«Contents Ü Foreword Elwyn Berlekamp and Tom Rodgers ½ I Personal Magic ¿ Martin Gardner: A “Documentary” Dana Richards ½¿ Ambrose, Gardner, ...»
Meanwhile, Gardner was playing amateur scientist himself. He bought a cheap egg timer, freed the glass from the wooden frame, and found a transparent cylinder into which it would ﬁt. “I ﬁlled the cylinder with water,” he wrote to Hein, “then I wrapped copper wire around the middle of the hourglass. By snipping off the ends of the wire I was able to make its weight such that it slowly rises in the cylinder. It worked just like the big version.”
THE FLOATING HOURGLASS PUZZLE 141Sanity was restored and the correction was never printed. We may never know what Hein got from the “horse’s mouth.” Perhaps Dieterman was leading him along, or perhaps there was an honest misunderstanding about the use of different liquids. There certainly was plenty of room for misinterpretations. The Dane and the Czech had to speak German because Dieterman knew no Danish or English, and his German was better than his French, and Hein’s French was worse than his German.
What fascinated me about the hourglass puzzle was how it led a mind like Piet Hein’s to come up with such brilliantly incorrect theories. They may have been wrong, but they were creative products of human thought, and deserved to be prized for that alone. Let others measure a refractive index or a freezing point, Hein wanted to think the problem through. He wanted to search for alternate, beautiful explanations. He wanted to expand his “perpendicular thinking.” I received over 1200 letters about the hourglass after publishing the puzzle in Omni. As I read them, sorting them into different piles, I found the largest single category was always the “correct” theory. This proportion stays at about 40% with each new batch of mail. The other 60% broke down into about 15 different theories.
About 40% of those readers with incorrect answers cited heat as a factor in the hourglass’s behavior. Falling sand generates heat, they said. Some argued that this warms the surrounding liquid so the hourglass stays down until the liquid cools again; others, that the hourglass ﬂoates up with the pocket of warm liquid surrounding the glass’s neck. But most in this category thought the heat warms the air in the glass, making it expand slightly and then rise.
More than 50 readers thought that the hourglass was ﬂexible. Some reasoned that when the sand presses down from the top, the hourglass widens and wedges itself into the cylinder. Others decided that the hourglass is ﬂexible only at the ends. “The top and bottom of the hourglass are so thin as to sag under the weight of the sand,” wrote B. G. of Los Altos Hills, California. “When enough sand falls into the bottom chamber, it ‘bubbles’ the bottom end out, increasing the hourglass’s volume,” reasoned D. Q. of Richmond Hill, Ontario, Canada.
Many correspondents blamed the “impact” of falling sand for keeping the glass down. Some even used mathematical formulas to show how much force a sand grain exerted, ﬁrst on the bottom of the glass and later on a 142 S. MORRIS mound of other sand grains. The theory may be correct, but the calculations have to consider the amount of time each grain of sand is falling and weightless; the two tiny opposing forces exactly cancel each other out.
Movements within the system don’t alter the weight of the system.
Another line of reasoning put the emphasis on the liquid: “The solution is in the solution!” wrote H. W. of Coweta, Oklahoma. If the liquid is naturally cooler and denser at the bottom, then the denser liquid is at the top when the tube is turned over. It eventually seeps down below the hourglass and buoys it up.
A surprising number thought it was all an illusion. “It just takes a long time for the hourglass to get started,” perhaps because the liquid is very viscous, wrote one reader. “The hourglass, as a system, is rising from the moment the column is inverted,” argued P. T. of Glendale, California. They concentrated on the air bubble that constantly rises, ﬁrst in the hourglass and then in the tube.
Many believed the air at the top of the hourglass lifts it to the top of the tube. “When enough air reaches the top chamber and exerts its pressure there, the hourglass begins to rise,” wrote T. H. of Chapel Hill, North Carolina. About 4% of those who wrote in thought that the shape of the hourglass affected its buoyancy. When the air is in the bottom half, the water below the glass can push up only on the circular end of the glass. When the air rises to the top half, water can push up all around the inverted cone, a greater surface area. “It’s the same principle that causes a snow cone to pop out of its cup when you squeeze the bottom,” explained D. A. N. of Tillamook, Oregon.
