Mathematical Reasoning and Problem Solving: Proving a Matchstick Puzzle
Mathematics is a powerful tool that helps us solve real-world problems and puzzles. In this article, we explore the intriguing question of how to use 2015 matches as efficiently as possible to build a specific figure. We will delve into the details of the mathematical solution and the underlying principles.
Understanding the Matchstick Challenge
Tackling a matchstick puzzle requires strategic thinking and a solid understanding of mathematical sequences. Imagine you have 2015 matches available, and your task is to create the largest possible figure. The challenge involves understanding the structure of the figure at each level, which consists of both vertical and horizontal matchsticks.
The Mathematical Structure
At each level k of the figure, the number of matchsticks (both vertical and horizontal) follows a specific pattern. For k1 vertical matches, there are k horizontal matches above and in between these vertical matches. Therefore, the total number of matches at each level k is calculated as 2k1.
At the very bottom level, k n, the total number of matches is defined as:
displaystyle n sum_{k1}^{n} 2k1 n 2 sum_{k1}^{n} k sum_{k1}^{n} 1 n 2 cdot dfrac{nn1}{2} n n^2 3n
The total number of matchsticks used to build the figure is given by the function n^2 3n. Our goal is to find the largest integer value of n such that the figure can be built without exceeding the 2015 matches available.
Solving the Equation
To find the largest possible value of n, we need to solve the inequality:
n^2 3n le 2015
This simplifies to:
4n^2 12n le 8060
Further simplification leads us to:
4n^2 12n 3^2 le 8060 9
which becomes:
2n 3^2 le 8069
After rearranging, we get:
-3-sqrt{8069} le 2n le -3sqrt{8069}
Dividing by 2:
dfrac{-3-sqrt{8069}}{2} le n le dfrac{-3sqrt{8069}}{2}
This results in:
-46.41 le n le 43.41
The largest integer value of n within this range is 43.
The total number of matchsticks used for n 43 is:
43^2 343 1978
Therefore, the number of matchsticks left is:
2015 - 1978 37
Conclusion
Using mathematical reasoning and careful calculation, we have determined that the maximum number of levels in the figure that can be built with 2015 matches is 43. This leaves us with 37 matches unused. Such puzzles not only challenge our problem-solving skills but also enhance our understanding of mathematical sequences and inequalities.