Category: Uncategorized
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Suppose you have a fair 6 sided die, and you’re rolling it repeatedly. On average, how many rolls will it take to get a 1? The answer is 6. And I mean, that makes sense, right? There’s a 1/6 chance you roll a 1 each time, so it should happen about 1 in every 6…
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Consider a 5×5 square grid, where each square can be colored either black or white. We’re going to impose some rules on this grid.Rule 1: Each vertex must be adjacent to at least 1 and at most 3 white squares.Rule 2: No checkerboards (i.e. two white squares and two black squares diagonally opposite each other).Rule…
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Here’s a random proof I came up with, and I already had it written up so I thought I may as well post it here. Assume A and B are coprime. Lemma 1: AB is the smallest number you can express as a sum of A’s and B’s in two different ways.Proof: Suppose there was…
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This is a continued fraction: To evaluate it, just simplify from the bottom up. That one comes out to 225/157.For the rest of this, I’m going to use the following notation: 225/157 = [1,2,3,4,5].Each number has a unique continued fraction (as long as you don’t use 0), and the rational numbers all eventually terminate. But…
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First week of the new schedule! I already missed it. Anyways, take a basic six-sided die.You know, the one that has the numbers from 1 through 6 on it. Roll it twice, and add up the total.The result can be anywhere from 2 to 12, but it’s most likely to be 7. You can see…
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I thought of this as a potential math competition problem, but I think it’s better if I share it here!Consider the following setup:You have a circle of radius 1. Call it O. Two other circles of radius 1/2, A and B, are internally tangent to O and externally tangent to each other, like so: Now,…
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First, some background.The number of natural numbers (1, 2, 3, 4, …) and the number of integers (…, -2, -1, 0, 1, 2, …) are both infinite. But are they the same kind of infinite?The answer is yes, because you can match each integer to a natural number. Order the integers like this:0, 1, -1,…
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Let’s say we have a sequence, and we want to guess what polynomial generated it.For example:3, 10, 19, 30, 43, 58, 75, …Finite differences helps with determining that. How it works is, take the difference between each pair of terms.10 – 3 = 7, then 19 – 10 = 9, then 30 – 19 =…
