Packing Intuition Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard

Oranges of radius

4

cm are packed in a square grid on a flat tray that is

40\,\text{cm}\times 40\,\text{cm}

. A second layer is placed on top in hexagonal arrangement offset by

4

cm in each direction. How many oranges are in the first layer? How many extra fit in the second layer?

Solution

1
Step 1: Each orange diameter $= 8$ cm. First layer (square grid): $40/8 = 5$ per row and column. Total $= 25$ oranges.
2
Step 2: Second layer centres sit over the gaps between first-layer oranges. In the offset grid, centres are shifted by $4$ cm in $x$ and $4$ cm in $y$ . Effective grid: $4$ per row and $4$ per column $= 16$ oranges (they sit in the $4\times4$ interior gaps).
3
Step 3: Total oranges $= 25 + 16 = 41$ .

Answer

First layer:

25

oranges; second layer:

16

oranges; total:

41

Staggered (offset) layers exploit the gaps in the layer below, fitting more objects in three dimensions. This is why cannonballs and oranges are stacked in offset pyramidal arrangements in practice.

About Packing Intuition

Arranging objects of given shapes to fit as many as possible into a bounded region without any overlapping.

Learn more about Packing Intuition →

More Packing Intuition Examples

Example 1 easy

Circular coins of radius [formula] cm are packed into a [formula] square tray in a square grid arran

Example 2 medium

Compare the packing efficiency of square packing ([formula]) vs hexagonal close packing ([formula])

Example 3 easy

A box is [formula] cm long, [formula] cm wide, [formula] cm tall. How many [formula] cubes can fit i

← All Packing Intuition Examples Packing Intuition Concept

Related Concepts

Area Volume Tessellation