A hard question on pigeonhole principle for P5, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P3 to P6 qualifying rounds and finals. The trick is to imagine the unluckiest case.
The question
A bag holds 4 red, 9 blue and 1 green balls, alike except for colour.
Keung takes them out one at a time without looking.
What is the fewest he must take to be sure of 3 the same colour?
Show the hints (3)
💡Hint 1: With the worst luck, how many of each colour can he take and still not have 3 alike?
💡Hint 2: Careful: some colours don't even have 2.
💡Hint 3: With the worst luck he has 2, 2, 1 of the three colours.
Worked solution
Have a go first, then read the solution step by step.
1.Worst luck: 2 red, 2 blue, 1 green (all of them), 5 in all, still not 3 alike.
2.Every colour still in the bag already has 2 out; one more, whatever its colour, makes 3: 5 + 1 = 6.
⚠️Common mistake: 3 × 2 + 1 = 7, forgetting a colour has fewer than 2.
Play more like this in Numsprite (unlimited questions, instant marking) →A free maths game: every level free from Easy to Expert (★ to ★★★★), unlimited questions, no download