Purple & Gold: 11/1/17 – 12/1/17

Then we denote 7 the best runner-up, 8 the second best runner-up, and so on until 12, the lowest ranked runner-up. However, this bracket does not satisfy the group diversity constraint, defined in Section 1. Since the group labels are totally ignored, it might be that in one half of the bracket, some group winners and some runners-up come from the same group, for example if team 1 (the best group winner) and team 12 (the lowest ranked runner-up) come from the same group. Note that in this example the ideal bracket (reported in Fig. 9) does not satisfy the group diversity constraint:• Wales and Slovakia play against each other in the round of 16 but advanced from the same group (B). The other 2 runners-up play against each other (positions 8 and 9 of the ideal bracket). Then a rearrangement of the 6 runner-up positions and a rearrangement of the 4 third-placed team positions are performed as follows:• First we look at the groups of teams 1, 4, and 5 (left hand side). A first solution consists of slightly distorting the ideal bracket (Fig. 8) in a deterministic way. This first started with the playing field. In the unfavorable cases (6 cases out of 135), we suggest to slightly tweak the bracket as follows: instead of playing against the lowest ranked of the 3 right runners-up, team 6 (the lowest ranked group winner) would play against the middle-ranked right runner-up.

After playing well in back-to-back games, BJ Boston reverted, going 2-9 from the floor for eight points. Honduras claimed the second qualifying spot by virtue of a superior goal difference over Argentina, who missed a penalty in their match, with both sides finishing on four points. 3), Poland, who is Team 7 and comes from the same group as Germany, can only take position X8 (in the left half), and cannot take position X7 (in the right half), and Spain (Team 8) can only take position X7. • Several winners and runners-up from the same group (France and Switzerland, Group A; Germany and Poland, Group C; Italy and Belgium, Group E) are also on the same half of the bracket. It might also be that in some quarter of the bracket, several teams come from the same group, for example if teams 1 (best group winner) and/or 8 (second best runner-up) and/or 16 (fourth best third-placed team) come from the same group.

Besides, third-placed teams play against group winners in the round of 16. The ideal bracket is actually a perfectly balanced bracket, in the sense that the ranks (from 1 to 16) of any two opponents sum to 17 in the 8 matches of the round of 16, and then, assuming the best ranked team always advances to the next round, the ranks of any two opponents sum to 9 in the 4 quarterfinals, and to 5 in the 2 semifinals. The other two runners-up, England and Belgium, go to positions 7 and 10. Symmetrically, Iceland, the lowest ranked left runner-up, goes to position 12 (against Italy), and Poland and Spain play against each other in positions 8 and 9. The 4 best third-placed teams come from Groups B, C, E, and F. Since the upper right quarter would have contained England (Group B), Germany (Group C), and Belgium (Group E), the only possible opponent for Germany would have been Portugal (third of Group F). Eventually, the third-placed teams from Groups B and E (Slovakia and Ireland) could have been placed equally in the two remaining quarters of the bracket (upper left and lower right).

The three right runners-up come from the groups of France, Wales, and Italy (the left hand side group winners): Switzerland, England, and Belgium. Among those 135 configurations, there are only 6 unfavorable configurations where it is impossible to assign third-placed teams so as to satisfy the group diversity constraint: when the lowest ranked of the 3 right runners-up is from group G1 (the group of team 1), the lowest ranked of the 3 left runners-up is from group G2 (the group of team 2), and 3 of the 4 best third-placed teams come from groups G1, G4, G5 or from groups G2, G3, G6.- When this does not happen (129 cases out of 135), one can easily check (e.g., running a simple computer code) that there exists 2 (in 112 cases out of 129) or 4 (in 17 cases out of 129) admissible allocations of the 4 best third-placed teams. To enforce group diversity, the runners-up of these 3 groups can only be placed on the right hand side of the bracket.

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