Question 6
Players P, Q, R and S played each other once in a competition where there were no draws.
Only the following results are known.
- Player P defeated players Q and R.
- Player Q defeated two players.
- Players R and S each defeated one player.
Based on these results, a dominance matrix $N$ was partially constructed as shown.
$$N=\begin{array}{c|cccc}&\text{P}&\text{Q}&\text{R}&\text{S}\\\hline\text{P}&0&1&1&0\\\text{Q}&\square&\square&\square&\square\\\text{R}&0&0&0&1\\\text{S}&1&0&0&0\end{array}$$
The completed matrix $N$ is
Not quite. This has Q defeating P, but P defeated Q. Q’s row needs a 0 in the P column. In 2024, 3.7% of Queensland students chose (A).
That’s it. Step through the hints below to see why. In 2024, 93.1% of Queensland students chose (B).
Not quite. This gives Q only one win, and the game between Q and R has no winner. There were no draws. In 2024, 1.4% of Queensland students chose (C).
Not quite. The 1 on the diagonal would mean Q defeated Q. Diagonal entries are always 0. In 2024, 1.5% of Queensland students chose (D).
Fill in Q’s row yourself. Tap a cell to switch it between 0 and 1.
In a dominance matrix, the entry in row X and column Y is 1 when X defeated Y, and 0 otherwise. A player can’t play themselves, so every diagonal entry is 0.
Each game has exactly one winner, so for every pair of players exactly one of the two matching entries is 1.
Use the rows you already have.
P defeated Q, so Q’s entry under P is 0. R’s row shows R only defeated S, so R lost to Q. S’s row shows S only defeated P, so S lost to Q too.
Check the count. Q’s row should have exactly two 1s, because Q defeated two players. Then check the rows for R and S: each has exactly one 1, matching one win each.
P defeated Q, so Q’s P entry is 0. The diagonal entry is 0. R and S each defeated only one other player, and neither of those was Q, so Q defeated both of them.
$$N=\begin{bmatrix}0&1&1&0\\0&0&1&1\\0&0&0&1\\1&0&0&0\end{bmatrix}$$
How Queensland students answered in 2024
From the QCAA Specialist Mathematics subject report for 2024. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2024 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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