Each integer is assigned to a category labelled by its remainder on division by 5. How many category labels are possible?

To use all files as pigeons and the folders as holes, which assignment condition is needed?

• Some files are left outside all available folders.
• Every file is assigned to all available folders.
• Only folders already containing two files are counted.
• Every file is assigned to exactly one folder.

Four single-colour socks are drawn from a drawer whose socks come in 3 colours. What must happen?

• Each colour appears at least once
• All four differ
• Two share a colour

In any group of 13 people, two share a birth month. True or false?

Twenty tokens occupy 6 boxes. Five specified boxes each contain at most 2 tokens. What is the greatest lower bound for the number in the remaining box?

Seven distinct integers are chosen from 1 through 12. Which pigeonhole argument proves that two sum to 13?

• The twelve integers form twelve holes, and seven selected numbers overfill those twelve holes.
• The six disjoint pairs summing to 13 give six holes for the seven selected numbers.
• The seven selected numbers form seven holes, and the twelve available integers overfill those holes.

Seven people each receive one badge numbered 1 through 6. Which assumption starts a contradiction proof that two badge numbers agree?

• Assume each badge number is used by at least one person.
• Assume each badge number is used by at most one person.
• Assume one badge number is used by all seven people.
• Assume every person has a badge numbered from 1 through 6.

Twenty objects in 6 boxes guarantee that some box contains exactly 4 objects. True or false?

Let m and n be positive integers, and let q and s be the integer quotient and remainder when m is divided by n: m = qn+s, 0 <= s < n. Which placement uses all m objects in n boxes and keeps every box at or below ceiling(m/n)?

• Put q+1 objects in every one of the n boxes, with no boxes receiving fewer objects.
• Put q+1 in s boxes and q-1 in each of the other n-s boxes, leaving no objects outside.
• Put q+1 in s boxes and q in each of the other n-s boxes.

Repeated values are allowed in seven selections from 1 through 12. Which selection disproves the claim that two selected entries must sum to 13?

• 1, 12, 2, 3, 4, 5, 6
• 2, 11, 3, 4, 5, 6, 7
• 1, 1, 1, 1, 1, 1, 1
• 3, 10, 4, 5, 6, 7, 8