Puzzle Socks Giveaway 3

Hey, puzzlers, our friends at Soxy are offering their comfy socks in a new puzzle pattern and want to share 5 pairs with you. Solve the puzzle below, post the answer on our Facebook wall, and you can be the lucky winner of a whole set of fun socks. Click the banner below to check Soxy’s other cool items.


Last week, I got from Soxy 1 pair of brown socks, 3 pairs of brown shoes, 2 pairs of black socks, 2 pairs of black shoes, and put them all in a wooden chest. How many times should I pick a random item from the chest, so that I end up with all-matching shoes and socks to wear on Comic-Con?

ANSWER ON FACEBOOK

Blue and Red Points

You have 100 blue and 100 red points in the plane, no three of which lie on one line. Prove that you can connect all points in pairs of different colors so that no two segments intersect each other.

Connect the points in pairs of different colors so that the total length of all segments is minimal. Now, if any two segments intersect, you can swap the two pairs among these four points and get a smaller total length.

Seeing Theory

Seeing Theory is a beautifully designed website, which aims to educate people about probability theory via series of visual and interactive lessons. If anyone is struggling to grasp some of the basic concepts in this field of mathematics or is just getting into it, the website can be a very useful learning tool. Seeing Theory was designed by Daniel Kunin as an undergraduate project in Brown University and has won numerous awards. To visit the website, click the banner below.

Close the Loop

Alex and Bob are playing a game. They are taking turns drawing arrows over the segments of an infinite grid. Alex wins if he manages to create a closed loop, Bob wins if Alex does not win within the first 1000 moves. Who has a winning strategy if:

a) Alex starts first (easy)
b) Bob starts first (hard)

Remark: The loop can include arrows drawn both by Alex and Bob.

In both cases, Bob wins. An easy strategy for part a) is the following:

Every time Alex draws an arrow, Bob draws an arrow in such a way, that the two arrows form an L-shaped piece and either point towards or away from each other. Since every closed loop must contain a bottom left corner, Alex cannot win.

For part b), Bob should use a modification of his strategy in part a). First, he draws a horizontal arrow. Then, he splits the remaining edges into pairs, as shown on the image below. If Alex draws one arrow on the grid, then Bob draws its paired arrow, such that the two arrows point either towards or away from each other. The only place where a loop can have a bottom left corner is where Bob drew the first arrow. However, if a loop has a bottom left corner in this positio, then it must have at least one more bottom left corner, which is impossible.