SOME MONTHS AGO WE EXPRESSED the opinion that amateur aerodynamics enjoys the smallest following among all the scientific avocations. No one challenged the statement, and a number of professional aerodynamicists wrote that the lack of amateur interest in the study is all too apparent in this nation, where the airplane was invented and carried to its highest development.

"The reason," writes J. J. Cornish of the department of aerophysics at Mississippi State University, "may lie in the nature of aerodynamics and the invisibility of the air itself. Perhaps the solution to the problem lies in making the flow of air visible. The words 'I don't see' have become synonymous with 'I don't understand.' And surely with understanding comes interest.

"If so complicated a device as a cloud chamber, which renders cosmic-ray tracks visible, can be reduced to the simple apparatus shown in your April, 1953, issue, surely aerodynamics like wise can be simplified.

"The aerodynamic smoke tunnel has long been used to gain a better understanding of various flow phenomena. Alexander Lippisch has made extensive use of the smoke tunnel in designing delta-wing aircraft. The Forrestal Research Center at Princeton University has several smoke tunnels in operation. This aerodynamic tool, the smoke tunnel, is well suited for experiments by amateurs.

"I recently constructed a small smoke tunnel which was designed to make use of an ordinary tank-type vacuum cleaner for its source of power [see drawing in Figure 5]. The smoke for the tunnel is obtained from cigarettes.

"The tunnel itself is made of two-by-twos and Masonite. The pieces were assembled to form a duct approximately one foot high and two feet long. One end of the duct, the entrance, was flared out to a width of about six inches. The rest of the duct has a width of 1 1/2 inches. One face was then covered with a pane of window glass. A 'screen door' made of fine screening a couple of layers of window screen will do) was made to cover the entrance so as to smooth out the entering air. Toward the other end of the tunnel a hole was cut in the back wall to allow the air to be exhausted by the vacuum cleaner. The cover from a two-inch spool of adhesive tape makes the fitting to attach the vacuum cleaner tube.

"The smoke is introduced into the tunnel through a 'rake' made from several pieces of 1/8-inch copper tubing attached to a pipe about half an inch in diameter [drawing in Figure 6]. To this rake is attached, by means of rubber tubing, the smoke generator. The smoke generator is made of two narrow cans and some balsa plugs. I started off with a generator that 'smokes' a single cigarette but soon worked up to a 'three-holer' for a denser smoke. From the other side of the generator another rubber tube is led to the exhaust side of the vacuum cleaner to produce the pressure differential necessary to burn the cigarettes.

"After sealing all joints and corners with plastic wood and smoothing the inside of the duct, I painted everything except the glass with flat black paint.

"Wonder of wonders, the thing worked on the first try. However, later I became dissatisfied with the smoke filaments and found that, as in mirror grinding, the product improves with attention to detail. I found that the smoke filaments were sharpest when the tubes of the rake were lined up accurately with the center line of the tunnel and when no air leaked in but all the flow came in by the front door. With only a little care sharp, crisp smoke lines were obtained.

"Models to be tested are mounted in the middle of the tunnel behind the glass face. With a knob projecting from the back of the tunnel I control the angle between the model and the flow. For best results the models should be big enough to span the 1 1/2-inch tunnel and should be about three or four inches long.

"It might be mentioned that the use of cigarettes as the smoke source has the advantage that the exhaust from the tunnel need not be conducted out-of-doors, as in the case of smokes from burning rotten wood or vaporized kerosene, often used in the larger tunnels. Even after a long evening of running, my kitchen is no worse off than after an evening with several cigarette-smoking guests. Supplying the tunnel with three cigarettes at a time has cut down on my own smoking, incidentally.

"Photographing the flow patterns obtained in the tunnel is one of the most interesting aspects of this hobby. When you have pictures of the flows, you can study the details at your leisure and compare different conditions. The black background and white smoke streams form contrasting patterns which are easily photographed. If you make pictures, you must pay some attention to lighting and reflectivity. The back wall of the tunnel must be almost absolutely flat black for the best results. Ordinary rough black paper such as is used in photograph albums is not quite black enough to get good contrast, as it reflects about 10 per cent of the incident light. Black velvet or velveteen reflects less than 1 per cent. [Roger Hayward, the illustrator of this department, points out that a sheet of glass painted black on the rear face and lighted from the front at a relatively low angle of incidence would give a much "blacker" black.]

"The models should be painted so that they show up in the pictures. At first I painted the models all white but this reflected too much light. Later I found that if the models were given a coat of white paint, then a coat of black paint, and then some of the black paint was scraped away around the edges, the white paint that showed through outlined the model very well.

