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30MSC 2023 #21: Counting

Counting is, perhaps, everyone’s first lesson in Math. It is used to determine the number of elements or objects in a group or set. But there are techniques for counting without really counting.

This question from the Junior IVMO 2021 illustrates one of those techniques: In a knock-out tennis competition, each match has a winner and a loser and a winner goes through to the next round. Rounds continue until there are two players left in the final. In the women’s singles at Wimbledon there are 128 players at the start. How many matches are there altogether?

We can actually count the number of games:

  1. First round involving 128 players: 128 ÷ 2 = 64 matches
  2. Second round involving the 64 winners: 64 ÷ 2 = 32
  3. Third round involving the 32 winners of the second roud: 32 ÷ 2= 16
  4. Round of 16 involving the 16 winners of the third round: 16 ÷ 2= 8
  5. Quarter finals with the 8 winners of the round of 16: 8 ÷ 2= 4
  6. Semifinals with the 4 quarterfinal winners: 4 ÷ 2 = 2
  7. Final match with the 2 semifinalist winners = 1

Total = 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127 matches.

An easy solution is to employ logic. Since each match has a loser, in order to have a champion, 127 matches are needed to eliminate 127 players.

Counting techniques in combinatorics are discussed in  Chapter 1 of our forthcoming book, 30 Master Strategies in Computing Vol III.

Follow us on our Facebook page, MATH-Inic Philippines at https://www.facebook.com/MATHInicPhils to see tomorrow’s 30MSC 2023 #22: Sequences

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