The following example was brought up in two different classes that I am taking, within a couple days of each other. It is the classic example of an immeasurable set on the interval [0,1]. Now in math, words like measure and measurable have technical definitions, and lead to bizarre results like the Banach-Tarski paradox. I’m going to give a fairly informal explanation here. Continue reading
Here are a number of misleading sequences. I have given the first few numbers of each sequence, and your mission, if you choose to accept it, is to guess the next number.
1. A Doubling Dilemma
1, 2, 4, 8, 16, ___?
Did you get 32 from doubling?
The answer is actually 31. Continue reading
A link to this article in pdf format: [pdf]
Here is a difficult probability question:
Suppose you are standing on an infinitely large square grid at the point (0,0), and suppose that you can see infinitely far but cannot see through grid points. Given a random grid point z = (x,y), where x and y are integers, what is the chance you can see z?
Abstract: The main statistical result of this paper is that of the reddit users who have upvoted at least one post, 31.4% of them have upvoted only one post and, moreover, have been the only person to upvote it. Conversely, 22.6% of posts which have been voted on at all have received only downvotes and no upvotes. In addition, 67.4% of users are connected in a giant component of upvoting common posts, while 91.4% of posts are connected by having been upvoted by common users.
In the Algebraic Number Theory course that I am taking, we covered the problem of when a number is the sum of two squares. I wrote a short expository paper [pdf] to go with it. Here are the positive integers up to 15 which can and cannot be written as the sum of two squares:
- 1 = 1² + 0²
- 2 = 1² + 1²
- 4 = 2² + 0²
- 5 = 2² + 1²
- 8 = 2² + 2²
- 9 = 3² + 0²
- 10 = 3² + 1²
- 13 = 3² + 2²
Can you find the pattern? (Note: This is a very hard question. You may need to write down more examples to start noticing it.)
I have also posted some hints.