What's the Number on My Hat?

A goblin forewarns N gnomes as follows: "Tomorrow morning, I shall place one hat on each of you. Each hat shall be labeled with some number drawn from the range [0, N-1]. Duplicates are allowed, so two different hats might have the same label. Any gnome shall be able to see numbers on other hats but not his own! When a bell rings, all gnomes shall simultaneously announce one number each. If any gnome succeeds in announcing the number on his own hat, all gnomes shall be set free." The gnomes have assembled in the evening to discuss their predicament. Can you help them devise a strategy?

Tiling a Chessboard with Dominoes

(A) An 8x8 chessboard has had two of its diagonally opposite squares removed, leaving it with sixty-two squares. Can we tile it with 31 non-overlapping 2x1 rectangles (dominoes) such that all squares are covered?

(B) Under what circumstances would removal of two squares from an 8x8 chessboard allow such a tiling? For example, if we remove an arbitrary white square and an arbitrary black square, can the remaining board be tiled?

Tiling a Chessboard with Trominoes

Show that a chessboard of size 2^n by 2^n can be tiled with L-shaped figures of 3 squares, such that only one square remains uncovered. In fact, the uncovered square may be any square — for every choice, there exists a tiling. In fact, the puzzle may be extended to 3D: Eight unit cubes make a cube with edge length two. We will call such a cube with one unit cube removed a "piece". A cube with edge length 2^n consists of (2^n)^{3} unit cubes. Prove that if one unit cube is removed from T, then the remaining solid can be decomposed into pieces.

Fuel Dumps on a Circular Racetrack

A set of fuel dumps on a circular racetrack have just enough gasoline for one car to make one round trip. Prove that there exists a fuel dump from which one car, starting with an empty gas tank, can complete the round trip.

Red Card

Alice repeatedly draws a card randomly, without replacement, from a pack of fifty-two cards. Bob has a one-time privilege to raise his hand just before a card is about to be drawn. Bob must execute his privilege before the last card is drawn. If the card drawn is Red just after Bob raises his hand, Bob wins; otherwise he loses. Is there any way for Bob to be correct more than half the times he plays this game with Alice?

My Cap Color

At the Secret Convention of Logicians, the Master Logician placed a band on each attendee's head, such that everyone else could see it but the person themselves could not. There were many, many different colours of band. The Logicians all sat in a circle, and the Master instructed them that a bell was to be rung in the forest at regular intervals: at the moment when a Logician knew the colour on his own forehead, he was to leave at the next bell. Anyone who left at the wrong bell was clearly not a true Logician but an evil infiltrator and would be thrown out of the Convention post haste; but the Master reassures the group by stating that the puzzle would not be impossible for anybody present. How did they do it?

Twelve Coins

One of twelve coins is counterfeit: it is either heavier or lighter than the rest. Is it possible to identify the counterfeit coin in three weighings on a beam balance?

Grid Infection

In an n by n grid of squares, two squares are neighbors if they share an edge. Initially, some squares are "infected". At successive clock ticks, an uninfected square gets infected if at least two of its neighbors are infected. How many squares must initially be infected so that all squares eventually get infected?

Duplicate Integer

An array of length n+1 is populated with integers in the range [1, n]. Find a duplicate integer (just one, not all) in linear time with O(1) space. The array is read-only and may not be modified. Variation: what if the array may be written into but must be left unmodified by the algorithm?

Find the Angle

Find the measure of angle "a" in the diagram.

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