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  • The evil wizard MoldeVort has been trying to kill you for years,

  • and today it looks like he's going to succeed.

  • But your friends are on their way, and if you can survive until they arrive,

  • they should be able to help stop him.

  • The evil wizard's protective charms ward off every spell you know,

  • so in an act of desperation you throw the only object in reach at him:

  • Pythagoras's cursed chessboard.

  • It works, but with a catch.

  • MoldeVort starts in one corner of the 5x5 board.

  • You have a few minutes to choose four distinct positive whole numbers.

  • MoldeVort gets to say one of them, and if you can pick a square on the board

  • whose center is exactly that distance away,

  • the curse will force him to move to that spot.

  • Then he'll have to choose any of the four numbers,

  • and the process repeats until you can't keep him inside the board

  • with legal moves.

  • Then he'll break free of the spell and almost certainly kill you.

  • What four numbers can you choose to keep MoldeVort trapped by your spell

  • long enough for help to arrive? And what's your strategy?

  • Pause the video to figure it out yourself.

  • Answer in 3

  • Answer in 2

  • Answer in 1

  • The trick here is to keep MoldeVort where you want him.

  • And one way to figure out how to do that

  • is to play out the game as MoldeVort would:

  • always trying to escape.

  • You're dealing with a relatively small board,

  • so the numbers can't be too big.

  • Let's start by trying 1, 2, 3, 4 to see what happens.

  • Moldevort could escape those numbers in just three moves.

  • By saying 2, then 3,

  • he would force you to let him into one of the middle points of the grid,

  • and then a 4 would break him free.

  • But that means you'll need to allow a number larger than 4,

  • which is the distance from one end of a row to another.

  • How is that even possible?

  • Through diagonal moves.

  • There are, in fact, points that are distance 5 from each other,

  • which we know thanks to the Pythagorean Theorem.

  • That states that the squares of the sides of a right triangle

  • add up to the square of its hypotenuse.

  • One of the most famous Pythagorean triples is 3, 4, 5,

  • and that triangle is hiding all over your chessboard.

  • So if MoldeVort was here, and he said 5, you could move him to these spaces.

  • There's another insight that will help.

  • The board is very symmetrical: If MoldeVort is in a corner,

  • it doesn't really matter to you which corner it is.

  • So we can think of the corners as being functionally the same,

  • and color them all blue.

  • Similarly, the spaces neighboring the corners behave the same as each other,

  • and we'll make them red.

  • Finally, the midpoints of the sides are a third type.

  • So instead of having to develop a strategy

  • for each of the 16 spaces on the outside of the board,

  • we can reduce the problem to just three.

  • Meanwhile, all the inside spaces are bad for us,

  • because if MoldeVort ever reaches one,

  • he'll be able to say any number larger than 3 and go free.

  • Orange spaces are trouble too, since any number except 1, 2, or 4

  • would take him to an inside space or off the board.

  • So orange is out and you'll need to keep him on blue and red.

  • That means 2 is bad,

  • since it could take MoldeVort to orange on the first turn.

  • But the four other smallest numbers, 1, 3, 4, and 5, might work.

  • Let's try them and see what happens.

  • If MoldeVort says 1, you can make him go from blue to red or red to blue.

  • And the same works if he says 3.

  • Thanks to our diagonals, this is even true if he says 5.

  • If he says 4, you can keep him on the color he's already on

  • by moving the length of a row or column.

  • So these four numbers work!

  • Even if your friends don't get here right away,

  • you'll be able to keep the world's most evil wizard

  • contained for as long as you need.

The evil wizard MoldeVort has been trying to kill you for years,

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    jeremy.wang posted on 2020/05/22
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