I think you mean a tournament with 13 rounds something like this:
(L E) (A R) (H Y) (N P) (T K) (Q D) (I W) (Z X) (O V) (B G) (M U) (C F) (J S)
(A S) (O P) (Q R) (T W) (Z Y) (B V) (H E) (C D) (M K) (L U) (J X) (I G) (N F)
(T D) (Z G) (M F) (Q U) (B S) (J W) (O X) (L P) (C E) (N Y) (I R) (A K) (H V)
(I Y) (H D) (T X) (A V) (O U) (Z F) (N K) (M S) (Q G) (C W) (B E) (J P) (L R)
(B W) (L Y) (I K) (J R) (N V) (H X) (C P) (T U) (A D) (Q F) (O G) (Z S) (M E)
(O F) (M W) (B P) (Z E) (C R) (L K) (Q S) (I V) (J Y) (A X) (N D) (H U) (T G)
(H R) (Q V) (Z D) (O K) (M X) (C G) (T Y) (J F) (B U) (I E) (L S) (N W) (A P)
(N X) (T F) (O S) (H G) (Q E) (M P) (A U) (B R) (Z W) (J K) (C Y) (L V) (I D)
(J U) (N S) (A E) (I F) (H W) (O R) (L G) (Q Y) (T P) (M V) (Z K) (B D) (C X)
(Q P) (B K) (C V) (M Y) (J D) (I U) (Z R) (N G) (L X) (H S) (A F) (T E) (O W)
(Z V) (C U) (J G) (B X) (L F) (N E) (M D) (A W) (I S) (T R) (H P) (O Y) (Q K)
(C K) (I X) (N U) (L D) (A G) (T S) (J V) (O E) (H F) (Z P) (Q W) (M R) (B Y)
(M G) (J E) (L W) (C S) (I P) (A Y) (B F) (H K) (N R) (O D) (T V) (Q X) (Z U)
where the rows could be rounds and the columns the 13 games. It may look random, but it has been made from
two orthogonal Latin squares, meaning that all games will be between 2 distinct groups of teams. These are:
(A B C H I J L M N O Q T Z) and (D E F G K P R S U V W X Y)
If you don't tell the teams about this pattern, then I guess they are unlikely to notice it.
