# Boxeslayers to the rescue

This is about layering boxes, not about slaying them. We have 1,830 2×5 boxes to stack safely as 10 alternating contiguous layer patterns of 183 boxes each. Layers have identical silhouettes that fit squarewise on a rectangular 2,000-square floor. The pair of layer patterns is to be solved along with the floor’s dimensions.

Background and guidelines

Once upon a time in packing to move we had a supply of identical boxes. One challenge was, in a land of earthquakes and elbows, to stack full boxes stably on the garage floor. As width and length of a box had some irrational ratio, we came up with an amusing pattern of alternating layers (that turns out to be suboptimal). This layer pair is exemplified with 2×3 boxes.

We defined safe stacking as similar to bricklaying, where no internal abutments align between layers.

Back to 2×5 boxes of the present puzzle, here are safe layer pairs of 4, 6, 7, 8, 9 and 10 boxes. Each pair is annotated with how efficiently it uses rectangular floor space. Bounty to anyone who improves on these efficiencies or finds a safe layer pair of 5 boxes.

As an example of what to avoid as unsafe, a tempting pair of 6-box layers has internal abutments that align between layers.

How can safely alternating contiguous 183-box layers fit onto a 2,000 square-unit rectangular area?

• Ugh... I have only 182 boxes per layer, though I managed that in a rectangular area of 1,980 square units. Mar 11, 2023 at 23:21
• You're well on the way, @Daniel Mathias! 183 boxes and 2,000 squares is where one (surprisingly in the long run) efficient pattern overtakes another. Guessing that your approach of comment could fit 183 boxes onto a 2,002-square floor. Feel free to show and tell in any case. Don't have to render the entire solution as long as a pattern is demonstrated.
– humn
Mar 12, 2023 at 0:20
• I believe all your proposed constructions are optimal, assuming that a layer has to have a single connected component. Mar 12, 2023 at 1:25
• The layer pattern I referred to is expandable with up to 96.97% efficiency. Another expandable pattern has up to 97.22% efficiency and allows for 184 boxes in an area of 2001 square units. Unfortunately, neither of these work with an odd number of boxes per layer. I will eventually share details, but for now here are 12- 14- and 16-box layers: image Mar 12, 2023 at 17:05
• @DanielMathias I get a slightly better solution ($11 \times 15$) for $n=16$. Mar 13, 2023 at 2:55

Here is one way to do it, showing the pattern along with its mirror image. The rectangular area is 83 x 24 units, or 1992 square units.

And the two combined, to show that this is a safe stacking pattern.

I had to be carefully inefficient, as the base pattern had 180 boxes in an area of 1909 square units, and I was able to layer 186 boxes within the 1992 square unit area before hitting the magic number of 183 boxes.

• F yeah! More beautiful than the intended solution.
– humn
Mar 14, 2023 at 10:19
• Gotta double check.@Daniel Mathias, which usually comes up positive. Hang in there and enjoy the well deserved +1s.
– humn
Mar 16, 2023 at 4:48
• Dear @Daniel Mathias , had to take leave and might again, still reeling from solutions. Stay tuned. (Same to Retudin.)
– humn
Mar 24, 2023 at 9:54

(This is not an answer to the 183 problem asked here)

High efficiency packing (arbitrary close to 100%) for far larger numbers of boxes

• I don't understand what this diagram is depicting. What does level 2 look like? Mar 17, 2023 at 13:02
• All horizontal are not shown (the inside white is filled, as well as the blue with orange layer horizontal blocks, and the orange with blue layer horizontal blocks ) , i.e. of the 20 rows, odd layers form a massive block of 5x10 in row 5..14 , while even layers have 4 10x2 massive subblocks in row 1..19 (in column 3..2, 15..24, 27..36 and 39..48) Mar 17, 2023 at 13:33
• Ah, like this? Mar 18, 2023 at 4:04
• Indeed, like that Mar 18, 2023 at 11:10
• Dear @Retudin , had to take leave and might again, still reeling from solutions. Stay tuned. (Same to Daniel Mathias.)
– humn
Mar 24, 2023 at 9:53