Circles and Squares

circles and squares 1

Here are three circles and two squares, inscribed successively as shown.

If the diameter of the largest circle is 10, what is the diameter of the smallest circle?

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There are a number of ways to solve this. The simplest is to rotate the smaller square 45 degrees: Now the radius of the largest circle (red), which we know to be 5, equals the side of the smaller square (green), because they’re the two diagonals of a square. So the diameter of the smallest circle is 5.

June 19, 2012February 5, 2013 | Puzzles

Heads of State

Ten senators are about to enter Congress when a barrage of snowballs knocks off their tophats. Each retrieves a hat at random. What is the probability that exactly nine of them receive their own hats?

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Zero. If nine receive their own hats, then the tenth must as well.

June 15, 2012February 5, 2013 | Puzzles

Four in Three

four in three

Can a square be inscribed in any triangle?

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Yes. Sit the triangle atop a square and extend its sides: Now we have a similar triangle that does enclose a square. Shrink it down to its original size and we’re done. (Note that we must orient the triangle so that an altitude drawn to its bottom side lies in its interior.)

June 14, 2012February 4, 2013 | Puzzles

Black and White

andrade chess problem

By J. da C. Andrade, Empire Review, 1923. White to mate in two moves.

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1. Re6 and White will mate by 2. Qxe5, 2. Qa2, or 2. Qe4.

June 13, 2012February 4, 2013 | Puzzles

“A Tradesman in a Difficulty”

A puzzle by Angelo Lewis, writing as “Professor Hoffman” in 1893:

A man went into a shop in New York and purchased goods to the amount of 34 cents. When he came to pay, he found that he had only a dollar, a three-cent piece, and a two-cent piece. The tradesman had only a half- and a quarter-dollar. A third man, who chanced to be in the shop, was asked if he could assist, but he proved to have only two dimes, a five-cent piece, a two-cent piece, and a one-cent piece. With this assistance, however, the shopkeeper managed to give change. How did he do it?

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1. The purchaser gives his one-dollar and his two-cent piece to the tradesman and his three-cent piece to the stranger. 2. The tradesman gives his half-dollar to the purchaser and his quarter-dollar to the stranger. 3. The stranger gives his two dimes and his one-cent piece to the purchaser and his five-cent piece and his two-cent piece to the tradesman. “Each has therefore received the precise amount to which he was entitled.”

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June 11, 2012February 4, 2013 | Puzzles

Testimony

From the 2000 Indiana College Mathematics Competition:

Four suspects, one of whom was known to have committed a murder, made the following statements when questioned by police. If only one of them is telling the truth, who did it?

Arby: Becky did it.
Becky: Ducky did it.
Cindy: I didn’t do it.
Ducky: Becky is lying.

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If Cindy is innocent, then her statement is true and the other three suspects are lying. But that immediately implies that Ducky is also speaking the truth, a contradiction. So Cindy can’t be innocent — she must be the guilty party.

June 10, 2012February 4, 2013 | Puzzles

The Five Rooms

five rooms

Here’s the floor plan of a house with five rooms. Can you draw a continuous line that passes through each of the 16 wall segments once and once only? If it’s possible, show how; if it’s not, explain why.

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It’s impossible. Three of the rooms have five wall segments, an odd number. A continuous line that starts outside such a room and crosses all five wall segments will be “trapped” inside the room, with no way to exit. We can start the line in one five-walled room and end it in another, but the third presents an insoluble problem. See The Bridges of Konigsberg and Knotty Thinking.

June 9, 2012February 4, 2013 | Puzzles

Alphabet Blocks

We have 27 wooden cubes. The first is marked A on every face, the second B, and so on through the alphabet to Z. The 27th cube is blank. Is it possible to assemble these cubes into a 3×3×3 cube with the blank cube at the center, arranging them so that cube A adjoins cube B, cube B adjoins cube C, and so on, forming a connected orthogonal path through the alphabet?

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No, it’s not possible. Imagine painting the large cube with a checkerboard pattern. As we move through the alphabet we’ll change colors at each step; if we’re to succeed there must be 13 red cubes and 13 black ones (omitting the center cube). But the surface of a large checkerboard cube contains 12 cubes of one color and 14 of the other. So it can’t be done.

June 8, 2012November 2, 2022 | Puzzles

Black and White

jakab chess problem

By A. Jakab, Good Companion, 1922. White to mate in two moves.

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1. Kc3 Kxc1 2. Qf1#. If 1. … Ka1 then 2. Kc2#.

June 7, 2012February 4, 2013 | Puzzles

Black and White

schumer chess problem

By J. Schumer, Transvaal Leader, 1906. White to mate in two moves.

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1. Bc8 Kxc8 2. Qa8#. If 1. … Ka7 then 2. Qb7#.

June 1, 2012February 4, 2013 | Puzzles