I promised copies of the book Omni Games to the ﬁve “most interesting” entries. Correct answers to this puzzle aren’t very interesting because they’re all virtually alike. Therefore, I awarded books for incorrect answers
¿º “The hourglass is composed of a ﬂexible material such as Nalgene,” wrote Timothy T. Dinger, Ph.D., and Daniel E. Edelstein, Ph.D. The two share a prize for having the conﬁdence to submit their incorrect theory on company stationery: IBM’s Thomas J. Watson Research Center in Yorktown Heights, New York.
º Cliff Oberg of Clarkdale, Arizona, theorized that the tube’s caps are hollow and that the ﬂuid must ﬂow from the tube through a hole into the cap below before the glass can rise.
º Finally, a theory about the density gradient of the liquid was signed “Bob Saville, Physics Teacher, Shoreham-Wading River High School, Shoreham, New York.” He added, “P.S. If this is wrong, then my name is John Holzapfel and I teach chemistry.” The ﬁfth book, therefore, goes to Holzapfel.
Binomial Puzzle A new, rather amusing, combinatorial puzzle can be constructed by acquiring twenty-seven uniform cubes of size ¢ ¢ and gluing them together
to form the following eight pieces:
One “Smally” of size ¢ ¢ (i.e., a single cube).
One “Biggie” of size ¢ ¢ (where ¾ ).
Three “ﬂats” of size ¢ ¢.
Three “longs” of size ¢ ¢.
Now color each piece with six colors according to the scheme in Figure 1.
The problem then is to arrange the eight pieces into a cube so that opposite faces have the same color.
This puzzle, if colored as above, has exactly two solutions — each with a different set of three colors. With proper insight it can be solved in a few minutes. Without this insight it typically takes several hours to arrive at a solution if one can be found at all. Before proceeding with this discussion the reader is urged to build a puzzle and attempt its solution.
Most Martin Gardner aﬁcionados will recognize that the eight pieces model the situation represented by the binomial expansion
´ · µ¿ ¿·¿ ¾ ·¿ ¾· ¿´½µ Gardner calls such models of mathematical theorems “look-see” proofs.
In fact he has recommended that every teacher of algebra construct a set of eight pieces for classroom use. Usually there is an “aha” reaction when students see that a cube can actually be constructed from the pieces. For more advanced students it would be reasonable to ask in how many essentially different ways can the cube be constructed from the eight (uncolored) pieces. This will depend on just what is meant by the words “essentially 146 J. FARRELL
Figure 1. A, B, C, 1, 2, and 3 are any six distinctive colors.
different” but one interpretation could be to orient the cube to sit in the positive octant in space with Biggie always occupying the corner (0, 0, 0). There are 93 solutions in this case. If, additionally, each of the 48 faces of the pieces is colored with a unique color, then there are ´ ¿µ´ µ´¿ µ¾,½ ¼,½½,½ distinct ways of constructing the cube.
The combinatorial puzzle uses only six colors in its construction, and the solver has the additional clue that a face is all of one color; but there still remain a great many cases that are nearly right. Trial and error is not a very fruitful way of trying to solve this puzzle.
Solution Hints. These hints are to be regarded as progressive. That is, after reading a hint, try again to solve the puzzle. If you cannot, proceed to the next hint.
½º Call the six colors A, B, C, 1, 2, and 3. A, B, and C will turn out to be a solution set. Notice that the eight pieces must form the eight corners of the completed cube — one piece for each corner. Therefore, each CUBE PUZZLES 147 piece must have on it a corner with the three colors A, B, and C in some order. (It is a fact, but not necessary for the solution, that four pieces will have the counterclockwise order A–B–C and the other four the order A–C–B.)
¾º To determine the colors A, B, and C, take any piece (Biggie is a good choice), and notice that the three pairs of colors A–1, B–2, and C–3 oppose each other. This means that A and 1 cannot appear together in a solution. Likewise B and 2 cannot, nor can C and 3. Of the 20 possible sets of three colors (out of six) we are left with only eight possibilities: A–B–C, A–B–3, A–2–C, A–2–3, 1–B–C, 1–B–3, 1–2–C, and 1–2–3.
