**Chaos, Bounds, and Randomness Practice Problems**

**Practice 1**

Does the tendency of athletic performance to occur in ''spurts'' mean that a gradual improvement can never take place in any real-life situation? For example, is it impossible that the curve for the swimmer's times (Fig. 7-7) could look like either the solid or dashed lines in Fig. 7-9 instead?

**Fig. 7-7. **Monthly best times (in seconds) for a swimmer whose specialty is the 100-m freestyle, plotted by month for a hypothetical year. Illustration for Practice 1 and 2.

**Fig. 7-9. **Illustration for Practice 1 and 2.

**Solution 1**

Gradual, smooth improvement in athletic performance is possible. Either of the graphs (the straight, dashed line or the solid, smooth curve) in Fig. 7-9 could represent a real-life situation. But orderly states of affairs such as this are less common than more chaotic types of variations such as that shown in Fig. 7-7.

**Practice 2**

What is the fastest possible time for the 100-m freestyle swimming event, as implied by the graphs in Fig. 7-7 or Fig. 7-9?

**Fig. 7-7. **Monthly best times (in seconds) for a swimmer whose specialty is the 100-m freestyle, plotted by month for a hypothetical year. Illustration for Practice 1 and 2.

**Fig. 7-9. **Illustration for Practice 1 and 2.

**Solution 2**

Neither of these graphs logically implies that there is a specific time that represents the fastest possible 100-m swim. It can be mathematically proven that there exists a *maximum unswimmable time* for this event. But determining the actual time quantitatively, and then proving that the determination is valid, is another problem.

**Practice 3**

Can the MUST scenario, in which there is a greatest lower bound, apply in a reverse sense? Can there be, for example, a *minimum unattainable temperature* (MUTT) for the planet earth?

**Solution 3**

Yes. The highest recorded temperature on earth, as of this writing, is approximately 58°C (136°F). Given current climatic conditions, it's easy to imagine an unattainable temperature, for example, 500°C. We then start working our way down from this figure. Clearly, 400°C is unattainable, as is 300°C, and also 200°C (assuming runaway global warming doesn't take place, in which our planet ends up with an atmosphere like that of Venus). What about 80°C? What about 75°C? A theorem of mathematics, called the *theorem of the least upper bound*, makes it plain: ''If there exists an upper bound for a set, then there exists a *least upper bound* (LUB) for that set.'' This means there is a MUTT for our planet, given current climatic conditions. Figuring out an exact number, in degrees, for the MUTT is another problem.

**Practice 4**

Name two poor sources of pseudorandom digits, and two good sources, that can be obtained by practical means.

**Solution 4**

The digits of a known irrational number such as the square root of 20 represent an example of a poor source of pseudorandom digits. These digits are predestined (that is, they already exist and are the same every time they are generated). In addition, they can be produced by a simple machine algorithm.

The repeated spinning of a wheel or rotatable pointer, calibrated in digits from 0 to 9 around the circumference, is another example of a poor way to get a pseudorandom sequence. The wheel can never be spun forcefully enough to randomize the final result. In addition, friction is not likely to be uniform at all points in the wheel bearing's rotation, causing it to favor certain digits over others.

A good source of pseudorandom digits can be obtained by building a special die that has the shape of a *regular dodecahedron*, or geometric solid with 12 identical faces. The faces can be numbered from 0 to 9, with two faces left blank. This die can be repeatedly thrown and the straight-upward-facing side noted. (If that side is blank, the toss is ignored.) The results over time can be tallied as pseudorandom digits. The die must have uniform density throughout, to ensure that it doesn't favor some digits over others.

Another good source of pseudorandom digits is a machine similar to those used in some states' daily televised lottery drawings. A set of 10 lightweight balls, numbered 0 through 9, are blown around by powerful air jets inside a large jar. To choose a digit, the lid of the jar is opened just long enough to let one of the balls fly out. After the digit on the ''snagged'' ball is noted, the ball is put back into the jar, and the 10 balls are allowed to fly around for a minute or two before the next digit is ''snagged.''

Practice problems for these concepts can be found at:

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