Problem of the Week #65: Monday September 30th, 2024
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.
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This problem is inspired by this twitter post.
We have a bracelet made of yellow and green beads, and we want to divide each coloured beads evenly among friends with the least number of cuts. Eg, here is a way to divide this bracelet between 2ppl with 2 cuts.
a). How many cuts do we need to divide this bracelet among 3ppl?
b). What’s the least number of cuts that guarantees we can divide any bracelet with 6 yellow beads and 6 green beads evenly among 3 people?
c). Notice these 2 segments were exactly 1 person’s portion each when dividing among 3 people.
Prove or disprove: We can always find at least 2 non-overlapping segments, containing exactly 1 person’s portion each, when dividing a bracelet with km yellow beads and kn green beads among k people.
d). Derive a formula for the number of cuts needed to divide a bracelet with km yellow beads and kn green beads evenly among k people.
e). What’s the least number of cuts that guarantees we can divide a bracelet with 6 yellow beads, 3 green beads, and 3 blue beads evenly among 3 people?
f). Is it possible to derive a formula for the number of cuts needed to divide a bracelet with beads in c different colours among k people if the number of beads in each colour is a multiple of k?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.