# The Multiverse Rescue Mission Riddle

It was a normal Tuesday at the superconductor, until a bug in the system caused your team to be trapped in 11 separate dimensions. Fortunately, there’s a half-finished experimental teleportation robot that may be able to get you all home… if you can figure out how to work it. Can you work out the robot’s design quirks and get your team back home safely? Dan Finkel shows how.

### Transcript:

It was a normal Tuesday at the superconductor, until a bug in the system created a small situation. Now your team is trapped in eleven separate pocket dimensions. Luckily for you, there’s a half-finished experimental teleportation robot that may be able to get you all home, if you can figure out how to work through the quirks of its design.

Over interdimensional radio, your engineers explain that the robot can teleport into the alternate universes you’re trapped in, but it’ll do so completely at random. The robot has two levers and one big button. When it appears, you just switch the position of one of the levers from A to B or vice versa, and then the robot will note your dimensional position and teleport to another of the eleven dimensions at random. If it shows up again, you’ll have to pull a lever before it’ll teleport away. When anyone presses the button, the robot will bring everyone who pulled a lever back home. Anyone who didn’t will be lost in the multi-verse forever.

The challenge is to make sure everyone has pulled a lever before anyone hits the button. While you can talk to each other now over the interdimensional radio and agree on a plan, the robot’s teleportation technology will interfere with all attempts at communication once it arrives. You won’t be able to attach messages to the robot or scratch notes into its superstrong alloy body. Your only way to communicate information is to change the position of exactly one lever or hit the button.

What plan will make sure everyone gets home?

It would be nice if you could set different combinations of the levers to indicate who’s already been visited by the robot. But it has only two levers. That gives four combinations— far too few to communicate about 11 people, especially when you’re forced to flip one to send the robot onward. There must be another way.

The critical insight is that not everyone has to know when every pocket dimension has been visited. If one person accepts responsibility ahead of time for hitting the button, then only they need to know who the robot has visited. In fact, they don’t even need to know exactly who’s been visited… just how many people have been.

You volunteer to be the person in charge of pressing the button when the moment is right, and give the following directions to everyone else. Your plan is simple: you’ll use the left lever to count visits, and the right lever will have no meaning, so there’s no harm in moving it up or down. Each of the others will pull the left lever from position A to position B exactly once. If the robot appears with the left lever already pulled down, or if an individual has previously pulled the left lever down at any point in the past, then they should move the right lever. You, meanwhile, will be the only one who ever resets the left lever from position B to position A. This gives you a way to count how many people have been visited by the robot. Everyone needs to pull the left lever down exactly once, and you’re the only one to pull it back up. So you know that the tenth time the robot visits you with its left lever in the down position, it must have visited all ten of the others. And that means you’re safe to press the button and teleport everyone home. It may take a while– most likely the robot will need to teleport around 355 times; but better that than leave anyone behind. Your teammates phase back into your home dimension one at a time. The mission proves a great success. Well…mostly.

### Ali Kaya

This is Ali. Bespectacled and mustachioed father, math blogger, and soccer player. I also do consult for global math and science startups.