| | | The 3x+1 problem |
Here is an example of a Maple procedure
with an if ... then ... else structure.
Collatz:=proc(n::posint) if type(n,even) then n/2; else 3*n+1; fi; end;
This Maple code defines a function named
Collatz of one argument n that sends n into
n/2 if n is even, and sends n into 3n+1 if
n is odd. The declaration proc(n::posint) checks that
the argument n is a positive integer and issues an error
message if it is not.
There is a famous unsolved problem--often called the 3x+1 problem--about the iterates of the Collatz function. Is it the case that for every positive integer n, iterating the Collatz function eventually leads to the cycle 1->4->2->1?
For example, if you start with the number 25, successive
applications of the Collatz function lead to the sequence
25, 76, 38, 19, 58, 29, 88, 44,
22, 11, 34, 17, 52, 26, 13, 40,
20, 10, 5, 16, 8, 4, 2, 1. Here
is a procedure iterateCollatz that can be used to
produce this list of numbers via iterateCollatz(25);.
iterateCollatz:=proc(n::posint) local c; c:=n; while c>1 do print(c); c:=Collatz(c): od; end;
This procedure implements a while loop that prints a value, applies the Collatz
function to the value, and iterates as long as the value
exceeds 1. No one knows if this procedure is guaranteed to
terminate for every positive integral input n. For more
about this problem, see a paper by Jeff Lagarias.
Exercise
Write a Maple procedure that takes one argument, a positive integer, and returns the number of iterations of the Collatz function required to reach the value 1 from the given starting point (assuming, of course, that the number of iterations is finite). What is the result of applying this function to your telephone number?
Write a Maple procedure that takes one argument, a positive integer, and determines the largest number attained by repeatedly applying the Collatz function to that value (assuming, of course, that there is such a largest number). What is the result of applying this function to your student identification number?
Now that you have tried your hand at writing a procedure to implement the Collatz function, find out how the experts do it. Visit the Maple Application Center and search on "Collatz".
The Math 696 course
pages were last modified April 5, 2005. | | | The 3x+1 problem |