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Category Archives: sequences
Predicting pi: pretty graphs and convergents
Recall the challenge I posed in a previous post: given the sequence of integers , what can you learn about (assuming you didn’t know anything about it before)? The answer, as explained in another post, is that you can learn … Continue reading
Posted in convergence, famous numbers, pattern, sequences
Tagged approximation, convergents, graphs, pi
5 Comments
Predicting Pi: solution
Now for the solution to the question in my previous post, which asked what you can learn about , given the sequence of integers . Nick Johnson commented: Well, the obvious thing one can learn given just (10^n)r is the … Continue reading
Posted in convergence, pattern, sequences, solutions
Tagged approximants, approximation, floor, pi, sequence
5 Comments
Predicting Pi
Inspired by a recent post over at Foxmaths!, here’s an interesting challenge question for you to think about: Suppose I give you the sequence of integers , and so on, where denotes the greatest integer less than or equal to … Continue reading
Challenge #12 solution, part III
And now for the solution to problem #3 from Challenge #12, which asked: how many ways are there to write a positive integer n as a sum of powers of two, with no restrictions on how many powers of two … Continue reading
Recounting the Rationals, part II (fractions grow on trees!)
Today I’d like to continue my exposition of the paper “Recounting the Rationals”, which I introduced in a previous post. Recall that our goal is to come up with a “nice” list of the positive rational numbers — where by … Continue reading
Posted in infinity, number theory, pattern, recursion, sequences
22 Comments
Recounting the Rationals, part I
This is the first in a series of posts I’m planning to write on the paper “Recounting the Rationals“, by Neil Calkin and Herbert Wilf, mathematicians at Clemson University and the University of Pennsylvania, respectively. I’m really excited about it, … Continue reading
Posted in counting, infinity, number theory, pattern, sequences
19 Comments
Perfect numbers, part I
Today I’d like to talk about perfect numbers, which touch on some clever algebra and some neat topics in number theory. At least, I’ll start talking about them, since it will probably take me three or four posts to say … Continue reading
Posted in famous numbers, number theory, sequences
3 Comments
A few related problems…
Here is a collection of interesting math problems. Despite appearances, they all have something in common. Can you figure out what it is? A single pair of baby rabbits is placed on an island. They take one month to grow … Continue reading
Posted in famous numbers, fibonacci, pattern, sequences
3 Comments
The Fibonacci sequence: an introduction
The Fibonacci sequence is probably one of the most famous — and most widely writtenabout — number sequences in all of mathematics. So why write about it, if it’s been written about so much already? Well, because I can, and … Continue reading
Posted in famous numbers, fibonacci, iteration, pattern, sequences
11 Comments