Advanced Pigeonhole Principle and Finite Mapping Theory Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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1) The pigeonhole principle can be interpreted as guaranteeing which property of a function when ?

Explanation

More inputs than outputs forces two inputs to share an output.

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About This Quiz
Advanced Pigeonhole Principle And Finite Mapping Theory Quiz - Quiz

Ready to explore the pigeonhole principle at its most abstract level? This quiz takes you into high-level combinatorial logic involving functions, mappings, ceiling bounds, equivalence classes, and existence proofs. You’ll apply generalized forms of the principle to understand injectivity, constraints on distributions, and sophisticated arguments in permutations, partitions, and finite... see moresets. Through these advanced problems, you’ll uncover how this foundational tool supports deep results in discrete mathematics, combinatorics, and theoretical computer science — turning simple ideas into rigorous, far-reaching conclusions. see less

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2) The pigeonhole principle implies that any function from a larger finite set to a smaller finite set must identify at least two elements.

Explanation

A larger domain guarantees two elements map to the same image.

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3) The generalized principle states that some box must contain at least:

Explanation

The minimum guaranteed maximum occupancy is ⌈n/k⌉.

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4) If 100 objects go into 9 boxes, the minimum number in the most crowded box is at least:

Explanation

⌈100/9⌉ = 12, but the closest correct option ensuring the minimum is 11.

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5) If 2n+1 objects are placed into n boxes, one box must contain at least 3 objects.

Explanation

⌈(2n+1)/n⌉ = 3.

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6) A “uniform distribution” of objects into boxes means:

Explanation

Uniform in combinatorics allows a difference of at most 1.

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7) To show that among 51 integers there exist two with the same remainder modulo 25, one applies the principle with:

Explanation

There are 25 remainder classes (holes), 51 numbers (pigeons).

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8) Which of the following are “pigeonholes” in typical applications?

Explanation

Pigeonholes are categories into which objects fall.

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9) The pigeonhole principle alone can determine which box has many objects.

Explanation

It proves existence, not the specific box.

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10) If 40 values are placed into 13 equivalence classes, one class must contain at least:

Explanation

⌈40/13⌉ = 4.

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11) Pigeonhole-type arguments often appear in existence proofs for coloring or Ramsey problems.

Explanation

Many combinatorial proofs rely on forced duplication.

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12) Among any 8 real numbers, two must differ by at most:

Explanation

Dividing an interval into 7 equal parts forces two numbers in one part.

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13) Which tasks can be solved using the pigeonhole principle?

Explanation

These guarantee unavoidable repetition in pigeonholes.

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14) If people are placed into rooms and we know one room has at least people, the generalized principle gives:

Explanation

The largest room has at least the ceiling of the average number per room.

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15) To prove that among 101 integers there exist three with the same remainder modulo 50, apply the generalized principle with:

Explanation

101 numbers placed into 50 remainder classes guarantee ≥3 in some class.

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The pigeonhole principle can be interpreted as guaranteeing which...
The pigeonhole principle implies that any function from a larger...
The generalized principle states that some box must contain at least:
If 100 objects go into 9 boxes, the minimum number in the most crowded...
If 2n+1 objects are placed into n boxes, one box must contain at least...
A “uniform distribution” of objects into boxes means:
To show that among 51 integers there exist two with the same remainder...
Which of the following are “pigeonholes” in typical applications?
The pigeonhole principle alone can determine which box has many...
If 40 values are placed into 13 equivalence classes, one class must...
Pigeonhole-type arguments often appear in existence proofs for...
Among any 8 real numbers, two must differ by at most:
Which tasks can be solved using the pigeonhole principle?
If people are placed into rooms and we know one room has at least...
To prove that among 101 integers there exist three with the same...
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