Surely you’ve heard of the Rubik’s Cube-Off, the annual speed solving cube competition? You know, the one that takes place on Pi Day, i.e. March 14, and that you qualify for by reciting as many digits of pi as you can? No? Well, that’s because my frequent math mostly co-host, Jed Rendleman, hasn’t reported on it yet. Or it could be because it doesn’t exist. Jed is one of my many excellent students this semester at Wesleyan. He’s been working with me on the radio versions of the last few puzzles and is really great at asking questions and being funny and all the other stuff that makes for a wildly successful math radio show. Anyway, this weeks puzzle story is about Jed’s spring break and how I helped him cheat at the cube-off. Which of course, just to be clear in case Jed is applying for grad school (or someone is background checking me!) didn’t actually happen. No wire was worn into a no-communication-allowed tournament. No windowless van was employed. No digits of pi were recited. But it does make for some kind of puzzle.
So, the story is that Jed, being a journalist, wanted to go behind the scenes to report on the Cube-Off for the Wesleyan Argus. He decided to enter as a contestant to get a real taste of the action at this, the bloodiest and most contentious Rubik’s Cube competition in the world. The qualifying round, which consisted of reciting at least 120 digits of pi in under 1 minute, was not problem for Jed. Being a bit of a savant, he had no problem memorizing the digits in a very short time, and handily qualified for the tournament. He took his place among 10,040 other pi stars hoping to be the one winner of the Cube-Off. So how does this competition work, you ask? Well, first of all, beyond the qualifying round, it has nothing to do with pi. Except they serve pie at the winner’s banquet, in celebration of the hallowed date. The tournament itself is just your basic single elimination tournament, in which contestants go head-to-head in many rounds of cube-solving. In each match, two contestants are given scrambled cubes. They compete to solve the cube as quickly as possible. The one who solves it faster stays in the running, while the slower solver is out of the tournament.
Because of the nature of the qualifying round, every year a different number of people qualify to compete in the tournament. That means the organizers can’t plan out the structure in advance. They don’t even know how many matches to schedule. So they have to figure it all out very quickly after the qualifying round, so that the Cube-Off can begin. Alright, so the first question is, how many matches are there in the tournament? The tournament judges decide to offer a cash prize to the first contestant who can tell them.
Here’s where the ethics violation comes in. You see, Jed was afraid that he would blow his cover at the Cube-Off because it turns out that he is not really nerdy enough to be at one of these things. He doesn’t know any Star Trek jokes, has no idea how many sides the D&D dice have, can’t extensively discuss the merits of the old TI-85 versus the new TI-82, let alone take a side in the TI versus HP debate. That last is all about graphing calculators, by the way. Hopefully you, like Jed, have no idea what I’m talking about. But the Cubers do! Jed was afraid that the other contestants would find him out as a fake nerd and he would be a tournament outcast before he had a chance to gather enough material for his story. So he’d asked me to be a sort of nerd coach, on call in case he needed some explanation or tips in passing for the kind of person who would be into all that stuff. We decided that he should wear a wire and that I should be on call outside, in a van, ready to help in emergencies. This was totally illegal, because outside help, like all calculating devices, are banned from the tournament. They don’t want anybody programming some kind of cube solver into their TI-85s.
Okay, so this is getting a little out of hand long, so I will get to the point. Jed used his wire to ask me how many matches would need to be played in this years tournament of 10,041 people (including Jed) to get a Cube champion. As a math puzzler, I immediately knew the answer. I did one incredibly easy mathematical operation in my head and told him how many matches it would take. He answered, and won the cash prize. And immediately forgot all about the story he was writing.
So this week’s puzzle is: How many matches are there in a single elimination tournament of 10,041 people, and what is the insight that let me figure it out incredibly quickly with a single mathematical operation? If you know the answer, write me an email at email@example.com. The best submission I get by Wednesday, April 4 will earn its writer an excellent t-shirt!
Solution to the planes puzzle: Ah! So, this puzzle got a few answers despite the spring break lull. Here’s one solution. You can do it with 3 planes. Call them A, B, and C. Airplane A will be the one to fly all the way around the world. Let’s say it takes 8 hours to fly around the world, just to keep it straight. A, B, and C take off to fly “clockwise” around the world. 1/8th of the way (or after 1 hour), plane C gives planes A and B each enough fuel to go an additional 1/8th of the way. Then C turns around and goes back to base. 1 hour later, 1/4 of the way around the world, plane B gives plane A enough gas to go an additional 1/8th of the way around the world, then turns around to go back to base. Plane A is now 1/4 of the way around the world, has a full tank of gas, and 2 hours have passed. Now, after 2 hours later, when B arrives back at base, B and C fuel up and take off to go around the world in the opposite direction. They go 1/8th of the way around the world that way, taking one hour, then C gives enough gas to go an additional 1/8th of the way around. C turns around and goes back to base. 5 hours total have passed. After another hour passes and B has gone another 1/8th of the way around the world, A meets B. B gives A enough gas to go 1/8th of the way around (I should have named a unit for this!!), then turns around and proceeds with A back towards base. 6 hours have passed. At this moment, C leaves base with a full tank of fuel. After an hour, when 7 hours have passed (and A has made it 7/8ths of the way around the world), C meets A and B and gives them each enough fuel to get back to base. Everybody gets home safe, at the same time!