How many students are there at Hagun Academy? After watching Rakudai Kishi no Cavalry, I found myself confused by what seems to be a plot hole. Specifically, the anime depicts the qualifying matches at Hagun academy for the Seven Stars Sword Art tournament. The qualifying matches seem to be a 20-round single-elimination tournament with 6 winners. This is never outright stated, but many points of dialogue suggest this rather strongly, and at the very least it is stated explicitly in the last episode of the anime that all of the competitors on the last day had won 19 matches.

The 6 winners:


   


By basic math, a single-elimination tournament with 6 winners and 20 rounds would imply that there are about 6.3 million entrants to the qualifying tournament. That number of students at the school seems to be effectively impossible. For comparison, the population of New York City is about 8 million people, and the total number of high school students in Japan in 2011 was about 3.5 million. So this seems unreasonably large as an absolute number.

In addition, the number is too large relative to the size of the school. The area of Hagun Academy is said to be more than 10 times the size of the Tokyo Dome (according to http://rakudai-kishi.wikia.com/wiki/Hagun_Academy); using that figure, the average population density would be about 5 people per square meter (more than 100 times as dense as the most dense city in the world). In addition, we've gotten aerial shots of the school, and while large, it isn't nearly big enough to contain that many people.



So, I'm wondering, is this really the right number of students at the school? If not, is there some explanation for how the tournament has 20 rounds, or is this just a plot hole?
Hagun Academy (破軍学園 Hagun Gakuen) is one of the seven Mage Knight schools -> there are only 7 schools.
We have a rule



  With the competition, six students defeat 10 different students earning them their spot as Hagun Academy’s representatives for the festival.


So looks like we have less students in Hagun. 

BTW, it's LN logic !!!