Investigate whether the ability to pack shapes into progressively smaller squares implies packability in a unit square | Step-by-Step Solution

TinyProf's Step-by-Step Solution 🎯

1. What We're Solving:

We have an infinite collection of shapes {r_m} that can be packed into squares of size (1 + 1/n) for any positive integer n. The question is: does this guarantee we can pack these same shapes into a unit square (size 1×1)?

This is a beautiful question about limits and compactness in geometric packing theory!

2. The Approach:

We need to think about what happens as n approaches infinity. As n gets larger, (1 + 1/n) gets closer and closer to 1. So we're essentially asking: "If we can pack shapes into squares that get arbitrarily close to size 1, can we pack them into a square of exactly size 1?"

This might seem obviously true, but in mathematics, limits can be tricky! Let's explore both possibilities systematically.

3. Step-by-Step Solution:

Step 1: Understanding the given condition

• For n = 1: We can pack into a 2×2 square
• For n = 2: We can pack into a 1.5×1.5 square
• For n = 10: We can pack into a 1.1×1.1 square
• For n = 100: We can pack into a 1.01×1.01 square
• And so on...

Step 2: Consider what "packing" means When we pack shapes, we need to fit them without overlap inside the boundary. The total area of shapes must not exceed the area of the container.

Step 3: Analyze the limit behavior The key insight is that we have infinitely many shapes. Even if each individual shape is tiny, their total area could still be significant.

Step 4: Construct a counterexample Let's think of a specific case:

• Suppose our shapes are squares with areas: 1/2, 1/4, 1/8, 1/16, ...
• Total area = 1/2 + 1/4 + 1/8 + ... = 1 (geometric series)

For a square of side (1 + 1/n):

• Available area = (1 + 1/n)² = 1 + 2/n + 1/n²
• This is always > 1, so packing is possible

But for the unit square:

• Available area = 1² = 1
• Required area = 1
• This is exactly at the boundary!

Step 5: The critical realization The issue isn't just area—it's also about the geometry of fitting shapes efficiently. Even if total area equals available area, perfect packing might be impossible due to boundary effects and shape constraints.

4. The Answer:

No, the ability to pack into squares of size (1 + 1/n) does NOT guarantee packability into a unit square.

The counterexample shows that while we might have exactly enough area, the geometric constraints of perfect packing can make it impossible to achieve 100% efficiency at the limit.

5. Memory Tip:

Think of it like this: "Almost fitting" doesn't guarantee "exactly fitting" when you have infinitely many pieces! The extra space 2/n + 1/n² might be crucial for maneuvering the shapes into position. 🧩

Great question! This problem beautifully illustrates why limits in geometry can be counterintuitive—what works for arbitrarily close approximations might fail at the exact limit!