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The Nightmare Puzzle
The Nightmare puzzle was conceived as Johnny Carson was delivering his monologue during the Tonight Show one night in 1984. As with most of my puzzle ideas, this one came to me fully formed and complete. I keep tools and wire handy for just such events having learned that three-dimensional ideas are difﬁcult to decipher and duplicate from two-dimensional scribbling on a piece of paper. After making the prototype, as I sat playing with it, my wife joined me in critiquing my latest design. Sometime later one of us stated that we probably would have nightmares about it that night. We didn’t have any nightmares about it, but the name stuck. The Nightmare 134 R. IRBY has more than lived up to its name with a convoluted three-dimensional shape that exacts extreme effort and concentration from all who attempt it.
The Nightmare puzzle is made from one continuous strand of wire. There are two outer and two inner loops, with the wire ends making small rings that are wrapped around the wire in such a manner as to eliminate any usable ends. A cord encircles the two inner loops of the puzzle, and the object is to remove the cord completely from the puzzle. In addition to the difﬁculty in conceptualizing the convoluted shape of the puzzle, the ﬂexibility of the cord allows one to make mistakes not possible with rigid pieces. Any wrong moves, not promptly corrected, quickly compound into a tangled mass of knots soon precluding any progression toward the solution. On a scale of 1 to 10, I rate the Nightmare an 8. The difﬁculty may be increased by, after the cord is removed, adding a ring to the cord that will not pass through either of the small end rings then attempting to replace the cord.
Beautiful But Wrong:
The Floating Hourglass Puzzle Scot Morris
One of the problems in Martin Gardner’s August 1966 “Mathematical Games” column was The Floating Hourglass.
An unusual toy is on sale at a Paris shop: a glass cylinder, ﬁlled with water, and at the top an hourglass ﬂoats. If the cylinder is inverted a curious thing happens; the hourglass remains at the bottom of the cylinder until a certain quantity of sand has ﬂowed into its lower compartment. Then it rises slowly to the top. It seems impossible that a transfer of sand from top to bottom of the hourglass would have any effect on its overall buoyancy. Can you guess the simple modus operandi ?
I gave the problem some thought but couldn’t come up with any good theory. I assumed it had to do with some law that I had forgotten since high school. The next month, when the answer came, I was delighted. It was so simple, so absurdly obvious, that I not only could have thought of it myself, I should have. The effect was like seeing a good magic trick or hearing a good joke.
I read Martin Gardner’s columns religiously, and corresponded with him occasionally from 1963 on, as a college student, as a graduate student, and then as an editor of Psychology Today. In 1978, in the months before a new science magazine, Omni, was to be launched, I had the pleasure of ﬁnally meeting the Master Explainer. I was going to write a column on “Games” so I made a pilgrimage to Hastings-on-Hudson to visit the Master. There on a shelf was the infamous Floating Hourglass itself. I could ﬁnally try out the curious toy I had read about so many years before. I turned it over and the hourglass stayed at the bottom, just like Martin said it would.
136 S. MORRIS At the 1991 Puzzle Collectors Party in Los Angeles, I saw my second Floating Hourglass and knew I could ﬁnally write about it, since I only published puzzles in Omni that I knew my readers could ﬁnd. Tim Rowett had brought one from England, made by Ray Bathke of London. I immediately ordered some. My September 1992 column introduced the Floating Hourglass 25 years after I ﬁrst heard about it. I asked my clever readers to submit theories to explain it. The results appeared in the January 1993 issue.
When Martin allowed me to look through his ﬁles, I found a thick folder on the Hourglass, a treasure trove of letters and drawings. For years I have itched to tell this story, but no magazine article could possibly contain it.
This book ﬁnally gives me the chance to tell the history of the Floating Hourglass Puzzle.
The First Theories Piet Hein, the Danish sculptor/inventor (the Soma Cube, the Super Egg) and poet/artist (Grooks), had visited Martin in early 1966. Hein told him about a toy he had seen in the Paris airport. He didn’t bring one back, but his description was clear enough; Gardner knew how it must work and wrote about it in his August and September columns without even having
THE FLOATING HOURGLASS PUZZLE 137seen one. Martin’s theory relied on friction between the glass and the cylinder, but Pien wasn’t convinced. He thought the inside of the cylinder he had seen was too smooth to offer much resistance. He felt there must be something more, something to do with the falling sand.
