Problem
Count games in a knockout tournament
Seven players play knockout table tennis. Two play at a time and the loser is out for good. The last player standing is champion. Count how many games get played.
Reasoning · Grade 4-2 Leagues and Tournaments
Your answer
Watch out Everyone-plays-everyone gives 21, but a knockout ends the moment you lose.
How to solve
Strategy Draw a Diagram — The problem asks for a bracket, so I draw one and count the games round by round — that gives a concrete answer I can trust. Then I change focus and count losers instead of games, which shows the same total comes out no matter how the bracket is drawn, and gives a rule that works for any number of players.
1STEP 1
Draw the bracket and play round 1
Round 1 has 3 games and one player sits out.
7 = 2 + 2 + 2 + 1 → 3 games
Pairing players off is the same as sharing 7 counters into groups of 2 and finding 3 groups with 1 left over.
4.OA.A.3Draw A Diagram2STEP 2
Play the remaining rounds
The rest are 2 games then 1.
4 → 2 games → 2 → 1 game → 1
Each round just halves the field, which is repeated division by 2 — a Grade 4 idea.
4.OA.A.3Draw A Diagram3STEP 3
Add up the games
That adds to 6.
3 + 2 + 1 = 6
Listing the games round by round and adding is straightforward once the bracket is drawn.
4.OA.A.3Make A Systematic List4STEP 4
Check it another way: count the players knocked out
One loser per game gives 7 − 1 = 6 too.
7 - 1 = 6
Counting the losers instead of the games works because games and losers come in matching pairs — one loser for each game.
4.OA.C.5Change Focus Count The ComplementCounting the players knocked out counts the games, because every game removes exactly one player.
Why?
One game produces one loser and one loser comes from one game, so games and knocked-out players match up one for one.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Everybody but the single champion is eliminated, so the number of losers is fixed without drawing the bracket at all.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
5STEP 5
Compare with a league, to see how different the two formats are
A league would need 21 — a big difference.
7 × 6 ÷ 2 = 21
The knockout only has to remove players, while a league has to arrange every possible pairing — so the counts are nowhere near each other.
4.OA.A.3Make A Systematic ListAnswer
6 games
7 − 1 = 6
6 games is between the smallest sensible answer (you cannot decide a champion among 7 people in fewer than 6 games, because each game removes only one player) and the 21 games a full league would need — so the size is right. The round-by-round count 3 + 2 + 1 = 6 and the knock-out count 7 - 1 = 6 agree, and the answer is a whole number of games, as it must be.
Takeaway
In a knockout, every game sends exactly one person home — so count the people who go home, not the games!
- Draw the bracket and play round 1
- Play the remaining rounds
- Add up the games
- Check it another way: count the players knocked out
- Compare with a league, to see how different the two formats are