I've been thinking about sorting $6$ elements with the minimal possible number of comparisons. I can do it in $10$ comparisons but I've no idea if this is optimal. Or is there a better algorithm ?  Algorithm1. Sort $a_1, a_2, a_3$ and $a_4, a_5, a_6$.Number of comparisons: $3+3=6$.2. Merge two subarrays.Number of comparisons: $3 + 3 - 2 = 4$.Total number of comparisons: $6 + 4 = 10$.  
Minimal number of comparisons - sorting $6$ elements
algorithms;sorting;lower bounds
According to A036604, 10 comparisons are indeed optimal. The link probably contains a citation to a paper proving this.