# Heavy Box Stacking

You have a bunch of heavy boxes and you want to stack them in the fewest number of stacks possible. The issue is that you can't stack more boxes on a box than it can support, so heavier boxes must go on the bottom of a stack.

## The Challenge

Input: A list of weights of boxes, in whole kg.

Output: A list of lists describing the stacks of boxes. This must use the fewest number of stacks possible for the input. To be a valid stack, the weight of each box in the stack must be greater than or equal to the sum of the weight of all boxes above it.

### Examples of Valid stacks

(In bottom to top order)

• [3]
• [1, 1]
• [3, 2, 1]
• [4, 2, 1, 1]
• [27, 17, 6, 3, 1]
• [33, 32, 1]
• [999, 888, 99, 11, 1]

### Examples of Invalid stacks

(In order from bottom to top)

• [1, 2]
• [3, 3, 3]
• [5, 5, 1]
• [999, 888, 777]
• [4, 3, 2]
• [4321, 3000, 1234, 321]

## Example Test Cases

### 1

IN: [1, 2, 3, 4, 5, 6, 9, 12]
OUT: [[12, 6, 3, 2, 1], [9, 5, 4]]


### 2

IN: [87, 432, 9999, 1234, 3030]
OUT: [[9999, 3030, 1234, 432, 87]]


### 3

IN: [1, 5, 3, 1, 4, 2, 1, 6, 1, 7, 2, 3]
OUT: [[6, 3, 2, 1], [7, 4, 2, 1], [5, 3, 1, 1]]


### 4

IN: [8, 5, 8, 8, 1, 2]
OUT: [[8, 8], [8, 5, 2, 1]]


## Rules and Assumptions

• Standard I/O rules and banned loopholes apply
• Use any convenient format for I/O
• Stacks may be described top to bottom or bottom to top, as long as you are consistent.
• The order of stacks (rather than boxes within those stacks) does not matter.
• You may also take input boxes as a presorted list. Order is not particularly important for the input, so long as the general problem isn't being solved by the sorting itself.
• If there is more than one optimal configuration of stacks, you may output any one of them
• You may assume that there is at least one box and that all boxes weigh at least 1 kg
• You must support weights up to 9,999 kg, at minimum.
• You must support up to 9,999 total boxes, at minimum.
• Boxes with the same weight are indistinguishable, so there is no need to annotate which box was used where.

Happy golfing! Good luck!

• May we take the weights in sorted order? (either ascending or descending) – Arnauld Aug 30 at 21:55
• "You must support up to 9,999 total boxes, at minimum." How is "support" interpreted here? Does it merely mean the program should be able to take such size of input, or does it mean that the program should actually provide the answer in a reasonable amount of time? If it is the latter, there should be much larger test cases provided. – Joel Aug 30 at 23:49
• Suggested test case: [8, 8, 8, 5, 1] -> [[8, 8], [8, 5, 1]] – Hiatsu Aug 31 at 1:05
• Or even better: [8, 5, 8, 8, 1, 2] -> [[8, 8], [8, 5, 2, 1]] – Hiatsu Aug 31 at 1:16
• @Arnauld, since otherwise that would add uninteresting sorting code to an answer, I'm going to say yes, you can take in the inputs in sorted order. – Beefster Sep 2 at 17:13

# Jelly, 19 bytes

Œ!ŒṖ€ẎṖÄ>ḊƲ€¬ȦƊƇLÞḢ


Try it online!

Obvious -3 thanks to Nick Kennedy...

Top to bottom.

Explanation:

Œ!ŒṖ€ẎṖÄ>ḊƲ€¬ȦƊƇLÞḢ  Arguments: S (e.g. [1, 2, 3, 4, 5])
Œ!                   Permutations (e.g. [..., [4, 1, 5, 2, 3], ...])
€                Map link over left argument (e.g. [..., [..., [[4, 1], [5], [2, 3]], ...], ...])
ŒṖ                  Partitions (e.g. [..., [[4, 1], [5], [2, 3]], ...])
Ẏ               Concatenate elements (e.g. [..., ..., [[4, 1], [5], [2, 3]], ..., ...])
Ƈ     Filter by link (e.g. [..., [[1, 3], [2], [4], [5]], ...])
€           Map link over left argument (e.g. [[1], [], [0]])
Ṗ                  Remove last element (e.g. [4])
Ä                 Cumulative sum (e.g. [4])
Ḋ               [Get original argument] Remove first element (e.g. [1])
>                Greater than (vectorizes) (e.g. [1])
¬          Logical NOT (vectorizes) (e.g. [[0], [], [1]])
Ȧ         Check if non-empty and not containing zeroes after flattening (e.g. 0)
Þ   Sort by link (e.g. [[[1, 2, 3], [4, 5]], ..., [[5], [4], [3], [2], [1]]])
L     Length (e.g. 4)
Ḣ  Pop first element (e.g. [[1, 2, 3], [4, 5]])

• Any chance at a less compact version with explanation? I learn a ton from those. – John Keates Sep 2 at 8:00
• @JohnKeates Added one. – Erik the Outgolfer Sep 2 at 13:03

# JavaScript (Node.js),  139 122  116 bytes

Expects the input sorted in ascending order.

f=(A,s=[],[n,...a]=A,r)=>n?s.some((b,i,[...c])=>n<eval(b.join+)?0:f(A,c,a,c[i]=[n,...b]))?S:r?0:f(A,[...s,[]]):S=s


Try it online!

### Commented

f = (                        // f is a recursive function taking:
A,                         //   A[] = input array
s = [],                    //   s[] = list of stacks, initially empty
[n,                        //   n   = next weight to process
...a] = A,             //   a[] = array of remaining weights
r                          //   r   = recursion flag
) =>                         //
n ?                        // if n is defined:
s.some((b, i,            //   for each stack b[] at position i in s[],
[...c]) => //   using c[] as a copy of s[]:
n < eval(b.join+) ?  //     if n is not heavy enough to support all values in b[]:
0                    //       abort
:                      //     else:
f(                   //       do a recursive call:
A, c, a,           //         using A[], c[] and a[]
c[i] = [n, ...b]   //         with n prepended to c[i]
)                    //       end of recursive call
) ?                      //   end of some(); if successful:
S                      //     return S[]
:                        //   else:
r ?                    //     if this is a recursive call:
0                    //       do nothing
:                      //     else:
f(A, [...s, []])     //       try again with an additional stack
:                          // else:
S = s                    //   success: save the solution in S[]


# Python 3.8 (pre-release), 178 bytes

f=lambda b,s=[[]]:(a for i in range(len(s))if b[0]>=sum(s[i])for a in f(b[1:],s[:i]+[[b[0]]+s[i]]+s[i+1:]+[[]]))if b else[s]
g=lambda a:(c:=sorted(f(a),key=len)[0])[:c.index([])]


Try it online!

Now works on all possible inputs. (It times out on TIO with more than ten or so boxes, but it calculates a correct answer)

• list(reversed(sorted(a))) can be written as sorted(a)[::-1] for golfing purpose. – Joel Aug 30 at 23:51
• You would think I would know that by now, especially since I do so much other indexing. Thanks. – Hiatsu Aug 30 at 23:53
• Just as a side remark, if not for golfing it would be better write sorted(a, reverse=True) instead. – Joel Aug 30 at 23:58