Exercise 1. This software determines whether the two permutations $$(1x)$$ and $$(123 ... n)$$ generate $$S_n$$ for various choices of $$x$$ and $$n$$. (That is, whether every element of $$S_n$$ can be expressed as some product of these permutations. ) For $$n = 4$$, run the program for $$x = 2, 3,$$ and $$4$$. For $$n = 5$$, run the program for $$x = 2, 3, 4,$$ and $$5$$. For $$n = 6$$, run the program for $$x = 2, 3, 4, 5,$$ and $$6$$. For $$n = 8$$, run the program for $$x = 2, 3, 4, 5, 6, 7,$$ and $$8$$. Conjecture a necessary and sufficient condition involving $$x$$ and $$n$$ for $$(1x)$$ and $$(123 ... n)$$ to generate $$S_n$$.

Please enter $$n$$ and $$x$$, the result will show below.