¿º Since it is also true that, on any other piece, colors that oppose each other cannot appear in the same solution, we may choose another piece, say a ﬂat, and use it to reduce further the possible solution colors. For instance, it may happen that on that ﬂat, 2 opposes A. We would then know to eliminate anything with A and 2. This would force A and B to be together. One or two more tests with other pieces lead to A–B–C (or 1–2–3) as a candidate for a solution. When eliminating possibilities, it is convenient to turn the three ﬂats to allowed colors, using Biggie as a guide.
º Place Biggie as in the diagram so that A–B–C is a corner. Place all three ﬂats so that A, B, and C are showing. These three ﬂats must cover all or part of the colors 1, 2, and 3 of Biggie. Of course, keep A opposite A, etc. It is easy to visualize exactly where a particular ﬂat 148 J. FARRELL must go by looking at its A–B–C corner. Then place the longs and, ﬁnally, Smally.
The solver will notice that the completed cube has a “fault-free” property.
That is, no seam runs completely through the cube in any direction and therefore no rotation of any part of the cube is possible. This will assure that only one solution is attainable with the colors A–B–C (the proof is left to the reader). There is another solution to this puzzle using the colors 1– 2–3. If only one solution is desired, take any piece and interchange two of the colors A, B, and C (or two of 1, 2, and 3). This will change the parity on one corner of the cube so that a solution is impossible using that set of three colors.
We like to have three Smallys prepared: one that yields two solutions, one that gives only one solution, and one to slip in when our enemies try the puzzle that is colored so that no solution is possible!
Magic Die Figure 3 shows the schematic for a Magic Die. The Magic Die has the amazing property that the sum of any row, column, main diagonal (upper left to lower right), or off-diagonal around all four lateral faces is always 42.
You can construct a Magic Die puzzle by taking twenty-seven dice and gluing them together into the eight pieces of our combinatorial cube puzzle— being sure the dice conform to the layout shown in Figure 3. There will be only one solution to this puzzle (with magic constant 42), and, even with the schematic as a guide, it will be extremely difﬁcult to ﬁnd. (Alternatively, you can copy Figure 3 and paste it onto heavy paper to make a permanent Magic Die.) The Nine Color Puzzle Sivy Farhi The nine color puzzle consists of a tricube, with each cube a different color, and twelve different dicubes with each cube of a dicube a different color.
Altogether there are three cubes each of nine different colors. The object of the puzzle is to assemble the pieces into a cube with all nine colors displayed on the six faces. A typical set is shown below in Figure 1, where each number represents a color. Typical solutions are shown at the end of this paper.
According to Reference 1, the nine color puzzle was ﬁrst introduced during 1973, in Canada, as “Kolor Kraze.” I learned of the puzzle in 1977 from Reference 2, and, using the name “Nonahuebes,” included it among the puzzles I produced in New Zealand under the aegis of Pentacube Puzzles, Ltd. I was intrigued by the puzzle’s characteristics but did not, until recently, analyze it thoroughly.
This puzzle is of particular interest because it is of simple construction, has an easily understood objective, has numerous variations and solutions, is a manipulative puzzle, and requires logical decision-making. Included in its analysis are the concepts of transformations, parity, backtracking, combinatorics, ordered pairs, sets, and isomorphisms. A complete analysis requires the use of a computer, but after reading this article it should not be 152 S. FARHI a difﬁcult exercise for a moderately proﬁcient programmer to verify the results obtained.
The ﬁrst observation noted is that the cube can be assembled with the tricube along the edge or through the center of the cube, but not in the center of a face. A proof is given in Section 2.
The second observation is that for a cube to be assembled with the nine colors on each face, none of the nine planes may contain two cubes of the same color. A proof is given in Section 3. Thus, when two cubes of the same color are in position, the location of the third cube is determined. Since during the course of trying to solve this puzzle a conﬂict often occurs, this rule then tells the experimenter to backtrack.
The third observation, obtained after some experimentation, is that there are numerous solutions, some with the tricube on the edge and some with the tricube through the center of the cube. The question then arises: How many solutions are there?