"I used two 200-watt bulbs in reflectors placed at 45-degree angles to the glass to light the smoke lines. The camera was a pre-World War I Kodak, circa 1905. I used plus-X film and a setting of f/8 at one quarter second.

"The pictures on the preceding page show the effects on wind flow of an inclined flat plate, a cylinder and an airfoil section. The airplane model shows how the lift on the wings of an airplane causes the flow to curl up into wing-tip vortices as it streams past the wing. Of course many other experiments can be performed. I have done some preliminary work on boundary-layer control, both by suction and by blowing, in the tunnel. The problems of laminar separation and boundary-layer transition are readily studied.

"The speed of this tunnel is on the order of two to eight feet per second and may be regulated by choking off the flow out of the exhaust end of the vacuum cleaner. Incidentally, a V-shaped baffle should be installed in the exhaust end of the tunnel [see above] to prevent the flow pattern from necking down into the exhaust hole. The speed is best measured at the exhaust hole, and the speed in the test section is computed from the ratio of the area of the test section to that of the exhaust hole.

"Although not very fancy and certainly not expensive (less than $3, exclusive of the vacuum cleaner and camera) this aerodynamic smoke tunnel can afford many an hour of entertainment. Model airplane builders have used mine to test new airfoil sections. There are doubtless many other uses."

Letters like the following have filled our mail ever since. "Please inform Mr. von Auw," wrote Fred Lathrop of Portland, Ore., "that he has wrecked the serenity of our engineering department. His problem has cost three Ph.D.'s, five M.S.'s and a dozen lesser lights like myself a shameful amount of sleep-both on and off the job. Please let us know if the enclosed solution is correct. Incidentally, why not dig into the mathematical origin of this problem, if it has a literature, and publish the result? I am sure the majority of your readers would enjoy the story of this blister on the heel of progress!"

Digging into the background of a classic puzzle is something like trying to uncover the roots of a giant sequoia, and we did not feel up to the job. But J. S. Robertson, a researcher on radiation sickness at the Brookhaven National Laboratory, who makes a hobby of recreational mathematics, came to our aid. He wrote: "Although you referred to Mr. von Auw's 12-ball problem as a 'classic,' I wonder if you know that it is of relatively recent vintage and that there is a fair amount of literature on it? It is usually referred to as the 'l2-coin' problem. Your presentation did not clearly indicate that there are two distinct types of solutions to the problem. In one, which can be called the 'contingent' type, the procedure to be used for the second weighing depends upon the results of the first weighing, and the third on the second. In the 'predetermined' type, the balls (or coins) to be used in each weighing are selected at the beginning. Howard D. Grossman, a professional mathematician and lecturer on recreational mathematics, has shown that the balls can be so numbered that the results of the three weighings indicate the number of the odd ball directly. In a paper in Scripta Mathematica for March, 1950, I showed that all predetermined solutions can be derived from seven primitive solutions by such operations as renumbering, switching pans and changing the order of the weighings.

"Incidentally, Mr. von Auw's limitation on the capacity of the pans to five balls seems unnecessary, as none of the solutions uses more than four balls on a pan."

A check of references suggested by Robertson shows that the problem is indeed new. It appears to have been invented about 1945. Within six months of publication it had circled the globe and appeared in mathematical journals in all the major languages. No one cares to estimate how many millions of man-hours it has cost engineering departments. Most foreign authorities, particularly the British, agree that it was invented by an American. Grossman seems to have been the first American to publish it-in Scripta Mathematica of December, 1945. He writes, however: "I did not invent the 12-ball problem. I always had the impression that it started in England about 1945. May I give you a skeleton gestalt of the whole thing as I see it? The problem seems to be a happy composite of two diverse themes.

"The first is illustrated by Claude Gaspar Bachet's weight problem, probably the granddaddy of all weight problems, which appeared in his second edition of Problèmes Plaisants et Delectables in 1624. Bachet asked: With what five weights can you weigh any whole number of pounds from 1 to 31, placing the weights on only one side of a scale? Or, with what four weights can you weigh any number from 1 to 40 pounds, using the weights on both sides?

"The second theme involves concepts recently formalized by Claude Shannon's information theory, particularly that of the 'bit,' or binary digit. In the binary system of notation all numbers are based on powers of 2, and they are expressed in terms of the two digits 0 and 1. The system lends itself naturally to the description of situations involving 2n choices.

"The version of Bachet's problem which specifies the use of five weights on only one side of the beam balance is based on the binary system-the weights being 2⁰, 2¹, 2², 2³ and 2⁴, or 1, 2, 4, 8 and 16 pounds, respectively. The second version of his problem, which adds the choice of placing weights on the other side of the balance, is a three-choice situation that can be described by 'ternary' number system. The four weights to be used are 3⁰, 3¹, 3² and 3³ or 1, 8, 9 and 27 pounds. All weighing problems involving triple choices-for example, 'up,' 'down' and 'balance'-how their interesting properties to number based on the digit 8. Scores of such puzzles have been proposed.

