An expert question on pigeonhole principle for P5–P6, 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 12 red, 4 yellow, 7 blue and 13 green marbles, alike except for colour.
Ching takes them out one at a time without looking.
What is the fewest she must take to be sure of one colour with at least 4 and another colour with at least 2?
Show the hints (3)
💡Hint 1: Think of the worst luck: lots taken and still not there. In what different ways can she fall short?
💡Hint 2: Falling short comes in two kinds: (a) no colour has 4 yet; (b) one colour has 4 or more (even all of it), but every other has fewer than 2. How many can each kind hold at most?
💡Hint 3: (a) at most 3 of each; for (b), try emptying each colour that has 4 or more, the others at most 1 each. Compare the two.
Worked solution
Have a go first, then read the solution step by step.
1.Falling short can happen only two ways: (a) every colour has fewer than 4; (b) one colour has 4 or more but every other has fewer than 2.
2.(a) At most 3 of each (all, if fewer): 3 red, 3 yellow, 3 blue, 3 green, 3 + 3 + 3 + 3 = 12.
3.(b) Empty one colour with 4 or more, the others at most 1 each: emptying red gives 15; emptying yellow gives 7; emptying blue gives 10; emptying green gives 16. The most is emptying green: 1 + 1 + 1 + 13 = 16.
4.The worst luck is the larger of the two: 16 and still short. One more must do it: 16 + 1 = 17.
⚠️Common mistake: thinking of only one way to fall short and answering 13. In (b), emptying a whole colour holds more than stopping every colour at 3.
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