Thought That Counts

Chris Maslanka devised this brainteaser for the Gathering for Gardner held in Atlanta in April 2004:

A bouquet contains red roses, whites roses, and blue roses. The total number of red roses and white roses is 100; the total number of white roses and blue roses is 53; and the total number of blue roses and red roses is less than that.

How many roses of each color are there?

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Call the number of red roses, white roses, and blue roses r, w, and b, respectively. So r + w = 100; w + b = 53; and b + r = x, where x is less than 53. Adding these three equations gives 2r + 2b + 2w = 153 + x. The expression on the left is an even number, so 153 + x must be even as well. That means x is odd. Since 2w + 2r = 200, we know that 200 + 2b = x + 153 and thus x = 47 + 2b. We know that the bouquet contains blue roses (that is, b is not zero) and that x is less than 53, so x is an odd number higher than 47 and lower than 53. So it’s 49 or 51. If x = 49, then adding the three equations and dividing by 2 gives r + b + w = 101, and we have 1 blue rose, 52 white roses, and 48 red roses. But the problem states that the bouquet contains blue roses in the plural. So x must be 51, and we have 2 blue roses, 51 white roses, and 49 red roses.

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January 15, 2014January 26, 2014 | Puzzles

Black and White

meyer chess problem

By Heinrich Friedrich Ludwig Meyer. White to mate in two moves.

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1. Qh4! and the queen will mate from e1 or d8. From The Chess Amateur, April 1916.

January 10, 2014January 19, 2014 | Puzzles

Digit Jump

digit jump puzzle

Place the digits 1 through 8 into these circles so that no two successive numbers are connected by a line. If we don’t count rotations or reflections, the solution is unique.

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One way to solve this is to invert the task: Draw a new diagram in which each circle is connected only to the circles to which it had not previously been connected, and then seek a connected path along which the successive digits 1-8 can be placed. There are four of these, all rotations and reflections of the solution above.

January 9, 2014January 19, 2014 | Puzzles

Black and White

cumpe chess problem

By Josef Cumpe, 1908. White to mate in two moves.

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White wins by passing his turn with 1. Kd6! Now every Black move exposes some fatal vulnerability: If he moves the bishop then White will mate with queen or knight, and if he moves the knight then White can capture the bishop. Note that d6 is the only safe square for the king’s waiting move — every other option invites some troublesome mischief (1. Kc5 Bg1!).

January 3, 2014January 12, 2014 | Puzzles

Misterioso

Edward Elgar loved hidden meanings. The English composer filled his letters with wordplay and musical riddles, and he named one of his family homes Craeg Lea, an anagram of the names (C)arice (A)lice and (E)dward ELGAR.

Two of his puzzles have never been solved. One lies at the heart of the so-called Enigma variations, a set of 14 variations on a theme that Elgar said is “not played.” What does this mean? “The Enigma I will not explain,” he wrote in a program note for the first performance in 1899. “Its ‘dark saying’ must be left unguessed, and I warn you that the connection between the Variations and the Theme is often of the slightest texture; further, through and over the whole set another and larger theme ‘goes’, but is not played. … So the principal Theme never appears, even as in some late dramas … the chief character is never on the stage.” He took the solution to his grave, and music lovers have been searching for the hidden theme for more than 100 years.

The second mystery lies in a letter that Elgar wrote two years before Enigma premiered. On July 14, 1897, he sent the message above to 22-year-old Dora Penny, the daughter of his close friend Alfred Penny, rector of St. Peter’s, Wolverhampton. “It is well known that Elgar was always interested in puzzles, ciphers, cryptograms, and the like,” she wrote in her book Edward Elgar: Memories of a Variation. “The cipher here reproduced — the third letter I had from him, if indeed it is one — came to me enclosed in a letter from [Elgar’s wife] to my stepmother. On the back of it is written ‘Miss Penny.’ It followed upon their visit to us at Wolverhampton in July 1897. I have never had the slightest idea what message it conveys; he never explained it and all attempts to solve it have failed. Should any reader of this book succeed in arriving at a solution it would interest me very much to hear of it.”

