Calculated Risks
Just as the electronic calculator made the slide rule obsolete, sophisticated computer software like Mathematica, Maple, and Derive are trumping calculators. These programs slice through thorny equations with ease, inspiring illustrations, interactive diagrams, and . . . puzzles. The following problems give a taste of what can be dreamed up with this software.

Parenthetically Speaking Mathematica is complex but still basically a calculator—type in 3 + 5 and it returns 8. But unlike most calculators, Mathematica lets you add parentheses to your expressions. Following the rules of basic algebra, you get different results depending on where you place parentheses. For instance, (3 - 2) - 1 = 0, whereas 3 - (2 - 1) = 2.

1. [Easy] List all the different numbers you can make by adding parentheses to the expression 4 - 3 - 2 - 1.

2. [Difficult] Find four numbers, a, b, c, and d, that make x equal 1, 2, 3, and 4 when parentheses are used in different ways in the equation a - b - c - d = x.

3. [Difficult] Which of the numbers 32, 42, 52, 62, and 72 is impossible to make by adding parentheses to the expression 10 + 8 x 6 - 4/2?

4. [Very difficult] How many different ways are there to add parentheses to the five-variable expression a - b - c - d - e? How many of those ways make different numbers?

5. [Extremely difficult] How many different ways are there to add parentheses to the 10-variable expression a - b/c - d/e - f/g - h/i - j?



Tetrahedral Tangle This convoluted clump was made with Mathematica's tools for drawing three-dimensional shapes. The figure incorporates several tetrahedrons, each with four triangular faces built from six bars that connect four vertices. Study the shape closely, then answer the following questions.

1. [Easy] How many tetrahedrons are in this figure?

2. [Difficult] If each face of each tetrahedron is counted as a separate triangle, to how many other triangles is each triangle linked? Two triangles are considered linked if they cannot be separated without breaking a bar.

3. [Difficult] Examine any two of the tetrahedrons and you will find that they are linked. Can you unlink them by deleting one edge from each of the two tetrahedrons?

4. [Difficult] If you draw a line between each corner and its three nearest corners, what type of polyhedron is formed?

5. [Very difficult] Imagine each tetrahedron as a solid shape instead of an open framework. What shape is formed by the intersection of all the tetrahedrons? In other words, what shape is the set of points common to all the tetrahedrons?



The Plot Thickens Patterns in numbers are often easier to spot if they're color coordinated. For example, to more clearly understand the pattern represented by the numbers in the grid below, each can be changed to a corresponding color. The colorful new grid reveals a pattern not as easily perceived from numbers alone. Below are four more grids, labeled a, b, c, and d. The numbers along the margins of each grid indicate the number value for each square within a row, column, or diagonal. The numbers within these grids can be added to corresponding numbers in other grids. If you add the value of each square in grid a to the value of the corresponding square in grid c, taking only the last digit of each sum, you get the pattern shown in grid 1. The lower right square in grid a has a value of 7, and the lower right square in grid c also has a value of 7, so the lower right square in grid 1, which represents the sum of a and c, has a value of 4 (the last digit of the sum 7 + 7 = 14). Can you determine which pair of grids a, b, c, or d was added to create each of the grids 2 through 6?






Solution

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Let us know what you think of Bogglers: E-mail alternate solutions, comments, and suggestions to bogglers@discover.com.

Got new solutions for the puzzle? Want to see other people's solutions? Talk to the puzzle master in his discussion forum at www.scottkim.com.



For more on Mathematica, go to www.wolfram.com; for Maple, see www.mapleapps.com, and for Derive, go to www.chartwellyorke.com.

There are more puzzles and inventions by Scott Kim on his Web site, www.scottkim.com, where you will also find an archive of his monthly Bogglers columns, a Bogglers discussion board, and other amusements for your mind.

Also check out Scott's creative "inversions"—upside-down lettering—on his Web site and in his book Inversions: A Catalog of Calligraphic Cartwheels (Emeryville, Calif.: Key Curriculum Press, 1996). To order, go to www.keypress.com.



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