Problem of the Week #53: Monday July 8th, 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.
Starting with a natural number at the top, we can split it at each level such that the two lower neighbours always add up to the higher neighbour. Let’s repeat this process until a row can’t be split using whole numbers in the following row anymore.
For example, here’s one way to split the number 13:
a). How do we know when we have reached the bottom and can’t split anymore?
b). Can every natural number be split this way until there are only 1s and 0s in a row? For example, here’s one way to split 13 until there are only 1s and 0s in a row:
c). Given a natural number, in how many ways can it be split this way until there are only 1s and 0s in a row? Here are two ways to split the number 5:
d). Can we predict the number of times a natural number appears in Pascal’s Triangle? For example the number 6 appears 3 times, and the number 10 appears 4 times.
e). Which numbers can be split until there are all 1s in a row? What about alternating 1s and 0s?
f). Given a natural number, can we predict the shortest row in which it can be split into only 1s and 0s? For example how many rows do we need to split 33 into 1s and 0s?
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.