Individual puzzles

A finite number of lines are dropped onto the plane. How many more lines must be added to ensure that every enclosed region of the plane is triangular? Is this possible?

Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yesno questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which. (via)

The integer 8 can be written as the sum of two squares of integers, m^2 + n^2, in four ways, when (m, n) is (2, 2), (2, 2), (2, 2), or (2, 2). The integer 7 can’t be written at all as the sum of such squares. Over a very large collection of integers from 1 to n, the average number of ways an integer can be written as the sum of two squares approaches π. Why? (via)

Counter Play: A bag contains one counter, known to be either white or black. A white counter is put in, the bag shaken, and a counter drawn out, which proves to be white. What is now the chance of drawing a white counter? (via)

Seven Tails: Here are seven pennies, all heads up. In a single move you can turn over any four of them. By repeatedly making such moves, can you eventually turn all seven pennies tails up? (via)

Opposites Exact: Prove that, at any given moment, there are two points on the equator that are diametrically opposed yet have the same temperature. (via)

The Dwarfs Problem: Seven dwarfs are sitting at a round table. Each has a cup, and some cups contain milk. Each dwarf in turn pours all his milk into the other six cups, dividing it equally among them. After the seventh dwarf has done this, they find that each cup again contains its initial quantity of milk. How much milk does each cup contain, if there were 42 ounces of milk altogether? (via)

You are a hunter in the forest. A monkey is in the trees, but you don’t know where and you can’t see it. You can shoot at the trees, you have unlimited ammunition. Immediately after you shoot at a tree, if the monkey was in the tree, he falls and you win. If the monkey was not in the tree, he jumps (randomly) to an adjacent tree (he has to). Find an algorithm to get the monkey in the fewest shots possible. (via)

Three people enter the room, each with a hat on their head. There are two colors of hats: red and blue; they are assigned randomly. Each person can see the hats of the two other people, but they can’t see their own hats. Each person can either try to guess the color of their own hat or pass. All three do it simultaneously, so there is no way to base their guesses on the guesses of others. If nobody guesses incorrectly and at least one person guesses correctly, they all share a big prize. Otherwise they all lose. One more thing: before the contest, the three people have a meeting during which they decide their strategy. What is the best strategy? (via)
Collections
 Quant Interview Questions (p. 812)
 Various Problems of the Day
 Physics Problems and Solutions for Real World Applications
 Physics Questions/Problems (Yakov Kantor)
 Physics Problems to Challenge Understanding, emphasizing concepts, and insight (Donald Simanek)
 Problem of the Week (David Morin)
 Duke Physics Challenges
 A Collection of Quant Riddles
 Haidong’s Puzzles, Brain Teasers, and Interview Questions
 Macalester College Problem of the Week
 Ken’s Puzzle of the Week
 Geometry Project of the Month
 “Jewish Problems”
 Math Coffins
 Ponder This (IBM Research)
 Favorite Logic Puzzles (Quora)
 Missouri State University’s Problem Corner
 A Collection of Math Riddles (Henry Adams)