Pigeonhole principle — Consecutive selections

Question

Show that choosing six numbers from guarantees two consecutive numbers.

This practice question was inspired by Question 14(a) in the 2025 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.Partition the ten numbers into five consecutive pairs.
Simplify the previous line carefully, keeping exact values where possible.Choose six numbers.
Simplify the previous line carefully, keeping exact values where possible.Two choices must come from one pair.
Finish the calculation, then check that the result meets the question’s conditions.Those two numbers are consecutive.
State the final answer clearly in the form the question requested.Final answer: Use the five boxes .

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

Verification: Five choices can avoid this by selecting one from each pair, so six is sharp.

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