Puzzles

Carol organised a knockout competition in December 2024, which 6 people entered. There were 2 matches in the first round with the remaining two players given byes (so they went into the next round without playing a match). The second round was made up of two semi-finals, then one final match was played to decide the winner. In total 5 matches were played.

This year, Carol is organising the competition again, but it has become a lot more popular: 355 people have entered. While planning the tournament, she can decide which rounds to give people byes in. What is the smallest number of matches that could be included in the tournament?

In the annual tournament of Christmas puzzles, each player must play one puzzle match against each other player. Last year there were four entrants into the tournament (A, B, C, and D), and so 6 matches were played: A vs B, C vs D, A vs D, A vs C, D vs B, and finally B vs C.

This year, the tournament has grown in popularity and 22 players have entered. How many matches will be played this year?