"But rarely has a century produced one that aroused as much interest among amateur mathematicians as the 12-ball problem. The charm of the problem lies in the fact that it just does not seem to offer enough-nor even nearly enough- information for reaching a solution. Thus it resembles many of the 'easy to ask but hard to answer questions' found in nature. Solving the 12-ball problem demands something closely akin to the creative act-reasoning by elimination or induction rather than deduction. In this respect the 12-ball problem claims a feature in common with detective-story puzzles and even, on an elementary plane, with such profound problems as that posed by photosynthesis."

The earliest published account of an "odd-ball" problem that we could find in the literature was one proposed by E. D. Schell of Arlington, Va., in the August-September, 1945, issue of American Mathematical Monthly. Schell gives you eight coins and the traditional beam balance. One of the coins is counterfeit- underweight. You are asked to sort out the counterfeit in two weighings.

It was solved immediately by a number of readers of his article and was generalized for m coins and n weighings by Irving Kaplansky, G. H. Neugebauer and W. O. Pennell. They showed that for any set of m coins, where 3^n-1 m < 3ⁿ not more than n balancings are needed to sort out the counterfeit.

Grossman independently published a similar generalization of the eight-coin problem in his December, 1945, paper in Scripta Mathematica, and he introduced the more ingenious 12-coin version with three weighings, with the troublesome additional requirement that you tell whether the odd coin is lighter or heavier. Apparently his problem had access to air transportation, because in a matter of days R. L. Goodstein published it, along with a generalization, in the Mathematical Gazette, the British counterpart of Scripta Mathematica. The avalanche of papers touched off by these two ingenious author-mathematicians has not yet died out.

"Like the wood's colt," writes Grossman, "this problem encounters difficulty in tracing its sire. During the mid-1940s it was just 'in the air,' and many of those who make a hobby of recreational mathematics took a whack at refining it. Royal V. Heath, a New York stockbroker, amateur magician and amateur mathematician, passed it on to me. The fact that I published it first would seem incidental."

Among the intuitive generalizations of the problem, one of the easiest for laymen to follow was presented by Karl Itkin in Scripta Mathematica for March, 1943. He demonstrated by experiment that two weighings are enough to identify and describe the odd ball from a set of three balls; three weighings suffice for 12 balls; four for 39, and five for 120. From these results he showed that if n weighings handle m balls, n + 1 weighings can cope with a set of 3(m + 1) balls. The ternary nature of the problem had been suggested by F. J. Dyson and C. A. B. Smith in the Mathematical Gazette of October, 1946, and February, 1947, and had been refined and simplified by Grossman. Hence Itkin recognized that his experimental results conformed to the ternary series: 3 = 3¹, 12 = 3¹ + 3², 39 = 3¹ + 3² + 3³, 120 = 3¹ + 3² + 3³ + 3⁴, and so on. He expressed the generalized equation as: m = (3ⁿ - 3)/2. The equation states, for example, that with only seven weighings you can identify and describe the odd ball among

or 1,092 balls!

Most amateurs who lose sufficient sleep over the problem manage to come up with one or more solutions of the contingent type. Through trial and error-and guided by the happy faculty for guessing right that characterizes a good research man-they simply weigh various combinations of balls until they hit the jackpot. Estimates of the total number of ways in which the problem can be solved run from a few score to several hundred thousand, depending on how you define a distinct solution. But until last year all of the contingent solutions were largely products of the cut-and-try method. Then Paul J. and Dorothy Kellogg, of the Laboratory of Nuclear Studies at Cornell University, hit upon a way of eliminating the guess work by making an ingenious application of information theory. They told about it in a paper called "Entropy of Information and the Odd Ball Problem" in the Journal of Applied Physics for November, 1954. Shannon, author of the information theory, defined "information" in a rather special way. The amount of possible information depends on the range of choices afforded by the system of symbols. A one-letter alphabet, for example, would allow no freedom of choice, and the information you could transmit would approach zero. An alphabet of two letters would afford greater choice and hence make possible the transmission of more information. It turned out that the equations which Shannon developed for assessing the information potential of such systems were precisely the equations that define entropy in thermodynamic systems.

The Kelloggs analyzed the odd-ball problem by means of these equations and found that certain combinations of balls yield a greater change in entropy than others. Those combinations in which entropy change reaches maximum value lead to solutions.

