Imagine a die that exactly covers one square of a checkerboard. Place the die in the top left corner with the 6 uppermost. Now, by tipping the die over successively onto each new square, can you roll it through each of the board’s 64 squares once and arrive in the upper right, so that the 6 is exposed at the beginning and end but never elsewhere?
In a photograph, is there a way to distinguish between a landscape and its reflection?
n. the distance between two stopping places
Another puzzle by Sam Loyd: Two ferry boats ply the same route between ports on opposite sides of a river. They set out simultaneously from opposite ports, but one is faster than the other, so they meet at a point 720 yards from the nearest shore. When each boat reaches its destination, it waits 10 minutes to change passengers, then begins its return trip. Now the boats meet at a point 400 yards from the other shore. How wide is the river?
“The problem shows how the average person, who follows the cut-and-dried rules of mathematics, will be puzzled by a simple problem that requires only a slight knowledge of elementary arithmetic. It can be explained to a child, yet I hazard the opinion that ninety-nine out of every hundred of our shrewdest businessmen would fail to solve it in a week. So much for learning mathematics by rule instead of common sense which teaches the reason why!”
Sam Loyd devised this puzzle in 1898. Begin at the heart in the center and move three squares in any of the eight directions, north, south, east, west, northeast, northwest, southeast, or southwest. You’ll land on a number; take this as the length of your next “march,” which again can go in any of the eight directions. “Continue on in this manner until you come upon a square with a number which will carry you just one step beyond the border, thus solving the puzzle.”
Interestingly, Loyd devised this puzzle expressly to defeat Leonhard Euler’s method of solving mazes. “Euler, the great mathematician, discovered a rule for solving all manner of maze puzzles, which, as all good puzzlists know, depends chiefly upon working backwards. This puzzle, however, was built purposely to defeat Euler’s rule and out of many attempts is probably the only one which thwarts his method.” The original puzzle, as published in the New York Journal and Advertiser, contained a flaw that permitted multiple solutions. That’s been corrected here — there’s only one way out.
Two-move chess is just like regular chess except that each side makes two moves at a time. Prove that White, who goes first, can be sure of at least a draw.
Two poles stand vertically on level ground. One is 10 feet tall, the other 15 feet tall. If a line is drawn from the top of each pole to the bottom of the other, the two lines intersect at a point 6 feet above the ground. What’s the distance between the poles?
Nokes went, he thought, to Styles’s wife to bed,
Nor knew his own was laid there in her stead;
Civilian, is the child then begot
To be allow’d legitimate or not?
– Bon Ton Magazine, July 1794
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?
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.
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.
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.)
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?
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?
Prove that, if we choose nine vertices of a regular icosagon, some three of these will form an isosceles triangle.
Speaking of Lewis Carroll — and further to Wednesday’s logic exercise — here’s the king of all Carroll’s logic problems. What’s the strongest conclusion that can be drawn from these premises?
- When the day is fine, I tell Froggy “You’re quite the dandy, old chap!”
- Whenever I let Froggy forget that 10 pounds he owes me, and he begins to strut about like a peacock, his mother declares “He shall not go out a-wooing!”
- Now that Froggy’s hair is out of curl, he has put away his gorgeous waistcoat.
- Whenever I go out on the roof to enjoy a quiet cigar, I’m sure to discover that my purse is empty.
- When my tailor calls with his little bill, and I remind Froggy of that 10 pounds he owes me, he does not grin like a hyena.
- When it is very hot, the thermometer is high.
- When the day is fine, and I’m not in the humor for a cigar, and Froggy is grinning like a hyena, I never venture to hint that he’s quite the dandy.
- When my tailor calls with his little bill and finds me with an empty pocket, I remind Froggy of that 10 pounds he owes me.
- My railway shares are going up like anything!
- When my purse is empty, and when, noticing that Froggy has got his gorgeous waistcoat on, I venture to remind him of that 10 pounds he owes me, things are apt to get rather warm.
- Now that it looks like rain, and Froggy is grinning like a hyena, I can do without my cigar.
- When the thermometer is high, you need not trouble yourself to take an umbrella.
- When Froggy has his gorgeous waistcoat on, but is not strutting about like a peacock, I betake myself to a quiet cigar.
- When I tell Froggy that he’s quite a dandy, he grins like a hyena.
- When my purse is tolerably full, and Froggy’s hair is one mass of curls, and when he is not strutting about like a peacock, I go out on the roof.
- When my railways shares are going up, and when it’s chilly and looks like rain, I have a quiet cigar.
- When Froggy’s mother lets him go a-wooing, he seems nearly mad with joy, and puts on a waistcoat that is gorgeous beyond words.
- When it is going to rain, and I am having a quiet cigar, and Froggy is not intending to go a-wooing, you had better take an umbrella.
- When my railway shares are going up, and Froggy seems nearly mad with joy, that is the time my tailor always chooses for calling with his little bill.
- When the day is cool and the thermometer low, and I say nothing to Froggy about his being quite the dandy, and there’s not the ghost of a grin on his face, I haven’t the heart for my cigar!
Unfortunately, Carroll died before he was able to publish the solution — but he warned that it contains “a beautiful ‘trap.’”
1. Find an expression for the number 1 that uses each of the digits 0-9 once.
2. Do the same for the number 100.
3. Write 31 using only the digit 3 five times.
4. Express 11 with three 2s.
5. Express 10 with two 2s.
6. Express 1 with three 8s.
7. Express 5 with two 2s.