Saavedra Position

White to play. Is this game a win for White, Black, or a draw?

This game is a win for White.

1. c7 Rd6+
2. Kb5 Rd5+
3. Kb4 Rd4+
4. Kb3 Rd3+
5. Kc2! Rd4!
6. c8=R! Ra4
7. Kb3

Now Black will either lose the rook, or get mated in one. If White promoted a Queen instead of a Rook, then 6… Rc4+ would lead to 7. Qxc4, which is a stalemate.

Securing the Box

There are 5 people who possess a box. You are allowed to secure the box with as many different locks as you like and distribute any combination of keys for these locks to any people among the 5. Find the least number of locks needed, so that no 2 people can open the box, but any cannot people can open it.

For every subset of 2 people you pick among the 5, there should be a lock which none of the 2 can unlock, and each of the remaining 3 people can unlock. Clearly, the lock in question cannot be the same for any two different subsets of 2 people you choose. Therefore the number of locks you need is at least the number of different 2-element subsets of a 5-element set, which is 5!/(2!3!)=10. This number is sufficient as well – just give keys to a different group of 3 people for every lock.

Relabeling Dices

Can you relabel two 6-sided dices, so that every face has a positive number of dots, and also their sum has the same probability distribution?

Yes, you can do this. The easiest way is to use generating functions. Using simple polynomial algebra, you can see that

(x + x2 + x3 + x4 + x5 + x6)2 = (x + 2x2 + 2x3 + x4)(x + x3 + x4 + x5 + x6 + x8).

Therefore, if you take a dice with spots {1, 2, 2, 3, 3, 4}, and a dice with spots {1, 3, 4, 5, 6, 8}, their sum will have the same probability distribution.

The Lion and the Zebras

The lion plays a deadly game against a group of 100 zebras that takes place in the steppe (an infinite plane). The lion starts in the origin with coordinates (0,0), while the 100 zebras may arbitrarily pick their 100 starting positions. The lion and the group of zebras move alternately:

  • In a lion move, the lion moves from its current position to a position at most 100 meters away.
  • In a zebra move, one of the 100 zebras moves from its current position to a position at most 100 meters away.
  • The lion wins the game as soon as he manages to catch one of the zebras.

Will the lion always win the game after a finite number of moves? Or is there a strategy for the zebras that lets them to survive forever?

Source: Puzzling StackExchange

The zebras can survive forever. They choose 100 parallel strips with width 300m each, then start on points on their mid-lines. If the lion lands on some zebra’s strip, the zebra simply jumps 100m away from the lion, along its mid-line.