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Cell[CellGroupData[{
Cell["Project - An Integer Sequence", "Title",
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Cell[TextData[StyleBox["This is a project which deals with discrete \
mathematics and sequences that are defined via a recurrence. You are to \
examine the long-term behavior of these sequences.",
 FontWeight->"Bold"]], "Text"],

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Cell[TextData[{
 "Take any positive integer, say 57. Square the individual digits and add \
them to get a new integer:\n\n\t",
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 "\n\nNow repeat the procedure. What will happen? Will the sequence converge \
(i.e., settle down to a single value)? Will it cycle through a sequence of \
values? Does each starting number produce a different behavior?\n\nYou are to \
answer these questions for the integers from 1 to 100."
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Cell["\<\
Write a function which computes the next integer in the sequence from the \
given one, i.e.,\
\>", "Text"],

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Cell[TextData[{
 "Make sure this function checks for positive integers. You will need to use \
the ",
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integer n. IntegerDigits[n, b] gives a list of the base-b digits in the \
integer n. IntegerDigits[n, b, len] pads the list on the left with zeros to \
give a list of length len.\"\>", " ", 
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Cell[TextData[{
 "Write a function which will detect whether a given sequence converges, has \
a cycle, or does not settle down into a regular behavior (within the first \
200 steps). You should do this using a ",
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sequence. If the new value repeats the value just before it, then there is \
convergence, otherwise there is a cycle. To check for repeated values, use \
the ",
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 "Once you have found convergence or a cycle, your function should return the \
values between the repeated ones, including the repeated values. For example, \
your function should return\n\n\t{2, 2}                 for a sequence \
converging to 2\n\t{3, 4, 7, 2, 3}     for the cycle consisting of the \
repeated values 3, 4, 7, 2.\n\nYou can use the ",
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objects matching pattern appear in expr. Position[expr, pattern, levspec] \
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pattern, levspec, n] gives the positions of the first n objects found.\"\>", 
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Cell["\<\
In order to compare the cycles of different starting values, you would want \
to \"standardize\" the cycles. For example, the three different sequences \
below all reference the same cycle, except we are focusing on a different \
area in the repeated sequence.\
\>", "Text"],

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Cell["\<\
To find a \"canonical\" or \"standardized\" form, we identify the minimal \
element in the list, and then write the cycle with that element as the first \
one. Write a function that produces the standard cycle from a  given \
(possibly) non-standard one. In the example given above, the standard cycle \
would be \
\>", "Text"],

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Cell[TextData[{
 "Put all the previous steps together to produce a table for the integers \
form 1 to 100 (in general from 1 to ",
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  FontSlant->"Italic"],
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\"other\". \n\n\t1\t{1, 1}\n\t2\t{4, 16, 37, 58, 89, 145, 42, 20, 4}\n\t\
\[VerticalEllipsis]\t\t\[VerticalEllipsis]\n\t100\t{1, 1}\n\t\nYou might want \
to think about which numbers have the same behavior (such as 1 and 100, since \
0s do not change a thing!) How many different behaviors do you expect at \
most??"
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Cell["\<\
Finally, create a visual display of the information given in the table of \
Step 4. One possibility is to display the integers on the horizontal axis, \
and the sequence values on the vertical axis (or vice versa). Display cycles \
that are the same in the same color - you will be surprised by the graph (not \
really, since you have already looked at the table). You may also think of \
any other type of plot which summarizes the table visually - be creative!
\
\>", "Text"]
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integers from ",
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 " to ",
 StyleBox["n",
  FontSlant->"Italic"],
 ".) Write a usage statement for your function, and if needed, an error \
statement. Test you function to answer the question posed for the integers \
from 1 to 100."
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