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Earthquake Intensity

It is well known that periodically the open interval (0, 1) on a real line is rocked by major earthquakes. The quakes are composed of a series of tremors, all of equal strength. There may be a million or more tremors in a single quake. A tremor can be thought of as a function on the interval of the form A(x - x2). A is a positive real number, and it measures the intensity of the quake, the way the Richter scale does. Since the points are displaced internally within the interval, it follows that the intensity of the quake must be greater than 1 and less than 4. Each real number in (0, 1) is affected by each tremor -- some being perennially driven to specified deposit sites. A good way to analyze an earthquake is to examine its aftermath.

  1. If after the quake, all points in the interval (0, 1) are found congregated at the point 0.4 --
    1. what was the intensity of the quake?
    2. how many tremors must have occurred in the quake?
  2. What is the intensity of the strongest quake that eventually leaves all the points in the interval deposited at a single site?
  3. If the intensity of the quake is 3.5 --
    1. what single point stays fixed throughout the whole quake?
    2. what two points alternate positions throughout the whole quake?
    3. what is the fate of most of the points?
  4. What range(s) on the Richter scale is (are) indicative of a quake that deposits points at --
    1. two distinct sites?
    2. four distinct sites?
  5. Comment on the following theories about earthquakes:
    1. If the intensity of two earthquakes is nearly identical, then the results ( number and placement of deposit sites) are nearly identical.
    2. Neighboring points are driven to the same deposit site.
    3. The stronger the earthquake is, the more deposit sites there are.
    4. The more deposit sites there are, the rarer the quake is.
    5. There are earthquakes that have any number of deposit sites --3, 4, 5, 6, 7, 8, 9, 10, and so on.

(Reprinted with permission from Exploratory Problems in Mathematics, copyright 1992 by the National Council of Teachers of Mathematics. All rights reserved.)

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