friend Wesley and I would share maths teasers that we found interesting. Our cohort would specifically dub them Wesley Problems if they were simple to state, simple to answer, hard to solve, and attention-grabbing in some way. I appreciate that this is a specific name for a universal experience amongst mathematicians, but I always found his problems particularly noteworthy.
As such a big fan, I always promised him I would compile these problems somewhere, so I will begin the process here. Please bear in mind that we stumbled upon these problems as undergraduates - the more seasoned mathematical reader may not consider them quite as exotic as we once did. And these problems certainly aren't ours originally!
Maths Puzzles
Wesley Problems
Wesley Problem 1
Ten students are walking along the (one-dimensional) paddock path, keen to get back to their rooms and study. So whenever two of them bump into each other, they instantaneously turn around to avoid a conversation. It takes a minute to traverse the paddock. What is the longest amount of time a student could get trapped on the paddock for?
Wesley Problem 2
Ten moorhens are swimming in a pond. At the start, they all randomly and independently decide to swim clockwise or anticlockwise around the edge of the pond, at a rate of one lap a minute. To make fun of the students' embarrassing display, they too decide to turn around if they bump into each other. After a minute, they will occupy the same positions that they collectively started at (why?). What is the probability that every moorhen ends up in the same position as they individually started in?
Wesley Problem 3
Four frogs sit at the corners of a square. If the other frogs are sitting still, a frog can choose to hop over one of their friends, landing on the opposite side of the friend, the same distance away. By hopping however they like, can the frogs sit at the corners of a larger square?
Wesley Problem 4
Ten points are fixed in the plane. You are given 10 coins of radius 1. Does there exist a placement of the points such that there is no way to cover them with the coins without the coins overlapping?
Wesley Problem 5 (Requires basic knowledge of chess)
Wesley and I play a game of two-move chess, where instead of a turn consisting of one legal move, we instead make two legal moves. Deluding ourselves into thinking we are perfect players, can Black win?Â
(We decide to pick a reasonable ruleset, e.g. check ends your turn, checkmate now requires the enemy king not to be able to escape check after the first of their two moves, en passant is valid on both your first and second move in a turn, but the precise ruleset should not matter, so long as it's the same for White and Black).