This week the question is: Can You Root for the Underdog?

Once again, there are four teams remaining in a bracket: the 1-seed, the 2-seed, the 3-seed, and the 4-seed. In the first round, the 1-seed faces the 4-seed, while the 2-seed faces the 3-seed. The winners of these two matches then face each other in the regional final.

Also, each team possesses a “power index” equal to 5 minus that team’s seed. In other words:

• The 1-seed has a power index of 4.
• The 2-seed has a power index of 3.
• The 3-seed has a power index of 2.
• The 4-seed has a power index of 1.

In any given matchup, the team with the greater power index would emerge victorious. However, March Madness fans love to root for the underdog. As a result, the team with the lower power index gets an effective “boost” B, where B is some positive non-integer. For example, B could be 0.5, 133.7, or 2𝜋, but not 1 or 42. To be clear, B is a single constant throughout the tournament, for all matchups.

As an illustration, consider the matchup between the 2- and 3-seeds. The favored 2-seed has a power index of 3, while the underdog 3-seed has a power index of 2+B. When B is greater than 1, the 3-seed will defeat the 2-seed in an upset.

Depending on the value of B, different teams will win the tournament. Of the four teams, how many can never win, regardless of the value of B?

And for extra credit:

Instead of four teams, now there are 2⁶, 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 2⁶+1, or 65. If the stronger team always advances, then the sum of opponents’ seeds in the second round is 2⁵+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?

Highlight to reveal (possibly incorrect) solution:

Spreadsheet. Animation.

First off, let’s word the problem differently.

• In the first round the a-seed faces the (5-a)-seed.
• The power index for the a-seed is 5-a.
• Further, in a match the team with the lowest power index gets a boost B, so the power index will actually be 5-a+B.
• If the a-seed and the b-seed meet each other, then the a-seed will win if 5-a > 5-b+B.
• As B isn’t an integer, there will always be a winner.
• In a match where the 1-seed loses to the 4-seed, these all give the same result:
  • 5-1 < 5-4+B
  • 3 = 4-1 < B
  • B = 3.1
  • B = 3.9
  • ⌈B⌉ = 4
• We can therefore just look at 1 representative for each integer, to get through all options: 0+ε, 1+ε, 2+ε etc.

In order to figure out what will happen with 4 teams, we probably have to look at B going from 0+ε to 5+ε. Enter a spreadsheet. Revealing that 1, 3 and 4 can win, but not 2. 1 team can’t win.

And for extra credit:

I do the same thing in the same spreadsheet, just bigger. This reveals the winning teams 1, 2, 10, 6, 22, 18, 14, 46, 50, 54, 58 and 62, 12 teams in all. 64-12 = 52 teams can never win. After correcting some errors in the spreadsheet after getting a hint from a fellow fiddler, it now reveals the winning teams 1, 3-2 (because I found them in that order), 6-4, 12-8, 24-16 and 48-64. Therefore 7, 13-15, 25-47 can’t win. 1 + 3 + 23 = 27 teams.