MAT2612 Assignment 2 (872267) Due 2025

Math2612
Assignment 2
Unique No: 872267
Due 2025

, Math2612

Assignment 2

Due 2025

Unique no: 872267

Solutions Based on Chapters 3, 4 & 5 of prescribed textbook



Question 1: Pigeonhole Principle

1.1 (7 Marks)

Problem: Twenty cards numbered 1 to 20 are placed face down on a table. Cards are
selected one at a time and turned over. If 2 of the cards add up to 21, the player loses. Use
the pigeonhole principle to show that if 11 cards are chosen, the player can never win the
game. State clearly what the pigeons and the pigeonholes are.

Solution:

• , ,...,20}.
Pigeons: The 11 cards chosen from the set {1 2

• Pigeonholes: The 10 pairs of numbers that sum to 21, i.e.,
, 0) (, 2 1, 9) (, 3 1, 8) (, 4 1, 7) (, 5 1, 6) (, 6 1, 5) (, 7,14), (8, 13), (9, 12), (10, 11)}
{(1 2

• Explanation: Each card belongs to one of these pairs. There are 10 pigeonholes.
Choosing 11 cards (pigeons) ensures at least two cards are from the same pair (since
11 >10). Their sum is 21, so the player loses. Thus, it is impossible to choose 11 cards
without losing.




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