Euclidea

Which was your favorite part of the Mathematics you learnt back in high-school? Not sure about you, but we definitely loved the construction problems, where we had to draw some shape, using just a straightedge and a compass. Even though we haven't solved such problems for many years, we got very excited to discover the amazing game Euclidea, designed by the guys from HORIS International Ltd. If the description below seems intriguing to you, make sure to visit the game's website and test your skills by clicking the provided link.

Euclidea is all about building geometric constructions using straightedge and compass. About doing it the fun way. With Euclidea you don’t need to think about cleanness or accuracy of your drawing — Euclidea will do it for you. But it’s also a game. A game that values simplicity and mathematical beauty. Find the most elegant solution — the one, which is built in the least possible moves, — and you’ll get the highest score.

http://www.euclidea.xyz

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 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 helps them to survive forever?


Solution

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.


Friends and Enemies

Show that in each group of 6 people, there are either 3 which know each other, or 3 which do not know each other.


Solution

Let's call the people A, B, C, D, E, F. Person A either knows at least 3 among B, C, D, E, F, or doesnt know at least 3 among B, C, D, E, F.

Assume the first possibility - A knows B, C, D. If B and C know each other, C and D know each other, or B and D know each other, then we find a group of 3 people which know each other. Otherwise, B, C and D form a group in which noone knows the others.

If A doesn't know at least 3 among B, C, D, E, F, the arguments are the same.


Broken Clock

An old wall clock falls on the ground and breaks into 3 pieces. Describe the pieces, if you know that the sum of the numbers on each of them is the same.


Solution

The total of all numbers on the clock is 1 + 2 + ... + 12 = 78. Therefore each piece must contain numbers with total sum 26. The only way for this to happen is if the pieces are broken via 3 parallel lines - {1, 2, 11, 12}, {3, 4, 9, 10}, {5, 6, 7, 8}.


OK GO - "The Writing's On the Wall"

How much effort and innovation can a band put into producing their music videos? When it comes to the American rock-band OK GO, the answer is A LOT. Becoming a notable youtube presence around 2010 with their famous "Treadmill Dance", OK GO keep pushing the boundaries of creativity in each consecutive video they make. Check out this amazing optical illusion video by these so talented guys, and if you enjoy it, make sure to see their other projects as well. Truly brilliant!

Cover the Grid

You must cover a 7x7 grid with L-shaped triminos and S-shaped tetrominos, without overlapping (flipping and rotating is permitted). What is the minimum number of pieces you can use in order to do this?

Remark: All pieces must placed within the board.


Solution

Consider all cells in the grid which lie in an odd row and odd column - there are 16 of them. Since each of the two pieces can cover at most 1 of these cells, we need at least 16 pieces. Giving an example with 16 pieces is easy.


Pinned Men

The following game is played under very specific rules - no pinned piece checks the opposite king. How can White mate Black in 2 moves?


Solution

First, White plays f3 and threatens mate with Qxe2. Indeed, blocking with the black rook on d4 will not help, because it will become pinned, which means that the rook on d6 will become unpinned, which will make the bishop on b6 pinned, and that will unpin the knight on c7, resulting in mate. Below are listed all variations of the game.

  1. ... Rd5 2. Qxe2#
  2. ... Bxa5 2. Kc8#
  3. ... Bxc7 2. Nxc7#
  4. ... Bxe8 2. Kxe8#
  5. ... Qxe7+ 2. Kxe7#
  6. ... Rd2 2. Bxd2#
  7. ... Rxd6+ 2. Qxd6#