Caracterizaci´ on experimental del arrastre y deposici´ on de gotas de agua 81

One hundred and sixty years ago, the iconoclastic hermit of Walden Pond offered this advice: “simplify, simplify!” One can only wonder what Thoreau would have thought today. My new cell phone came with 50 pages of instructions, and Lasser’s latest income tax guide has over 800 pages, most of which are incomprehensible to me. Do things really need to be that complicated? Our natural yearning for simplicity might also apply to mathematical recreations, and more specifically to my special interest, which is geometrical and me-chanical puzzles. Historically, it has been the simpler amusements that have usually enjoyed the most enduring popularity. Tinker-toys and building blocks are likely to still be around long after all of today’s video games have been discarded.

Of course, to the aspiring inventor, it always seems as though most of the simpler ideas have long ago been conceived and brought to light, perhaps even copyrighted or patented. All the more satis-faction, then, when the explorer of ideas stumbles upon a simple amusement that appears to be new and original, as much as any-thing in this world can truly be so described. For this article, I have sifted though my 35-year accumulation of puzzle designs and picked out a few that best illustrate the concept of simplicity.

Some of my more satisfying geometrical dissection puzzles have involved fitting four pieces into a square or rectangular tray, while

Figure 1. A peudo-dissection puzzle.

leaving some empty spaces. Because traditional dissection puz-zles involve pieces packing solidly to fill all gaps, these puzpuz-zles are perhaps more properly called pseudo-dissection puzzles. Shown in Figure 1 is one that Mary and I frequently take on Elderhostel trips to entertain our companions, very few of whom are ever able to solve it, even after receiving hints. This puzzle exploits one’s over-powering natural tendency to begin by trying to fit square corners of pieces into square corners of the container, as we all have been doing habitually, practically from birth. Even when the hapless victims of this psychological trap are cautioned to try a different approach, they will almost invariably revert to this hopeless first step.

My most successful polyomino-type puzzles have involved fit-ting just four or five pieces into a square or rectangular tray, as in the example shown in Figure 2. The size of the square tray is such that the five pieces fit snugly when they are arranged symmetrically as shown. Again, the success of the design is based more on psy-chology than mathematics. We spend most of our lives immersed in a world of orthogonal arrangements—everything from city streets and building plans to printed pages and computer screens. Thus the unfortunate puzzle solver has much difficulty ignoring this pre-disposition long enough to place the first piece skewed at an angle to the tray. I call this general class of puzzles “Square Root Type,”

and I call this subclass “Square Root of Five,” referring to the

Figure 2. A “Square Root of Five”

puzzle.

Figure 3. A “Square Root of Ten”

puzzle.

ative dimensions of puzzle pieces and tray. Closely related is the

“Square Root of Ten” subclass, perhaps even more confusing, as in the example shown in Figure 3. Many other variations on this theme are possible.

In the world of cubic puzzles, the 3× 3 × 3 size was popularized by Piet Hein’s seven-piece Soma Cube. The puzzle is based on dividing a cube into 27 equal parts according to a 3× 3 × 3 grid and joining these parts to form the puzzle pieces. Perhaps because of the multiple solutions (over 200) for Piet Hein’s puzzle, the 27-block size is often overlooked by puzzle designers as more of a novel plaything rather than a real challenge. Indeed, the tendency among puzzle designers these days (myself included) has been to tinker with interlocking assemblies of greater size and complexity.

But of my designs in this class, my favorite is still the classic 3×3×3 Half-Hour Puzzle (see Figure 4). Here, my design objective was to discover a set of pieces, all dissimilar and asymmetrical, preferably all the same size, and with the maximum number of pieces that would assemble into a 3× 3 × 3 cube in only one way. Because some of these requirements are mutually exclusive, a compromise is required, resulting in the six pieces shown below. As the name suggests, a half hour is a reasonable time for discovering the one solution. Incidentally, many puzzle solvers now have access to computer programs that can solve puzzles of this sort with blinding speed, by a process that might be described as systematic trial and error. One of the benefits of solving such puzzles the old-fashioned

Figure 4. Half-Hour Puzzle.

Figure 5. Drop Out Puzzle.

way is that nothing the human brain does can truly be described as random. One is continually discovering educated tricks and clever shortcuts to the solution, whether consciously or otherwise, and that can be a recreation in itself.

In 1964, Martin Gardner wrote a Scientific American column en-titled “The Hypnotic Fascination of Sliding Block Puzzles”.¹ In it he discussed the classic nine-piece Dad’s Puzzle, a particular favorite of my childhood, although I don’t believe I ever solved it. In these

1Scientific American, 210:122–130, 1964. Reprinted in Martin Gardner’s Sixth Book of Mathematical Diversions from Scientific American, Chapter 7, pages 64–70 (University of Chicago Press, 1983).

many years, the hypnotic fascination hasn’t changed. Some of the recently published designs of sliding-block puzzles are exceedingly clever and complex, especially the three-dimensional ones. Even among the flat kind, most are larger and more complex than the 4× 5 Dad’s Puzzle. But I wondered: are simpler designs possible?

This question led to the creation of the Drop Out Puzzle, shown in Figure 5. There are six movable pieces. The 3× 4 tray has a clear plexiglass top with a circular hole at one end, through which the round disk can be dropped. The object is, by shifting the pieces about, to eventually drop the disk through a hole in the bottom of the tray at the opposite end. (An additional hole in the center of the cover, not shown, is merely to facilitate pushing the pieces around with the eraser-end of a pencil.) It may appear to be impos-sible, but it can be done. I don’t think you will find a puzzle much simpler than this. I wonder if Thoreau would have approved.