Worked examples

Orientation and tiny gaps can change a regular grid.

All measurements below are in millimetres. Counts describe the best among the tested uniform single-orientation grids only. They do not account for mixed layouts, physical tolerances or handling constraints.

1. An orientation worth comparing

Use internal container dimensions 600 × 400 × 300 mm, item L/W/H dimensions 220 × 180 × 120 mm, and gap 0. With item dimensions 220 × 180 × 120 along X/Y/Z, the axis counts are 2 × 2 × 2 = 8.

Rotating the global assignment to HWL puts 120 × 180 × 220 mm along the axes. The counts become 5 × 2 × 1 = 10 items, the best tested grid. The used spans are 600 × 360 × 220 mm and remaining axis spans are 0 × 40 × 80 mm. Those remaining spans are not a guarantee of another usable packing region.

2. A gap changes the best assignment

Keep the same container and item, but enter 10 mm between neighboring items. Assignment WHL puts 180 × 120 × 220 mm along X/Y/Z and gives 3 × 3 × 1 = 9 items. Used X span is 3 × 180 + 2 × 10 = 560 mm. Used Y span is 3 × 120 + 2 × 10 = 380 mm. One item along Z uses 220 mm with no internal gap.

The remaining spans are 40 × 20 × 80 mm. The previously best HWL assignment now gives 4 × 2 × 1 = 8, so checking the permitted orientations matters even when all dimensions stay the same.

3. An exact fit with no gap

A 100 × 100 × 100 mm container with 50 × 50 × 50 mm items and gap 0 has 2 items along every axis: 2 × 2 × 2 = 8. The used spans equal the container dimensions. The item is a cube, so all six symbolic permutations represent one distinct ordered dimension assignment.

4. A tiny positive gap at the boundary

For those same cubes, change the gap to 0.000001 mm. Two items now require 50 + 0.000001 + 50 = 100.000001 mm along an axis, which is above 100. The tested grid therefore becomes 1 × 1 × 1 = 1 item.

This is a deliberate numerical boundary example. It demonstrates exact input arithmetic; it does not imply that a real container can be measured or controlled to that tiny clearance.

5. Zero is a valid no-fit result

Use a 100 mm cube container and an item with edges 110, 20 and 20 mm. In every allowed global orientation, the 110 mm edge exceeds one 100 mm container edge. Each tested grid has 0 complete items. Counts on the other axes may be positive, but they do not create complete cuboids. Used spans and per-axis remainders are not applicable for those rows.

6. Large counts are kept as integers

As a software-boundary check, a container with all edges 1000000000 mm and an item with all edges 0.000001 mm at gap 0 gives 10¹⁵ per axis and 10⁴⁵ items. The calculator retains the entire integer and does not try to draw that many objects. This is not a practical crate size or a measurement-accuracy claim.