Impossible!

Is it possible the following chess position to occur in a game?

No, it is impossible. The White’s pawn from e2 should have captured the Black’s bishop from c8. In order for the bishop to get there, the pawn on c6 should have captured one of White’s rooks. It couldn’t be the rook from h1, so it should have been the rook from a1. But in order for the rook from a1 to get to c6, the pawns from b2 and c2 should have been moved to b3 and c4 respectively. However, in that case, the bishop from f1 couldn’t get to a4, since it has been blocked before the capture e2xf3.

Larger or Smaller

Alice secretly picks two different integers by an unknown process and puts them in two envelopes. Bob chooses one of the two envelopes randomly (with a fair coin toss) and shows you the number in that envelope. Now you must guess whether the number in the other, closed envelope is larger or smaller than the one you have seen.

Is there a strategy which gives you a better than 50% chance of guessing correctly, no matter what procedure Alice used to pick her numbers?

Choose any strictly decreasing function F on the set of all integers which takes values between 0 and 1. Now, if you see the number X in Bob’s envelope, guess with probability F(X) that this number is smaller. If the two numbers in the envelopes are A and B, then your probability of guessing correctly is equal to:

F(A) * 0.5 + (1 – F(B)) * 0.5 = 0.5 + 0.5 * (F(A) – F(B)) > 50%.

Wizards with Hats

There are 2 wizards and each of them has infinitely many hats on his head. Every hat has 50-50 chance to be white or black, and the wizards can see the hats of the other person, but not their own. Each wizard is asked to identify a black hat on his head without looking, and they win if both succeed to guess correctly. If the wizards are allowed to devise a strategy in advance, can they increase their chance of winning to more than 25%?

Each wizard guesses the position of the lowest black hat on the head of the other wizard. Then the chance of winning becomes 1/4 + 1/16 + 1/64 + … = 1/3. It can be shown that this is an optimal strategy as well.

Rubik’s Chess

Last week we found out that Puzzle Pranks Co. have invented a new type of puzzle – Rubik’s Chess. The goal is simple – you get a scrambled cube with various chess pieces on its sides, and you must unscramble it so that on each side there is one mated King, assuming the kings cannot capture the neighboring pieces (Queens, Rooks, Bishops, kNights).

We are usually good with this type of puzzles, but we spent our entire weekend trying to solve this one without any success. We even started wondering if it can be actually solved, so decided to share it with you and see if you can help us figure that out.

Below you can see the way the cube looks when seen from 8 different angles:

Remark: The orientations of the pieces are irrelevant to the final solution, i.e. they don’t need to be consistent on each side.

The Rubik’s Chess puzzle cannot be solved. You can see a detailed solution HERE.

Queen’s Death

On which spot was the white queen captured?

Since the pawns on e6 and h6 have taken 2 of the White’s pieces, and the only two white pieces which could get there are the knight and the queen, the answer is one of these two squares. Similarly, the pawn on b3 should have taken the Black’s c8 bishop, and this should have happened before the White’s queen was taken. Therefore first the white knight was taken on e6, then the black bishop on b3, and finally the white queen on h6.

Deadly Vodka

Two politicians go to a bar and order two glasses of vodka on the rocks. The first politician quickly empties his glass, then orders a second one, a third one… The second politician patiently drinks his own vodka, but about 20 minutes later, he drops down dead. The police discovered that the barman tried to assassinate both politicians, but how come the second one died and the first one lived?

The poison was in the ice cubes, so the second politician drank them when they melted in his drink.

Breakfast with Friends

One early morning, a group of friends meets in their favorite café.

Ash, the biggest in the group, remarked :

“Must have been millions of years since we were all together, uh?”

Affie, who was wearing her brand new elephant pants, nodded. Anthony kept on complaining about the weather back home.

“Yeah, I am glad to be here with you, it is so cold at my place!”

Eugenie, who was saddened by the loss of a friend, was looking at Samuel and Namur, who were arguing about the possible future election of Donald Trump.

“These two are really inseparable”, she said to herself.

Octavia, the smallest among her friends, stood and spoke :

“Guys, I have a surprise for you! We’re going to the opera tonight!”

The waiter, waiting for the orders, wondered why these customers reminded him of something. But all of a sudden, he said:

“Ladies and gentlemen, may I suggest some sliced bread with butter, slices of cheese and ham? We also have croissants and other pastries. And for drinking, is coffee fine? We also have tea, of course, and orange and apple juice for you.”

Which is this group of friends, and what came to the waiter’s mind?

These are the seven continents – Ash (Asia), Affie (Africa), Anthony (Antarctica), Eugenie (Europe), Samuel (South America), Namur (North America), Octavia (Oceania). Their entire conversation consists of various hints. In the end, the waiter was thinking about bringing them Continental Breakfast.

Source:

Puzzling StackExchange

Cork in a Glass

You have a jar filled with water and a glass. If you pour some water into the glass and place a cork in it, the cork will float towards the edges of the glass. What is the easiest way to make the cork float towards the center?

Since the liquid molecules adhere to the glass molecules on the sides of the glass, the water level there is higher and buoyancy makes the cork float in that direction. If you fill the glass all the way to the edge, then the water surface will be convex and the cork will float towards the center.

Magic Liquid

You buy a bottle with a letter from the merchant, the merchant tells you that when you drink the liquid in the bottle it grants you eternal life, he supposedly deciphered this from the letter.
After you get home you decide to study the letter if it really says what the merchant told you, can you figure out if the bottle really grants eternal life?

You come to a fork in the road.
To the left is an empty well made from stone.
On the right is a pirate’s buried treasure.
Ahead you only see a tall straight tree.
The night is dark with only a dying moon in the sky.

Source: Puzzling StackExchange

The objects described in the last paragraph have the following shapes:
fork in the road = T
empty well = O
buried treasure = X
straight tree = I
dying moon = C
The 5 letters form the word “TOXIC”, which suggests you shouldn’t drink from the bottle.