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Monthly Archives for April, 2013

Three cans problem

Posted on: April 19th, 2013 by
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You may have heard of the classic puzzle where you are asked to measure 4 liters of water, given a 8-liter can filled with water, and 2 empty cans: One with 5-liter capacity and the other with 3-liter capacity. Here is a generalization of the puzzle: You are given an empty A-liter can, an empty B-liter can and a filled C-liter can, where A and B Continue reading the story "Three cans problem"

Number of unique distances

Posted on: April 18th, 2013 by
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If (n+1)(n+4)/2 distinct points are chosen in n-dimensional space \mathcal{R}^n, show that there are at least 3 unique distances. The number of unique distances is the number of elements in the set \{d_{ij}: \text{ i,j are 2 of the chosen points in }\mathcal{R}^n\}. Here d_{ij} denotes the euclidean distance between points i and j. Example: An equilateral triangle in 2-d or a regular tetrahedron in Continue reading the story "Number of unique distances"

Butterfly theorem

Posted on: April 17th, 2013 by
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Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly. Then prove that M is the midpoint of XY. Butterfly theorem

Ratio of 2 random numbers

Posted on: April 16th, 2013 by

Two real numbers X and Y are chosen at random in the interval (0,1). Compute the probability that the closest integer to X/Y is even. Express the answer in the form r + sπ, where r and s are rational numbers. via Putnam exam

Four knights

Posted on: April 15th, 2013 by
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Four knights are positioned on a 3x3 chessboard as shown on the first chessboard below. Can they move to the positions shown on the second chessboard? chess via Washington math

Find the error in the arithmetic

Posted on: April 14th, 2013 by
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What's wrong with the following? 1+2+4+8+\dots \\ =(2-1)(1+2+4+8+\dots)\\ =(2+4+8+16+\dots)-(1+2+4+8+\dots)\\=-1+(2-2)+(4-4)+(8-8)+\dots\\=-1

Expected number of loops

Posted on: April 13th, 2013 by

Suppose you have n pieces of ropes, each of length 1m. Thus there are 2n ends. In each iteration, you randomly choose 2 ends and tie them together. After 2n-1 iterations, you have a bunch of loops. Find the expected number of loops.

Elevator wait time

Posted on: April 12th, 2013 by
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Suppose you are on the 13-th floor of a 15 floor office building. The elevator is programmed to go up and down continuously as 1,2,3,...14,15,14,13,....2,1,2,.... except that when a user presses a button the elevator stops momentarily for him/her. Assume the time to load/unload passengers is negligible. You complain that whenever an elevator approaches you, it is more likely to go up than to go Continue reading the story "Elevator wait time"

Birthday problem

Posted on: April 10th, 2013 by
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When you were in high school, did 2 of your classmates have the same birthday? Thought it was coincidence? Suppose there is a class of 57 students. What's the probability that 2 or more students have the same birthday? (Assume an year has 365 days). If your answer is surprising, you must read the Wikipedia article here.

Thinking outside the box

Posted on: April 9th, 2013 by
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You might have heard of the popular "Think outside the box" puzzle. Here are 9 dots arranged in a 3x3 grid: Your goal to connect the dots by drawing four straight, continuous lines that pass through each of the nine dots, and never lifting the pencil from the paper. Of course, it is well known that in the optimal solution is to think outside Continue reading the story "Thinking outside the box"

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