In a cave in Roswell, New Mexico, a group of scientists have discovered a collection of objects that appear to be (for lack of better words) from ``out of this world.'' Among the objects are a technical manual, metallic tokens, an electrical device (surprisingly similar in design to a PrimeCo phone), and a pair of rubber-like gloves.

Although the scientists fear these objects were placed in the cave by local teens as a hoax, they are taking their job seriously and they are enthusiastically studying the objects. Several of the scientists are focusing on a group of symbols, some of which appear on each of the objects that were discovered:

Since these symbols appear in a unique combination on the margin of each page in the technical manual, one scientist, Dr. Alfred Eisenstein, has suggested that they represent numbers. The first ten pages of the manual are ``numbered'' in the following way:

Dr. Eisenstein believes that these symbols are elements of a positional numeration system much like our own-each ``digit'' has a face value and a place value.

1. If Dr. Eisenstein is correct, what is the base of this strange numeration system? Make a key indicating the face value of each digit.
2. To test Eisenstein's theory, one researcher counted the pages until she reached the one labeled . How many pages did she count?
3. The manual had a total of 89 pages. How was the last page numbered?
4. The last page was actually numbered , but after carefully examining the manual, the researcher discovered that several pages had been torn out. How many pages were missing? (Write your result as a number in this strange numeration system.)
5. Dr. Eisenstein was asked by a colleague to examine the gloves that were found. He was not-at-all surprised when he saw them. How many fingers do you think each glove had? Explain your reasoning.
6. Several of the metallic tokens that were discovered were numbered. The small gray tokens were numbered , while a slightly larger, more ornate token was numbered . The largest and most detailed tokens were numbered . Develop a theory regarding these tokens.
7. Construct an addition table and a multiplication table for this numeration system.
8. (Challenge Problem) There were several circles drawn on one of the pages of the technical manual. The sequence of symbols appeared many times on this page. What does this sequence of symbols refer to? Carefully explain your reasoning.
9. (Now for something totally different) The two Voyager spacecraft each carried a 12-inch phonograph record containing images and sounds selected to portray the diversity of life and culture on Earth. Visit the website http://vraptor.jpl.nasa.gov/voyager/record.html. Follow the link to Images and then click on Mathematical definitions. Write a short essay explaining how this image describes our numeration system. You should address the following questions: Is it clear from the image that we use ten basic digits to write all numerals. Does the image show how we use the idea of place value? To what base is our base-ten system compared? What symbols are used as basic digits in the other base? How would have you described our numeration system?
10. (Now for something else totally different) Research the numeration system of the ancient Mayans. Write a short essay describing what you find. Include a key showing their basic digits and several examples of multi-digit numerals.