Ones in a Million

Of the integers from 1 to 1,000,000, which are more numerous: the numbers that contain a 1 or those that don’t?

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To list the numbers that don’t contain a 1, imagine six spaces and fill each with the digits 0, 2, 3, 4, 5, 6, 7, 8, or 9. The number of ways of doing this is 9⁶. There’s one exception: 000000 doesn’t fall between 1 and 1,000,000. So the number of integers without 1 is 9⁶ – 1 = 531,440, and the number with 1 is 468,560.

June 25, 2012February 5, 2013 | Puzzles

The Figure 8 Puzzle

figure 8 puzzle

Can this loop of string be freed from its wire? Stewart Coffin, who devised the puzzle in 1974, writes, “I soon became convinced that this was impossible, but being a novice in the field of topology, I was at a loss for any sort of formal proof.” He published the challenge in a newsletter and has been receiving requests for a solution ever since. Adding to the confusion, in 1976 a British puzzle editor mistakenly claimed with that Coffin’s creation was equivalent to another puzzle with a known solution, and Pieter van Delft and Jack Botermans published an amusingly bewildering “solution” of their own in their 1978 book Creative Puzzles of the World.

In the meantime, fans around the world have continued to experiment, and mathematicians Inta Bertuccioni and Paul Melvin have both offered proofs that the puzzle is unsolvable. “Whoever would have guessed that this little bent piece of scrap wire and loop of string would launch itself on an odyssey that would carry it around the world?” Coffin writes. “Will it mischievously rise again, perhaps disguised in another form, as topological puzzles so often do?”

June 23, 2012June 22, 2012 | Puzzles · Science & Math

Black and White

kubbel chess problem

By L.I. Kubbel, 1941. White to mate in two moves.

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The quiet 1. Rg1 forces 1. … Kh5, permitting 2. g4#.

June 22, 2012February 5, 2013 | Puzzles

Odd and Even

A puzzle by Noboyuki Yoshigahara:

“An odd number plus an odd number makes an even number. An even number plus an odd number makes an odd number. An even number plus an even number is an even number. Right?”

“Yes.”

“An odd number times an odd number is an odd number. An even number times an odd number is an even number. Right?”

“Sure.”

“An even number times an even number is an odd number. Right?”

“Huh?”

“You don’t think so? An even number times an even number is an odd number.”

“Why?”

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AN ODD NUMBER PLUS AN ODD NUMBER has 26 characters, etc. Yoshigahara notes, “This puzzle works in Japanese as well.”

June 21, 2012February 5, 2013 | Puzzles

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