While I was working on Ten distinct numbers in the table, I found that odd-sized tables will all work. Additionally, the technique @xnor used to prove a 10x10 doesn't work does not automatically rule out 4x4 and 8x8 tables from working. A 4x4 table doesn't work, but what about an 8x8 table?
Same rules as before - can you pick the numbers 0 through 7 such that you have one of each number, with every column and row being represented exactly once each?
If this doesn't work, are there any even-sized tables that do? If it does, is there a general rule for which ones work and which don't?
Parity check: 0-7 add up to 28, so numbers+rows+columns adds up to 84. Even number passes the parity check.