Although entropy equations take the guesswork out of contingent solutions, you will find the computation complicated unless you happen to be a designer of jet engines or a specialist in some other branch of thermodynamics.

To make the job easy for the rest of us, J. J. Cornish, the designer of the aerodynamic smoke tunnel discussed above, invites you to construct the little computer shown above. Its operation resembles that of a slide rule. You first position the hairline of the slider over the caret between the first four balls (1, 2, 3, 4) and the second four (5, 6, 7, 8) in the bottom tier of this rule. This represents the first weighing. Let us suppose that in an actual weighing the left side goes down, showing that the combination of the first four balls is heavier than the second four. Now for the second weighing go to the second tier and move the slider toward the light side-that is, the right side in this case. (The slider must always be moved to the light side.) Set the hairline on the caret toward the right in this tier-the caret between the balls 1, 2, 5 and 3, 6, 12. Say the weighing shows the left side again is heavier and that side goes down. So you again move the slider to the right and go to the third tier for the final weighing. Now the caret on which the hairline falls is between balls 1 and 2. Suppose the right side goes down. To read the result of the third weighing move the slider to the light side, over ball 1, and read the answer above: 2 H. The odd ball is 2, and it is heavier than the others. In this manner the odd ball, whichever of the 12 it happens to be, can be identified quickly in three weighings.

In our opinion the contingent solutions are less elegant and entertaining than those of the predetermined type. Grossman is credited with presenting the first comprehensive treatment of a predetermined solution in 1948. His approach employed the ternary system.

The ternary system, as we have seen, uses the powers of 8: 3⁰ = 1, 3¹ = 3, 3² = 9, and so on. Any decimal number can be stated in terms of powers of 3 and their negative counterparts: - 1, -3, -9 and so on. For instance, the number 5 can be written -1 - 3 + 9. With the first three powers of 3, and the negatives of the first two, we can construct a table from which all the numbers from 1 to 12 can be formed by such algebraic addition. Put down these powers as column headings and list in each column the numbers requiring the figure at the head in their ternary expression: for example, the number 5 is listed in the column under-1 and also under -3 and +9. The table is:

Such a table has remarkable properties. You can easily extend it to 100 numbers or more and astonish your friends with it. For instance, if a friend indicates in which columns his age is listed, you can immediately tell his age by adding the headings of those columns algebraically. A table of this kind can suggest card tricks and other puzzles that may be solved by means of ternary numbers.

Grossman made one that tells you how to weigh the balls in the 12-ball problem so that the column headings give you a solution of the predetermined type. His table has six columns-a pair for each of the three weighings. He rearranged the entries so that only four numbers appear in each column-corresponding to four balls for each pan during each weighing. He accomplished this by treating some of the numbers as though they were negative. The digit 5, for example, can be made negative simply by reversing the signs of its ternary components: + 1 + 3 - 9. Grossman observed that the columns would be equalized-that is, each would contain just four numbers-if he considered as negative all even numbers of the form (3n - 1)/2 (e.g., 4) and all odd numbers not of this form (e.g., 3). Numbers remaining positive (up to 12) are, then: 1, 2, 6, 8, 10 and 12. Those considered negative are 3, 4, 5, 7, 9 and 11.

Here is Grossman's table:

Each of the 12 balls is identified by a number. To use the table, weigh the balls under + 1 against those under -1, then those under +3 against those under -3 and finally those under +9 against those under -9. At each step note which pan is heavier, if either one is, and after the third weighing add up algebraically the numbers at the heads of the heavy columns. The sum will identify the odd ball, and if that number appears in the "heavy" columns, the odd ball is heavier than the others; if it appears in the "light" columns, it is lighter. For instance, suppose that 7 is the odd ball and it is light. On the first weighing the +1 pan will be heavier than -1; on the second, -3 will be heavier than +3; on the third, +9 will be heavier than -9. The sum is + 1 - 3 + 9 = 7. The number 7 appears in the light columns; therefore the ball is light.

If there are insomnophiles who wish to venture further into this fascinating territory, Octave Levenspiel of Bucknell University offers more sophisticated problems:

"Can you find the odd ball among 13 in three weighings?

"Given 15 coins, one identified as genuine, can you find the single counterfeit among them in three weighings?

"Is it true that in n weighings (with n greater than 3) one can find two balls heavier than the others among m balls if

m = 3^n/2 when n is even, or m=2[3^(n-1)/2] when n is odd?"

Bibliography

AERODYNAMICS FOR MODEL AIRCRAFT. Avrum Zier. Dodd, Mead & Company, 1942.

MATHEMATICAL RECREATIONS AND ESSAYS. W. W. R. Ball. The Macmillan Company, 1947.

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