Supreme Court Handshakes

The first judge shakes hands with all eight other judges. Then the second judge shakes hands with seven other judges because they already shook hands with the first judge. This pattern continues and it results in $8 + 7 + 6 + \cdots + 1$ handshakes. Quickly calculating this sum results in $\sum_{i = 1}^{8} = \frac{(8 + 1)(8)}{2} = \boxed{36}$

There's probably more than one way to solve this, however my solution was to assign each justice a letter of the alphabet (A-I) in column. Then I created a duplicate column next to the original so that A was directly across from A, B across from B and so on. Then I drew a line from A in the first column to every other letter in the second column excluding the A in the second column (No one shakes their own hand, that would be weird). These lines represent the "handshakes". There are 8 lines in total. Next I realized that that configuration would happen between each judge, thus 8 connections times 9 judges equals 72 handshakes. Half of those lines are redundant, i.e., B in the first column connected to H in the second column is the same relationship (handshake) as H in the first column connected to B in the second. Thus we have 72 divided by 2 which equals 36.