# Game against the Devil

While browsing through S.E you notice a spider crawl right out from under your keyboard! Startled, you twitch your hand and left clicked your mouse. You look up in shock as you notice that you just gave an unjustified down vote! You try to fix it as fast as possible, but before you can, the devil himself appears before you and takes you to his realm!

The devil challenges you to his game. The game will take place on a cylinder, in which you start solely on one base of the cylinder. The devil is then able to replicate any amount of you as he pleases to start on the other side of the cylinder (These replicates are obviously evil). Once the game starts your challenge is to get to the other side of the cylinder without being captured by your evil replicates!

The devil makes a promise that if you can beat him in this game, he will return you to your desk and banish his spider from your desk. But if you lose you will be stuck watching the devil abuse your account forever! Would you able to save your account?

• The cylinder is large enough to consider yourself (and the replicates) as zero radius points
• The replicates will play with the best possible strategy to make you lose.
• You and the replicates move at the same speed and at the same time.

If the devil makes you another offer to play, however your replicates receive a greater than 0 radius and in return he allows you to chose the dimensions of the cylinder, should you accept?

• Where on the base do you start? Do they know where you are and vice versa? May 1, 2015 at 19:32
• @kaine In the center or on the edge (which ever you decide), both should be equivalent. (same with the replicates) May 1, 2015 at 19:34
• Is this basically the same thing as puzzling.stackexchange.com/questions/12685/… ?
– JLee
May 1, 2015 at 19:35
• I don't understand the radius thing in the second answer May 1, 2015 at 19:41
• @leoll2 The replicates only have to be within radius length to get you [where radius can be theirs arms length]. (oppose to the exact same location when radius = 0) May 1, 2015 at 19:43

You can't win the first proposal, since the devil can just mirror your starting position and your evil clone© can just mirror your moves.

For the second proposal, I think you could win if you cheat a little on making the cylinder: make a cylinder with uneven base diameters and start on the smaller one. You can turn radially faster than the evil clones© (so, a bit like the duck in a pond question) and potentially escape. Unfortunately, the devil can negate your advantage by having extra evil clones© that will make your rotation ineffective, and have extra clones to chase you.

• You mean an irregular cylinder? May 1, 2015 at 19:49
• Yeah, an irregular cylinder May 1, 2015 at 19:50
• @JonTheMon Very clever May 1, 2015 at 19:59

You can't win the first round because the devil can always move along the circumference of his base, staying aligned with you and, so, not letting you to cross his circumference.

This applies of course to cylinders of any dimension (indeed, my strategy never mentioned sizes).

Do you think that devil would ever let you win?

• But you and the replicates are zero-radius points. It's not enough for the replicates to be at any point on that circumference, they need to be at the exact point that you are. May 1, 2015 at 19:42
• @Cubicon, not sure I get your objection. If you try to cross the circumference, then a replicant will be at the exact point that you are. May 1, 2015 at 19:44
• @Cubicon He is right and the problem is trivial. He moves faster than you in the $\theta$ direction so if at any point he gets the same $\theta$ value as you, you are doomed. As he can generate an infinite number of you, he can make them arbitrarily close so no matter what the dimensions of the cylinder you are doomed. May 1, 2015 at 19:45
• Ah, okay. I misread the scenario. May 1, 2015 at 19:51
• I think one problem with your answer is that it assumes the replicates are allowed to move -slower- than you, whereas in the problem they are required to move at the same speed. So a replicant couldn't actually move in concert with your angular position along the circumference of their base if you were moving diagonally. May 1, 2015 at 21:45