April 2025
Define a function \(f\) whose input and output are both lists of \(n\) nonnegative integers by \[f(a_1,a_2,\dots,a_n)=(|a_1-a_2|,|a_2-a_3|,\dots,|a_{n-1}-a_n|,|a_n-a_1|)\] where, as usual, \(|x|\) represents the absolute value of \(x\).
For example, \[f(1,2,3,4)=(|1-2|,|2-3|,|3-4|,|4-1|)=(1,1,1,3)\] and \[f(2,3,5)=(|2-3|,|3-5|,|5-2|)=(1,2,3).\] We will denote by \(f^k\) the function that iterates the application of \(f\) a total of \(k\) times. For example, \[f^4(1,1,1,3)=f^3(0,0,2,2)=f^2(0,2,0,2)=f(2,2,2,2)=(0,0,0,0).\] We will call a list \((a_1,a_2,\dots,a_n)\) smooth if there is some \(m\) for which \(f^m(a_1,\dots,a_n) = (0,0,\ldots,0)\). That is, a list is smooth if some number of applications of \(f\) will result in the list of all zeros. For example, \((1,1,1,3)\) is smooth since \(f^4(1,1,1,3)=(0,0,0,0)\), as demonstrated above.
Find a list of length \(5\) that is not smooth. Find a list of length \(7\) that is not smooth.
Show that for all odd integers \(n\geq 1\) there exists a list \(L\) of length \(n\) that is not smooth.
How many smooth lists \((a,b,c)\) are there with \(a\), \(b\), and \(c\) each no larger than \(100\)?
Suppose \(L\) is a list of length 4 consisting of only zeros and ones. Show that \(L\) is smooth.
Show that all lists of length 4 are smooth.