Fiddler on the Proof: March 28, 2025

David Ding

March 28, 2025

Can You Root for the Underdog?

Explanation:

This is part 2 of the March Madness (or should I say "Mathness") Fiddler. Unlike last week, this week, we are exploring how often underdogs win given any boost B. For the original puzzle, let's think critically.

For the 4-seed to win out, it needs to beat the 1-seed in the first round, which can be achieved with $B>3$. This boost $B$ will be sufficient to beat out the winner of the 2-3 matchup, since both of them will not receive any boosts when facing the 4-seed, and the 4-seed's boost of $B>3$ is sufficient to beat any opponent.

On the other hand, for the 1-seed to win out, it needs to have $B<1$. In this case, even the 2-seed cannot beat the 1-seed with similar analysis as above.

When $1<B<2$, the 3-seed will emerge victorious against the 2-seed, but will fall at the hands of the 1-seed who will advance against the 4-seed. When $2<B<3$, the 3-seed will once again be victorious against the 2-seed and in fact will win the entire tournament. In any case, the 2-seed will never win the tournament for any value of B (non-integer as given).

Therefore, the answer to the original Fidder is that one team never wins: the 2-seed.

Extra Credit

I wrote a MATLAB script to generate the desired bracket, calculate the eventual winner given any value of $B$, and visually present it in a GIF for values of $B$ going from 0.5 to 63.5 in increments of 1.

From the above animation, 27 teams never won. When the boost value $B$ reaches a critical value of $B>32$, it will steadily give seeds 48 and above the victory. For boost values less than 32, there are flunctuations in the lower-seeded teams and the higher-seeded teams don't have enough boost to win out.

Below is the complete list of winners for each boost value from 0.5 to 63.5. For values less than 1, the 1-seed always wins. For values greater than 63, the 64-seed always wins.

Winner for boost = 0.5 is the 1-seed
Winner for boost = 1.5 is the 1-seed
Winner for boost = 2.5 is the 3-seed
Winner for boost = 3.5 is the 2-seed
Winner for boost = 4.5 is the 6-seed
Winner for boost = 5.5 is the 5-seed
Winner for boost = 6.5 is the 5-seed
Winner for boost = 7.5 is the 4-seed
Winner for boost = 8.5 is the 12-seed
Winner for boost = 9.5 is the 11-seed
Winner for boost = 10.5 is the 11-seed
Winner for boost = 11.5 is the 10-seed
Winner for boost = 12.5 is the 10-seed
Winner for boost = 13.5 is the 9-seed
Winner for boost = 14.5 is the 9-seed
Winner for boost = 15.5 is the 8-seed
Winner for boost = 16.5 is the 24-seed
Winner for boost = 17.5 is the 23-seed
Winner for boost = 18.5 is the 23-seed
Winner for boost = 19.5 is the 22-seed
Winner for boost = 20.5 is the 22-seed
Winner for boost = 21.5 is the 21-seed
Winner for boost = 22.5 is the 21-seed
Winner for boost = 23.5 is the 20-seed
Winner for boost = 24.5 is the 20-seed
Winner for boost = 25.5 is the 19-seed
Winner for boost = 26.5 is the 19-seed
Winner for boost = 27.5 is the 18-seed
Winner for boost = 28.5 is the 18-seed
Winner for boost = 29.5 is the 17-seed
Winner for boost = 30.5 is the 17-seed
Winner for boost = 31.5 is the 16-seed
Winner for boost = 32.5 is the 48-seed
Winner for boost = 33.5 is the 49-seed
Winner for boost = 34.5 is the 49-seed
Winner for boost = 35.5 is the 50-seed
Winner for boost = 36.5 is the 50-seed
Winner for boost = 37.5 is the 51-seed
Winner for boost = 38.5 is the 51-seed
Winner for boost = 39.5 is the 52-seed
Winner for boost = 40.5 is the 52-seed
Winner for boost = 41.5 is the 53-seed
Winner for boost = 42.5 is the 53-seed
Winner for boost = 43.5 is the 54-seed
Winner for boost = 44.5 is the 54-seed
Winner for boost = 45.5 is the 55-seed
Winner for boost = 46.5 is the 55-seed
Winner for boost = 47.5 is the 56-seed
Winner for boost = 48.5 is the 56-seed
Winner for boost = 49.5 is the 57-seed
Winner for boost = 50.5 is the 57-seed
Winner for boost = 51.5 is the 58-seed
Winner for boost = 52.5 is the 58-seed
Winner for boost = 53.5 is the 59-seed
Winner for boost = 54.5 is the 59-seed
Winner for boost = 55.5 is the 60-seed
Winner for boost = 56.5 is the 60-seed
Winner for boost = 57.5 is the 61-seed
Winner for boost = 58.5 is the 61-seed
Winner for boost = 59.5 is the 62-seed
Winner for boost = 60.5 is the 62-seed
Winner for boost = 61.5 is the 63-seed
Winner for boost = 62.5 is the 63-seed
Winner for boost = 63.5 is the 64-seed