Hein believed the impact of the sand grains hitting the bottom of the glass exerts a downward force, an ‘effective change in mass’: “The hourglass is heavier while the sand is falling,” Hein wrote Gardner on September 16, 1966. “Imagine if the hourglass were opaque and you didn’t know it were an hourglass at all. There it stands at the bottom and changes its weight!...What keeps the hourglass down is the falling of the sand, not the amount of sand that has fallen or is left. The hourglass rises not because there is little sand left in the top chamber, but because the rate of falling sand has decreased. This seems to solve the whole problem.” Note that he wrote all this after the September Scientiﬁc American was out. He knew of Gardner’s “answer,” but he also knew that Martin had never examined an actual hourglass sample. For him this meant that the ﬁnal proof was not yet in. Until you could see and touch one, break it open or learn how it is made, all theories were valid contenders. In the absence of knowing the truth, the best criterion for a theory is its beauty. And Martin had already admitted that Piet’s impact theory was beautiful.
Hein was obsessed with hourglasses. He drew a cartoon of himself on an elevator with Einstein, pondering an oversized hourglass. He designed an hourglass-powered perpetual motion machine and created a fantasy ocean full of bobbing hourglasses. Since the glasses change their weight whenever the sand falls inside, Hein issued a mock warning: When mailing hourglasses, don’t weigh the package while they’re running, or you’ll have to pay a higher postage.
A Painful Paradigm Shift Just a couple of days after writing the letter and drawing the cartoon, Hein had an agonizing experience in Milan, Italy. In a shop there he saw a double-glass: two cylinders side by side, a ﬂoating hourglass in one and a sunken hourglass in the other. When turned over, the glasses stay in place at ﬁrst and then one rises while the other sinks. He knew immediately that his impact theory was doomed. A “sunken” hourglass that stays in place at the top of a tube can’t be explained by sand grains falling in the opposite direction.
Hein tried the double-glass in the shop a few times, just enough to see that it worked. “That was all I wanted to know,” he wrote “and all that was needed to make my intellectual headache come back much worse than 138 S. MORRIS the ﬁrst time.” Hein was forced to make a paradigm shift, and he found it painful. He knew that the sinking glass directly refuted his theory, but he didn’t buy one to take home. “This is not a question of fumbling one’s way to a solution, but of thinking,” he explained. “I couldn’t think of anything else but the principle. It really hurt.” He couldn’t bear to abandon his pet idea completely. He rationalized in jest: “My theory works, with the exception that the sand was running downward in both hourglasses, I must admit, but it would be easier to explain the symmetry of the phenomenon if it were falling upward in one of them!” Hein acknowledged Gardner’s appreciation of beauty in a theory. “I am glad you think my explanation is beautiful,” he wrote. “So do I, but let us be honest and not rate it any lower just because it is false. I admit, being false makes it less right. But it does not make it any less beautiful.” When Piet Hein left Milan on the afternoon of September 21, he was somber, “somewhat worried and depressed on behalf of my beautiful theory.” Then, as the plane soared over Mont Blanc, Hein had a sudden crystallizing insight - “At an altitude of 8500 meters, I found myself in possession of the solution.” He later wrote about the “Aha” experience. “When the Alps in the lite of the (superelliptic) setting sun and the growing, exactly half moon (D for dynamics) were left behind us and all was dark, there was a lull in my mind, a tabula rasa. And on that tablet lay, like a small ﬁsh on the center of a huge platter, the solution of the hourglass mystery. My dear Watson, it’s very simple.”
The Solution Is in the Solution
The key to the puzzle isn’t in the glass but around it, Hein realized, not in the falling of the sand but in the ﬂowing of the liquid. There must be two liquids of different speciﬁc weights that don’t mix completely, and are indistinguishable in color and transparency.
When the cylinder is inverted, the heavier liquid is on top pushing down.
