Pigeonhole principle
- Olympiad · IOQM
The pigeonhole principle says that if you put more objects than boxes into the boxes, at least one box must hold two or more objects.
Think of 10 pigeons and 9 holes. Each pigeon flies into a hole. However they spread out, some hole ends up with at least two pigeons. There is no way round it.
The idea sounds too plain to be useful. Its power is that it proves something exists without finding it. You only need to spot what the objects are and what the boxes are.
A small example
In any group of 13 people, two were born in the same month. The people are the objects. The 12 months are the boxes. Since 13 is more than 12, one month holds two people.
Now a harder one. Pick any 5 whole numbers. Show that two of them differ by a multiple of 4. Divide each number by 4. The remainder can only be 0, 1, 2 or 3, so there are 4 boxes. Five numbers go into 4 boxes, so two share a remainder. Their difference then leaves remainder 0, which means it is a multiple of 4.
The same trick works with any divisor. This link to remainders is where it meets congruence.
Where it fits
This goes beyond the Class 10 board syllabus. HBCSE's syllabus for the Mathematical Olympiad names it under combinatorics. The Mathematics Teachers' Association (India), MTA(I), conducts IOQM, the first stage of the Mathematical Olympiad Programme that HBCSE organises for NBHM.
Sources
Facts last checked against these sources on 30 September 2026.
- Syllabus for Mathematical Olympiad (Combinatorics) · HBCSE, TIFR
- Mathematical Olympiad 2026-2027: stages of selection · HBCSE, TIFR
- Brochure: Mathematical Olympiads 2026-2027 · HBCSE, TIFR
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