This is a mathematical problem involving the maximum number of pieces Lennie can produce from a rectangular pan of brownies with specific cutting rules.

Step-by-Step Analysis:

1. Cuts Parallel to the Sides:

   • Each cut must extend entirely across the pan, either horizontally or vertically.
   • This ensures that all resulting pieces are rectangular and uniform.

2. Equal Number of Interior and Perimeter Pieces:

   • The total number of brownies includes both the pieces along the perimeter and the interior ones.
   • To maximize the total number of brownies, Lennie must arrange the cuts such that the division is as uniform as possible.

3. Strategy:

   • Let’s assume Lennie makes $m$ horizontal cuts and $n$ vertical cuts. These cuts divide the rectangle into $(m+1) \times (n+1)$ pieces.

4. Constraints:

   • To balance the interior and perimeter pieces, Lennie needs to calculate how the pieces along the border relate to the total number of cuts.

Let me calculate this for a theoretical case, assuming no specific dimensions are provided for the pan. Define the function to calculate total pieces, perimeter, and interior def calculate_pieces(horizontal_cuts, vertical_cuts): # Total number of brownies (rectangular pieces) total_pieces = (horizontal_cuts + 1) * (vertical_cuts + 1)

# Perimeter brownies (on the border of the grid)
perimeter_pieces = (2 * (horizontal_cuts + 1)) + (2 * (vertical_cuts + 1)) - 4  # Corners are double-counted

# Interior brownies (non-border pieces)
interior_pieces = total_pieces - perimeter_pieces

return total_pieces, perimeter_pieces, interior_pieces

Test for various configurations of horizontal and vertical cuts

results = [] for h_cuts in range(1, 10): # Testing up to 10 horizontal cuts for v_cuts in range(1, 10): # Testing up to 10 vertical cuts total, perimeter, interior = calculate_pieces(h_cuts, v_cuts) if perimeter == interior: # Look for balanced perimeter and interior pieces results.append((h_cuts, v_cuts, total, perimeter, interior))

resultsFrom the calculations:

• If Lennie makes 5 horizontal cuts and 7 vertical cuts, or 7 horizontal cuts and 5 vertical cuts, she can divide the pan into 48 pieces, with exactly 24 perimeter pieces and 24 interior pieces.

This configuration ensures that the number of perimeter and interior pieces are equal, maximizing the total number of pieces under the given conditions.

Would you like details about the calculations or assumptions?

Related Questions:

1. How do you calculate the total number of pieces for any given number of cuts?
2. What happens if Lennie only wants a specific ratio between interior and perimeter pieces?
3. How does the shape of the pan affect the cutting strategy?
4. Can we generalize this method for non-rectangular pans?
5. What if Lennie introduces diagonal cuts—how would this impact the number of brownies?

Tip:

Always consider the total number of sections formed by the cuts to verify the balance of perimeter and interior pieces!