Prank the Professors

Three professors fell asleep under a tree. At some point a prankster passed by and painted their faces with black dye. When the professors woke up, each of them saw the others’ faces and started laughing at them. After a while though, they stopped laughing, realizing that their own faces were painted as well. How did they deduce that?

Let us denote the professors with A, B and C. The thought process of A would go like this: “If my face is not painted, then B will see that C is laughing at him and will realize immediately that he is being pranked. However, B was laughing for a while and therefore my I must being pranked as well.”. Then A will stop laughing and the same will happen with the other two professors B and C.

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Repetitive

You have two groups of words:

  1. black, word, English, brief, noun, grandiloquent, last
  2. white, number, Russian, long, verb, unpretentious, first

To which group does “repetitive” belong?

The first group contains self-explanatory words (known as autologicals), the second group does not. Therefore “repetitive” should belong to the first group.

Chocolate Bar

Louis has a bar of chocolate 4×6 which is marked into 24 little squares. At each step, he breaks up one of its pieces along any of the marked horizontal/vertical lines. Show that no matter how he does that, it will always take the same number of steps until the chocolate is broken into single 1×1 pieces.

Every time he splits the chocolate, the number of pieces increases by 1. Therefore it will always take him 23 steps to split it into single pieces.

Four Points in the Plane

Find all configurations of four points in the plane, such that the pairwise distances between the points take at most two different values.

All 6 configurations are shown below: a square, a rhombus with 60°-120°-60°-120°, an equilateral triangle with its center, an isosceles triangle with 75°-75°-30° and its center, a quadrilateral with 75°-150°-75°-150°, and a trapezoid with base angles of 72°.

Guess the Fruits

You are given 3 boxes – one labeled “Apples”, one labeled “Bananas”, and one labeled “Apples and Bananas”. You are told that the labels on the boxes have been completely mismatched, i.e. none of the three labels is put on its correct box. How can you open just one box and pick a random fruit from it, so that after seeing the fruit, you can guess correctly the contents of every box out of the three?

Open the box labeled “Apples and Bananas”. If you pick a banana from it, then the box labeled “Bananas” will contain apples, and therefore the box labeled “Apples” will contain apples and bananas. Similarly, if you pick an apple from it, then the box labeled “Apples” will contain bananas, and therefore the box labeled “Bananas” will contain apples and bananas”.

Thank You!

A cowboy walks into a bar and asks the barman for a glass of water. The barman pulls out a gun instead and points it at the man. The man genuinely says “Thank you” and walks out.

What happened?

The cowboy had hiccups and needed water. The barman shocked him with his gun instead and that cured the hiccups.

Twiddled Bolts

Two identical bolts are placed together so their grooves intermesh. If you move the bolts around each other as you would twiddle your thumbs, holding each bolt firmly by the head so it does not rotate and twiddling them in the direction shown below, will the heads:

(a) move inward
(b) move outward, or
(c) remain the same distance from each other?

One of the bolts will be screwing itself, and the other one will be unscrewing itself. This will happen at the same pace and the bolts will remain the same distance from each other. Thus the answer is (c).

The Missing Dollar

Three people check into a hotel room and each of them gets charged $10 – a total of $30. Later the clerk realizes that the bill is just $25, so he sends the bellboy to return $5 to the guests. On his way to the room, the bellboy decides to cheat and pockets $2 of the money, and gives the three men just one dollar each. Now the three men have spent $9 each, for a total of $27. Additionally, the bellboy took $2 for himself, which adds up to $27 + $2 = $29. Since the guests originally handed over $30, the question is what happened to the remaining $1?

The calculation is made the wrong way. The three men originally gave $30, but later $5 of them were sent back, which makes it $30 – 5 = $25 left at the clerk. Each of the men spent $9, so they gave $27 in total, $2 of which ended up in the bellboy’s pocket. $27 – $2 = $25, so no “missing dollar” here.

3 x $9 – $2 = $30 – $5.