Only when enough of it has seeped down does the glass begin to rise. The falling sand is just for misdirection, but what a clever ruse it is. The hourglass puzzle is caused by an hourglass effect, but the relevant displacement is in the liquid, not the sand. Hein was awed by the brilliance of it all, “I should like to meet the person who invented this effect and designed it so as to hide the solution so elegantly for us,” he wrote on September 22.
THE FLOATING HOURGLASS PUZZLE 139Gardner’s reply on October 5th was a gentle letdown: “You said you would like to meet the clever fellow who thought of the ‘two-liquid’ principle. Well, all you have to do is shake your own hand. You are the inventor.
The principle is simply too clever to be true.” The Hourglass Letters While Hein was having his epiphany in Europe, Gardner was getting letters in response to the September issue. One man wrote that he owned one of the Paris cylinders, but reported that his glass sometimes ﬂoated and sometimes sank, perhaps depending on the temperature.
On September 6th, Albert Altman wrote from the U.S. Naval Ordnance
Laboratory at Silver Spring, Maryland:
Another solution to the hourglass science teaser is that the momentum carried by the falling sand causes the hourglass Ô¾, where is the rate of the ﬂow of the sand, the accelplus sand to weigh more than its static weight by an amount eration of gravity, and the height through which the sand has fallen. The height decreases due to the buildup of sand on the bottom of the hourglass and at a critical value the net force on the hourglass acts upward and rises.
Gardner was beginning to wonder if his friction explanation told the whole story. Could temperature and sand impact also be factors? Would he
have to print a correction? He replied to Altman on September 12th:
I am embarrassed to admit that your explanation may be right.
I have not yet seen the toy; having relied (unfortunately) on an account given to me by a friend who examined the toy in Paris, but did not bring one back with him. It is possible that one version of the toy works on the principle you mention, and the other on the principle I suggested, or perhaps still another one. In short, at this point I am hopelessly confused.
Hopelessly confused? Martin Gardner?! That is something that surely doesn’t happen very often, and it didn’t last long. A letter dated September 29 came from Walter P. Reid, also from the U.S. Naval Ordnance Lab. “I am writing to put your mind at ease (on the impact theory posed by Altman), and to suggest that you not publish a correction. I am sure that your explanation was correct.” Reid went on to show mathematically how the impact of sand hitting the glass’ bottom is exactly balanced by the loss of the sand’s mass while it is in free fall. Reid later adapted his letter for publication and his short article “Weight of an Hourglass” appeared in American 140 S. MORRIS Journal of Physics, 35(4), April, 1967. This remains, to my knowledge, the only scientiﬁc writing on the subject. Gardner cites it in the hourglass puzzle reprinted in Mathematical Circus.
Hein and the Horse’s Mouth All was settled for a few months, but then in the spring of 1967 the glass rose again. Hein wrote that he went back to the shop in Paris where he ﬁrst saw the glass, and tracked down the maker. He turned out to be a Czechoslovakian glassblower named Willy Dietermann, and he conﬁrmed Hein’s two-liquid theory. When Hein asked about the liquid, Mr. Dietermann explained it was 90 to 95% water and 5 to 10% a combination of alcohol and formic acid.
Gardner wrote to Dietermann but received no reply. He decided he had to write a correction after all. Set into type and slated to appear in a summer 1967 column was this recreation of the moment of truth: “When Hein stated his two-liquid theory, the inventor staggered as if struck. It was the ﬁrst time his secret had been guessed: two liquids of different density, but which do not separate completely, so there is no visible division between the two.
A slight difference in refractive power is concealed by the curvature of the glass cylinder. The glasses do tip to the side when inverted, but that is just a holding operation until the liquids have had time to change places. The main principle is the liquid hourglass effect, completely invisible, which eventually sends one hourglass up, the other down.”
By this time, Gardner had his own hourglass tube. He loaned it to C.L.
“Red” Stong, the original “Amateur Scientist” columnist of Scientiﬁc American. In his lab, Stong shone polarized light through the cylinder and found that the refractive index of the liquid did not change from top to bottom.
He concluded the liquid inside was homogenous. He also reported it had a freezing point of eight degrees below zero.