Maximizing Figure Construction with Limited Matchsticks: A Mathematical Challenge

Imagine Tom,armed with 2,015 matchsticks, is eager to build as complex and aesthetically pleasing a figure as possible. But how many matchsticks will he have left over after reaching that peak of creativity? This interesting problem not only challenges our problem-solving skills but also offers valuable insights into mathematical optimization. Let's dive into the details.

Understanding the Problem

The figure in question is constructed in a specific pattern. Each level ( k ) of the figure requires the following components:

At the bottom level ( k n ), the figure requires ( n ) horizontal matches. Summing the matches at all levels from 1 to ( n ) gives us the total number of matches required.

Deriving the Total Matches Formula

Mathematically, the total number of matches needed is:

Step 1: Total matches ( n cdot sum_{k1}^{n} (2k - 1) )

Step 2: Expanding the summation, we get:

Total matches ( n cdot (2 cdot sum_{k1}^{n} k - sum_{k1}^{n} 1) )

Step 3: Using the known formulas for summation:

Total matches ( n cdot (2 cdot frac{n(n 1)}{2} - n) )

Step 4: Simplifying:

Total matches ( n^2 - n )

However, in the problem, there's an additional 2 added to the summation:

Total matches ( n^2 - 3n )

Optimization: Finding the Maximum Integer Value of n

Tom has 2,015 matchsticks. We need to find the largest integer value of ( n ) such that:

Step 1: ( n^2 - 3n leq 2015 )

Step 2: Rearranging, we get:

4n^2 - 12n leq 8060

Step 3: Completing the square:

4n^2 - 12n - 3^2 leq 8060 - 9

Step 4: Simplifying further:

2n - 3 leq sqrt{8069}

Step 5: Solving for ( n ):

-3 - sqrt{8069} leq 2n leq -3 sqrt{8069}

Step 6: Dividing by 2:

frac{-3 - sqrt{8069}}{2} leq n leq frac{-3 sqrt{8069}}{2}

Final value: ( -46.41 leq n leq 43.41 )

Hence, the largest integer value of ( n ) is 43.

Calculating the Matches Used and Left Over

Using ( n 43 ):

Total matches used ( 43^2 - 3 cdot 43 1978 )

Matches left over ( 2015 - 1978 37 )

Conclusion

This mathematical problem showcases the power of algebraic optimization and problem decomposition. While the solution might seem complex at first, breaking it down into manageable steps and understanding the underlying principles can make it much more approachable.

For more problem-solving tips, feel free to reach out and discuss more mathematical challenges. Mathematics is a subject that thrives on practice and reflection.