Cosmos Computed

A couple of pieces of good news: First, in this post will be the answer to the problem I gave called Flipping Quarters ; and second, I'll even work through the solution! We have to figure out the average amount of money that the man pays you, minus $5. To start out, let's forget about the money and look at the actual coin flipping. Since it's about as likely for the coin to give tails as it is to give heads, you would get heads 50% of the time. So on your 1st flip, there's a 50% chance you'll get heads - a 50% chance that you'll have to stop. What is the chance that you'll have to stop immediately after the 2nd flip? There's a 50% chance that you will actually get to the 2nd flip, and a 50% chance that you'll flip heads on it. Multiply those numbers, and you get 25%. Look at the diagram to the right if you don't understand. Now for the 3rd flip: there's a 25% chance there will be one (because you got tails on the 2nd flip), and a 50% chance y...

In a couple of my posts, you may notice some computer generated images or animations. They were created with Blender. In this post, I'll explain how I got it and started using it. A long time ago, I wrote a BASIC program that drew a circle; the user would give the program a number, and it would draw a circle that size. There was a problem, though: the circle had a lot of holes in it. I solved the problem by filling the circle, but that was like cheating. I told my dad about the problem, and he showed me a book about graphics. I looked through the book, but didn't find anything about my problem. I did find a section with nice, colorful computer-generated images, however - and some were so good they looked like photos! At one point, there was a series showing the construction of a photo-realistic image of a room, starting with the edges, then moving on to the surfaces, texturing, lighting, etc. Here's a similar series I made using Blender (but not photo-realistic): Edges Su...

Here's an interesting puzzle involving chance: A man in a park asks you to play a game with him. It's a form of gambling. To play, you must pay the man $5, then flip a coin repeatedly until you get heads. As soon as you get heads, you stop flipping. If you only flipped the quarter once, he'll give you $1. If you flipped it twice, you get $2. Three times, $4. Four times, $8. Each extra flip gets you twice as much money, so the longer it takes before you get tails, the more money you get. Should you play, if you have a lot of time and the man will play as many games as you want? How much money, on average, would you gain (subtracting the $5 fee)? I will give the solution in a later post .