Instead of four teams, now there are $2^6$, or $64,$ seeded from $1$ through $64$. The power index of each team is equal to $65$ minus that team’s seed.
The teams play in a traditional seeded tournament format. That is, in the first round, the sum of opponents’ seeds is $2^6+1,$ or $65.$ If the stronger team always advances, then the sum of opponents’ seeds in the second round is $2^5+1$, or $33$, and so on.
Once again, the underdog in every match gets a power index boost $B$, where $B$ is some positive non-integer. Depending on the value of $B,$ different teams will win the tournament. Of the $64$ teams, how many can never win, regardless of the value of $B$?
After setting up the $64$ team bracket, we can choose various values for $B$ and then compute the winners directly. Since the outcome will remain the same for all non-integer values in $(\lfloor B \rfloor, \lceil B \rceil)$, we need only select a total of $64$ values, e.g., $b+0.5,$ for $b = 0, 1, \dots, 63.$ The summary of the outcomes of the $64$ team tournament bracket are in the table below. We note that there are $27$ seeds (namely, $7$, $13$-$15$, and $25$-$47$) who never win the tournament. Most unlucky out of each of them are $7$, $15$ and $47$, each of which have sets of $B$ of measure $3$ for which they will end up the runner up of the tournament, though they will never win it.
| $B$ | Winning Seed | Runner Up |
|---|---|---|
| (0,1) | 1 | 2 |
| (1,2) | 1 | 3 |
| (2,3) | 3 | 1 |
| (3,4) | 2 | 1 |
| (4,5) | 6 | 5 |
| (5,6) | 5 | 3 |
| (6,7) | 5 | 3 |
| (7,8) | 4 | 3 |
| (8,9) | 12 | 11 |
| (9,10) | 11 | 9 |
| (10,11) | 11 | 9 |
| (11,12) | 10 | 9 |
| (12,13) | 10 | 9 |
| (13,14) | 9 | 7 |
| (14,15) | 9 | 7 |
| (15,16) | 8 | 7 |
| (16,17) | 24 | 23 |
| (17,18) | 23 | 21 |
| (18,19) | 23 | 21 |
| (19,20) | 22 | 21 |
| (20,21) | 22 | 21 |
| (21,22) | 21 | 19 |
| (22,23) | 21 | 19 |
| (23,24) | 20 | 19 |
| (24,25) | 20 | 19 |
| (25,26) | 19 | 17 |
| (26,27) | 19 | 17 |
| (27,28) | 18 | 17 |
| (28,29) | 18 | 17 |
| (29,30) | 17 | 15 |
| (30,31) | 17 | 15 |
| (31,32) | 16 | 15 |
| (32,33) | 48 | 47 |
| (33,34) | 49 | 47 |
| (34,35) | 49 | 47 |
| (35,36) | 50 | 49 |
| (36,37) | 50 | 49 |
| (37,38) | 51 | 49 |
| (38,39) | 51 | 49 |
| (39,40) | 52 | 51 |
| (40,41) | 52 | 51 |
| (41,42) | 53 | 51 |
| (42,43) | 53 | 51 |
| (43,44) | 54 | 53 |
| (44,45) | 54 | 53 |
| (45,46) | 55 | 53 |
| (46,47) | 55 | 53 |
| (47,48) | 56 | 55 |
| (48,49) | 56 | 55 |
| (49,50) | 57 | 55 |
| (50,51) | 57 | 55 |
| (51,52) | 58 | 57 |
| (52,53) | 58 | 57 |
| (53,54) | 59 | 57 |
| (54,55) | 59 | 57 |
| (55,56) | 60 | 59 |
| (56,57) | 60 | 59 |
| (57,58) | 61 | 59 |
| (58,59) | 61 | 59 |
| (59,60) | 62 | 61 |
| (60,61) | 62 | 61 |
| (61,62) | 63 | 61 |
| (62,63) | 63 | 61 |
| $\gt 63$ | 64 | 63 |
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