Olympiad topic · P3–P6
Inclusion–exclusion
Common in the Hua Xia Cup's P3 to P6 qualifying rounds and finals; harder ones have three overlapping groups, or ask who is in exactly two.
Try one
★★ MediumPupils decorate the classroom with 7 paper streamers, each 19 cm long.
They glue them end to end into one long streamer, each two neighbours overlapping by 1 cm.
How long is the finished streamer?
- Hint 1: Gluing 7 streamers in a line, how many overlaps are there?
- Hint 2: Each overlap counts 1 cm twice.
- 1.Ignoring overlaps: 7 × 19 = 133 cm.
- 2.There is one overlap between each two neighbours: 7 streamers make 7 − 1 = 6 overlaps, 6 × 1 = 6 cm.
- 3.The overlaps were counted twice, so take them off once: 133 − 6 = 127 cm.
- ⚠️Common mistake: thinking there are 7 overlaps and taking off 7 cm.
Answer: 127 cm
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsTwo groups, straight.
- ★★MediumFree2 question types · unlimited questionsSome in neither, overlapping ribbons.
- ★★★HardFree2 question types · unlimited questionsOverlapping multiples, three groups.
- ★★★★Expert (competition level)VIP3 question types · unlimited questionsExactly two of three, the fewest who pass all three.
Inclusion–exclusion: what it is and how to solve it
Two (or three) groups that overlap: find the total or the overlap without counting anyone twice.
For example: 10 keep hamsters, 12 keep rabbits, 6 keep both. How many keep at least one? 10 + 12 − 6 = 16.
The key idea: Two groups: total = A + B − both. Three: A + B + C − each pair's overlap + all three. Draw a Venn diagram and fill it from the middle outwards.
Practise this topic free →Easy to Hard (★ to ★★★) all free, unlimited questions, no download