Advanced Pigeonhole Principle and Finite Mapping Theory Quiz

1) The pigeonhole principle can be interpreted as guaranteeing which property of a function when ?

Injectivity

Surjectivity

Non-injectivity

Bijectivity

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

Explanation

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

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

True

False

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

Explanation

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

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

N/k

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

Explanation

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

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

10

11

12

15

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

Explanation

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

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

True

False

⌈(2n+1)/n⌉ = 3.

Explanation

⌈(2n+1)/n⌉ = 3.

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

Exactly equal numbers in each box

Numbers differ by at most 1

At least one box is empty

Each object chooses a box randomly

Uniform in combinatorics allows a difference of at most 1.

Explanation

Uniform in combinatorics allows a difference of at most 1.

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

25 pigeons, 51 holes

51 pigeons, 25 holes

26 pigeons, 2 holes

25 pigeons, 50 holes

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

Explanation

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

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

Residue classes

Countries

Buckets in a hash function

Subsets of a partition

Pigeonholes are categories into which objects fall.

Explanation

Pigeonholes are categories into which objects fall.

Submit

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

True

False

It proves existence, not the specific box.

Explanation

It proves existence, not the specific box.

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

2

3

4

5

⌈40/13⌉ = 4.

Explanation

⌈40/13⌉ = 4.

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

True

False

Many combinatorial proofs rely on forced duplication.

Explanation

Many combinatorial proofs rely on forced duplication.

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

1/7

1

2/7

Depends on additional data

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

Explanation

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

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

Showing two strings have the same prefix length

Proving two keys must collide in a hash table

Showing two subsets of the same size share an element

Showing two people share the same birthday month

These guarantee unavoidable repetition in pigeonholes.

Explanation

These guarantee unavoidable repetition in pigeonholes.

Submit

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

M≥⌈n/k⌉

M≤n-k

M=n/k

No relation can be deduced

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

Explanation

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

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

101 pigeons, 100 holes

50 pigeons, 101 holes

101 pigeons, 50 holes

3 pigeons, 50 holes

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

Explanation

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

Advanced Pigeonhole Principle and Finite Mapping Theory Quiz

1) The pigeonhole principle can be interpreted as guaranteeing which property of a function when ?

2)

What first name or nickname would you like us to use?

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

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

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

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

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

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

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

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

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

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

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

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

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

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