Sequence of Six Game

Book: A Walk through Combinatorics

Problem

• A soccer team played 30 games this year, and scored a total of 53 goals, scoring at least one goal in each game. Prove that there was a sequence of consecutive games in which the team scored exactly six goals.
• Prove that claim of part 1 does not hold for a team that scored 60 goals, with the other parameters unchanged
• Prove that the claim of part 1 does not hold for a team that scored 59 goals, with the other parameters unchanged

Solution

Proof by contradiction:

Let us assume