Problem of the Week #35: Monday September 4th, 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 trying to scramble a string of numbers, say 1, 2, 3, 4, 5, but with the requirement that none of the numbers can end up where they started. Let’s define this as a complete scramble. For example, the following is an incomplete scramble of the 5 numbers above because one of the numbers ends up where it started:
Notice in the above example, the number 1 is moved to the 4th place, then 4 is moved to the 5th place, 5 is moved to the 2nd place, and 2 is moved to the 1st place. This sequence of rearrangements (1, 4, 5, 2) where the numbers each moves to the next position is called a loop. This loop involves 4 numbers, so it has a length of 4.
a). What are all the possible lengths of loops in a complete scramble of 5 numbers?
b). How many loops can we possibly have in a complete scramble of 5 numbers?
c). How many complete scrambles of 5 numbers have exactly 1 loop of length 2?
d). How many complete scrambles of 5 numbers are there in total?
e). How many loop length combinations are there in a complete scramble of n numbers, where n is a natural number?
f). How many complete scrambles of n numbers have exactly 1 loop of length k, where n and k are natural numbers?
g). How many complete scrambles of n numbers are there in total, where n is a natural number?
h). Share your own problem inspired by this one.
i). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.