Parenthetically Speaking



1. -2, 0, 2, and 4

2. a = 4.5, b = 2, c = 1, d = .5 is one solution. The first number minus the second must be 2.5, and the second two numbers must be 1 or -1 and .5 or -.5, in either order.

3. You cannot make 62 by adding parentheses to the expression 10 + 8 x 6 - 4/2. Here are the ways to make the other numbers: 32 = 10 + (((8 x 6) - 4)/2)
42 = 10 + (8 x (6 - (4/2)))
52 = (((10 + 8) x 6) - 4)/2
72 = (10 + 8) x (6 - (4/2))

4. There are 14 ways to add parentheses to the five-variable expression a - b - c - d - e, of which eight make different numbers. Adding parentheses to a - b - c - d - e has the effect of essentially allowing us to choose whether the last three variables c, d, and e are added to or subtracted from the quantity (a - b). Multiplying together the two choices (parentheses or no parentheses) for the three variables gives us 2 x 2 x 2 = 8, or eight choices altogether. Normally, each of these eight choices makes a different number, unless we choose exceptional values like c = d = e = 0.

Here are the 14 ways to add parentheses to the expression a - b - c - d - e. The list is divided into four parts: adding parentheses to a and bcde, ab and cde, abc and de, and abcd and e.

a - (b - (c - )d - e))) a - (b - ((c - d) - e)) a - ((b - c) - (d - e)) a - ((b - (c - d)) - e) a - (((b - c) - d) - e) (a - b) - (c - (d - e)) (a - b) - ((c - d) - e)

(a - (b - c)) - (d - e) ((a - b) - c) - (d - e) (a - (b - (c - d))) - e (a - ((b - c) - d)) - e ((a - b) - (c - d)) - e ((a - (b - c)) - d) - e (((a - b) - c) - d) - e

5. There are 4,862 ways to add parentheses to the 10-variable expression a - b/c - d/e - f/g - h/i - j. The numbers that count the ways to parenthesize an n-variable expression are called the Catalan numbers, after 19th-century mathematician Eugène Charles Catalan. The first few Catalan numbers are 1, 1, 2, 5, 14, 42, 132, 429, 1,430, and 4,862. Like the Fibonacci numbers, each Catalan number can be computed from the previous Catalan numbers. They also specify, among other things, the number of ways to dissect a polygon with n + 2 sides into triangles.



Tetrahedral Tangle



1. The shape is made of five tetrahedrons.

2. Each triangle is linked to six other triangles: two each for three of the four other tetrahedrons, and none for the fourth.

3. Here is how to unlink two tetrahedrons by deleting just one edge from each.

4. A dodecahedron, a 12-sided figure, shown below in red.

5. The intersection of all five tetrahedrons is an icosahedron, a regular shape of 20 triangular faces. The key is to realize that each triangular face of a tetrahedron lies in a plane perpendicular to a line from the center of the figure to the fourth corner of the tetrahedron. This means that the 20 planes containing the 20 faces of the tetrahedrons are arrayed in the same symmetry as the corners of the dodecahedron above. The regular dodecahedron and icosahedron have what is known as a "dual" relationship: The corners of one shape can be constructed as the centers of the faces of the other shape, and both have the same symmetries. In other words, there are as many faces in the icosahedron as there are vertices of the dodecahedron.



The Plot Thickens

1. a + c 2. b + c 3. c + d

4. a + d 5. b + d 6. a + b

One of the most far-reaching features of Mathematica is a built-in programming language that lets you create interactive applications that can work with other programs or across the Web. Thousands of add-on packages have been written, covering areas as diverse as statistics, number theory, optics, audio-signal processing, and blackbody radiation. Even artists have gotten into the act (check out www.graphica.com). Learn more about Mathematica at www.wolfram.com.





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