Consonants Are Free. What Does "Octahedron" Cost?

I love this problem. I love this problem so much. We're at the word store, the store where you buy and sell words. And the way that the word store works is consonants are free. You can stuff your pockets with as many C's and D's as you want. You can buy, you can't even buy. Consonants are free. Take them. The store can't get rid of them fast enough. But vowels, vowels will cost you. At the word store, each vow sells for a different price, but all consonants are free. The word triangle sells for $6. Square sells for 9. Pentagon sells for 7. Cube also seven. And tetrahedrin sells for $8. What is the dollar cost of the word octahedron? Okay, first of all, I don't feel like this is organized particularly well. Let's rewrite the problem slightly. Okay, that's a little bit better. So, if we want the word octahedron, what is it that we need to know? We need to know the cost of an O. A. We need to know the cost of an E. And then we need to know the cost of another o, right? So everything else in ocahedron is free, but o a eo o a eo. That's going to cost us some money. O a eo. Uh what can we figure out? You might be tempted here to set up a system of equations. As much as anyone has ever been tempted to set up a system of equations, I am tempted right now to set up a system of equations. What I mean is I want to say things like I + A + E equals 6. So that's a way in mathematical symbols to describe the cost of the word triangle. Cost only depends on vowels. Triangle has I, A, and E as vowels. And the cost here is $6. Right? Square has a U and an A and an E and it costs $9. Pentagon has an E and an A and an O. Now, one thing you notice me doing right now is rearranging this. There will be a reason for that in a moment, but I mean, most basically, I just want things to line up. I like when things line up. I'm OCD that way. So, O + A + E, that's going to be equal to $7. Cube, that's simple enough. U + E is going to equal $7 as well. And then tetrahedrin. We've got two E's, an A, and an O. And all of that adds up to $8. What are the things I'm learning right now? What am I paying attention to? Well, one thing that I really like right away, O + A + 2 E is equal to 8. But up here for pentagon, O + A + E was equal to 7. There's only one difference between these two equations. The second equation for tetrahedrin had an extra e and its cost of course was $1 more. That is, even though I've gone to the trouble of setting up some equations here, I don't necessarily want to dive into matrix theory and use linear algebra to solve this. I'm not going to set up some kind of 5x6, you know, reduced row echelon form thing going on. I just want to try and use some logic right now. If the only difference between these two equations is one of them has one E and the other one has two E's and the only difference in their price is a dollar, then that one extra E must be costing me $1. So at this store E cost a dollar. Now why am I happy about that? What's exciting? U plus E from cube was equal to $7. Wow. If only a dollar of that cost was coming from the E, then the other $6 of that cost must have been coming from the U. So there it is. A U is $6. Why do I care about that? Take a look here. U + A + E equals 9. Again, the U and the E I know are accounting for $7 of that cost. $1 for the E, $6 for the U. And so the A all by itself must be accounting for the other $2. That gets us to $9 there. And a by itself must be $2. Now we're almost to what we need cuz remember we wanted o a + e plus two o's. And we know every vow cost other than the o. Oh, poor little I. I is not going to matter at all in this problem. Isn't that always the way? Imaginary unit. So what are we going to do? We're going to go back to where O was involved. Pentagon sold for $7. O plus A plus E was supposed to be a $7 cost, but we know A is two and we know E is one together. That's three. And so our $7 cost for pentagon must have been $3 for the A and the E and the other $4 for O. And so there it is. We now know everything we need to know to be able to figure out the cost of the octahedron. two O's. That's 2 * 4 makes 8 plus an A. 8 + 2 is 10 plus an E. 10 + 1 is 11 makes a total cost of $12. And so it must be the case that at the word

store, ocahedron is going to cost us $12. Oo, I don't know. That feels shortish video today. The feed is sponsored by brilliant. org. Please check out brilliant. org/polymathmatic. Drop it in the comments. What did I do wrong? What did I take too long on? Please tell me. I love reading constructive criticism in the comments. And otherwise, I will see you all next time.