No satisfactory solution has ever been found. Elgar named Dora herself as the inspiration for the 10th variation of the Enigma, so some wonder whether this was a clue. When she asked him in later life about the musical puzzle, he said, “I thought you, of all people, would guess it.” But no one knows what this means, and if Dora ever found the answer she told no one before her death in 1964. Perhaps the solution is now beyond our reach.

January 2, 2014January 1, 2014 | Puzzles

Black and White

malmström chess problem

By Gustaf Emanuel Malmström. White to mate in two moves.

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1. Rf7! Kxd8 2. Rf8# From Folket, 1919.

December 27, 2013January 5, 2014 | Puzzles

Blowin’ in the Wind

You’re taking care of a friend’s house while he’s on vacation. One hot day you pull the chain on a ceiling fan, and when it doesn’t respond you realize the house has temporarily lost power. Unfortunately, you have to leave now, and you’ll be away for several days.

You know that the fan was in the “off” position before you pulled the chain, and that pulling the chain successively will cycle it through its remaining settings (“off,” “high,” “medium,” etc.). You don’t know how many settings there are, but you’re sure there aren’t more than 4.

How can you ensure that the fan will be in the “off” position when power is restored? (Assume that you can’t simply cut the fan’s power.)

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Pull the chain 11 more times (for a total of 12). This will restore the fan to the “off” position whether it has 2, 3, or 4 settings. (Thanks, Marc.)

December 23, 2013December 27, 2013 | Puzzles

Hat Check

One hundred people stand in a line, all facing in the same direction. Each is wearing a red or a blue hat, assigned at random. Each person can see all the hats before him in the line, but not his own or those of the people behind him. Starting at the back of the line, each person in turn must guess the color of his own hat. Each person can hear all the prior guesses. If the group are allowed to discuss strategy beforehand, how many can be sure of guessing correctly?

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The answer, surprisingly, is 99 — the first guess may be wrong, but everyone else can guess correctly. Before any guesses are made, each person in line counts the red hats before him. If the total is odd, he prepares the answer “red”; if it’s even (including 0), he prepares the answer “blue.” Now, while awaiting his turn to speak, each person listens to the guesses behind him. Each time he hears “red,” he switches the color of his own prepared guess from red to blue or back again. When his turn comes to speak, he gives his prepared guess. (Thanks, Jeff.)

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December 22, 2013December 27, 2013 | Puzzles

Black and White

friedlander chess problem

By Louis Friedländer. White to mate in two moves.

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1. Ra5! and Black will be mated along the sixth rank. From Chess Poetry, 1919.

December 20, 2013December 27, 2013 | Puzzles

A Letter Home

Young telegraph operator Joseph Orton Kerbey was enlisted as a spy for the federal forces during the Civil War. In 1861, laid up in a sick bed in Richmond, he needed a way to communicate his latest discoveries to his friends in the north. The message would have to appear innocent and contain the key to its own decipherment. Here’s what he sent:

He directed it, not to his father’s name and address, but to a friend in the telegraph office at Annapolis. What was the hidden message?

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“If the reader will look at an apparent flourish under the words, ‘My dear Father,’ as if underscored, he will observe three little dashes like this, `- - -` and a little further on a careless looking scratch of the pen, resembling `. - -`. This forms the key to the simple cipher, and the same characters are indifferently scattered about the sheet so as to attract only the eye of an operator. The three little dashes represent the Morse character for the figure five `- - -` (5), while the other signal, a dot and two dashes, is a W, which, when placed alone, is always understood to stand for word. Now the operator will be sure to see that 5, W, while the chances are that no one else but an operator would. The young friend to whom I had addressed this I knew would understand, from the tone of the letter, that it was a blind, and he would search for a different interpretation, and would soon discover the 5, W, which he would see referred to the fifth word. If the reader will read only every fifth word of this letter he will have the true meaning. “Translation. — Been all through Southern Army, again obliged to delay here account sickness Impossible Confederate advance are exhausted half army absent sick balance are demoralized look under front portion Blank’s house situated on hill road Manasses to Washington black roll of papers official proofs wish Friend Covode secure them officers are there night students Georgetown signal South from the dormitory will be home soon as can.” From Kerbey’s 1889 memoir The Boy Spy: A Substantially True Record of Secret Service During the War of the Rebellion.

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December 19, 2013December 27, 2013 | Puzzles