Pigeonhole principle — Boundary case

Question

13 teams, penalty for more than 3, exactly 39 over-age players. Must a team be penalised?

This practice question was inspired by Question 12(b) in the 2022 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: identify the objects and boxes, then show why avoiding a repeated box is impossible.39 can be each: no team forced over 3.
State the final answer clearly in the form the question requested.Answer: not necessarily.

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Equality does not force a pigeonhole overflow.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.