### a longest path problem

fivethirtyeight.com has a weekly column, "The Riddler". The Riddler poses two problems in each column. The first, "Riddler Express", is intended to be an easier problem that one can solve quickly, while the second, "Riddler Classic", is more difficult. Modulo vacations, it appears each Friday, and if you submit a solution by the end of the following Sunday (Eastern Time) , you will be among those who might be given credit for the solution in the next column.

The August 28th column's Classic problem is as follows: what's the longest sequence of integers x[i] for 1 ≤ i ≤ n such that

for all i, 1 ≤ x[i] ≤ 100for all i < n, either x[i+1] is a multiple of x[i] or x[i+1] is a factor of x[i]for all i, j, x[i] = x[j] iff i = j, i.e. the x[i] are all distinct (The last constraint avoids trivial sequences like 2, 4, 2, 4, 2, 4... which can go on forever if repeats are permitted.)

One way to characterize this problem is to look at it as a graph with a hu…

The August 28th column's Classic problem is as follows: what's the longest sequence of integers x[i] for 1 ≤ i ≤ n such that

for all i, 1 ≤ x[i] ≤ 100for all i < n, either x[i+1] is a multiple of x[i] or x[i+1] is a factor of x[i]for all i, j, x[i] = x[j] iff i = j, i.e. the x[i] are all distinct (The last constraint avoids trivial sequences like 2, 4, 2, 4, 2, 4... which can go on forever if repeats are permitted.)

One way to characterize this problem is to look at it as a graph with a hu…