Measuring Weights

(a) Customers at a grocer's shop always want an integral number pounds of wheat, between 1 pound and 40 pounds. The grocer prefers to measure wheat in exactly one weighing with a beam balance. What is the least number of weights he needs?

(b) Customers come to a pawn shop with antiques. An antique always weighs an integral number of pounds, somewhere between 1 pound and 80 pounds. The owner of the pawn shop is free to do as many weighings as necessary to ascertain the unknown integral weight by using a beam balance. What is the least number of weights he needs?

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Working Computer

A room has n computers, less than half of which are damaged, others are good. It is possible to query a computer about the status of any computer. A damaged computer could give wrong answers. The goal is to discover a good computer in as few queries as possible.

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Tiling With Calissons

A large regular hexagon is cut out of a triangular grid and tiled with diamonds (pairs of triangles glued together along an edge). Diamonds come in three varieties, depending on orientation; prove that precisely the same number of each variety must appear in the tiling.

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Loop in a Linked List

Given the head of a linked list, find if it contains a loop or not, in linear time. Can you accomplish it with only O(1) extra space? Can you do it without modifying the list in any way?

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Perplexing Polynomial

Alice is allowed to choose an arbitrary polynomial p(x) of any degree with nonnegative integer coefficients. Bob can infer the coefficients of p(x) by only two evaluations as follows. He chooses a real number a and Alice communicates p(a) to him. He then chooses a real number b and Alice communicates p(b) to him. What values of a and b helped Bob succeed and how?

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Poisoned Wine Barrels

An enemy spy has poisoned one out of 1000 barrels of wine in a king's cellar. Even a sip of the poisoned wine has potency to kill. However, the effects of the poison show only in 30 days. The king asks the royal jailor to identify the poisoned barrel within 30 days. What is the least number of prisoners the jailor must employ to identify the poisoned barrel?

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Two Eggs and a Building

With two eggs and a building with 100 floors, what is the optimal strategy for finding the lowest floor at which an egg will break when dropped? Followup: What if the number of floors of the building is unknown?

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Number Guessing Game

Shankar chooses a number between 1 and 10,000. Geeta has to guess the chosen number as quickly as possible. Shankar will let Geeta know whether her guess is smaller than, larger than or equal to the number. The caveat is that Geeta loses the game if her guess is larger than Shankar's chosen number two or more times. (A) How many guesses are necessary? (B) What if Shankar is allowed to pick an arbitrarily large positive number?

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Cutting a Cake with Icing

One day, grandma baked a cake with a square top and dimensions 30cm x 30cm x 10cm. What is a simple strategy for cutting the cake into 9 equal pieces? The next day, grandma baked another cake with the same dimensions. This time, she put a thin layer of icing on top and on all four sides but not on the bottom. What is a simple strategy for cutting such a cake into 9 pieces such that all pieces have the same amount of cake by volume and the same amount of icing by surface area?

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Number Guessing Game II

Shankar chooses a number uniformly at random between 1 and 1000. Geeta has to guess the chosen number as quickly as possible. Shankar will let Geeta know whether her guess is smaller than, larger than or equal to the number. If Geeta's guess is larger than the number, Shankar replaces the number with another number chosen uniformly at random [1, 1000]. What should Geeta's strategy be?

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