This challenge is a riff on Dion's challenge "Is this a rectangle?". The goal of this challenge is to write a program to decide whether or not some collection of tuples of integers represents a hypercube of some dimension.


A hypercube is a generalization of a square.

  • A \$0\$-cube is a single point.
  • A \$1\$-cube is a line segment.
  • A \$2\$-cube is a square.
  • A \$3\$-cube is an ordinary cube.
  • An \$n\$-cube is a connected geometric object consisting of pairs of parallel line segments, perpendicular to each other and of the same length.


For example, if you are given the input \$\{(0, 4, 0, 9), (2, 2, -4, 9), (-2, 0, -6, 9), (-4, 2, -2, 9)\}\$, then you should return a truthy value because these four points define a \$2\$-cube (a square).

You are allowed to input the data in any reasonable format—but the computation needs to work regardless of the input order of the points.

An \$n\$ cube has \$2^n\$ vertices, so if the list of numbers does not contain \$2^n\$ numbers, you must return a falsey value.


This is a challenge, so shortest code wins.

Test data





If you'd like more test data, or if you'd like to suggest more test data, let me know.

  • \$\begingroup\$ In the first paragraph one can read "to decide whether or some" \$\endgroup\$
    – RGS
    May 12, 2020 at 6:32
  • \$\begingroup\$ Are you sure [(0,4,0),(0,1,1),(1,0,1),(1,1,1)] is a cube? \$\endgroup\$
    – Adám
    May 12, 2020 at 7:24
  • 1
    \$\begingroup\$ May we assume that all points have the same number of coordinates? \$\endgroup\$
    – Adám
    May 12, 2020 at 7:33
  • 1
    \$\begingroup\$ @Adám, yes, all points have the same number of coordinates. \$\endgroup\$ May 12, 2020 at 7:59
  • 1
    \$\begingroup\$ Will the input points be distinct? \$\endgroup\$
    – xnor
    May 12, 2020 at 8:35

6 Answers 6


APL (Dyalog Unicode), 44 bytes


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the argument is a vector of coordinate vectors

,⍳2⍴⍨⌊2⍟≢⍵ build a hypercube as the cartesian product \$\{0,1\}^{\left\lfloor \log_2\left|\omega\right|\right\rfloor}\$

≡/(f)¨⍵(..) evaluate f for and the 01 hypercube, and test if they match

∘.(+.×⍨-)⍨ matrix of pairwise distances

(÷∘⊃⍨1↓⍋⌷¨⊂)⍤1 sort each row and divide by its second element


Python, 262 \$\cdots\$ 305 303 bytes

Saved a whopping 19 bytes thanks to dingledooper!!!

Added 118 bytes to fix a bug kindly pointed out by xnor, Peter Kagey and l4m2.

lambda l,R=range,L=len:(n:=L(l))<2or(d:=L(bin(n))-3)and(p:=sorted([sum((x-y)**2for x,y in zip(i,j))for i in l for j in l]))==[i*p[n]for i in R(d+2)for _ in R(2**d*math.comb(d,i))]and(K:=R(L(l[0])))and L({sum(([sum(l[i][j]for i in R(n))for j in K][j]-n*l[i][j])**2for j in K)for i in R(n)})<2
import math

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Inputs a list of points and returns True/False.


Calculates the square of the distances between all possible pairs of points (including self-pairs and both \$(p_i,p_j)\$ and \$(p_j,p_i)\$ for all points \$p_j\$ and \$p_i\$ where \$i\neq j\$) and normalises them by the smallest non-zero square distance. For an \$n\$-cube we should then see a pattern of integers \$i = 0,1,\dots, n\$ each occurring \$2^{n}{n\choose i}\$ times. This corresponds with the \$0\$s for all the self-pairs, and the square of the lengths of all the sides being \$a^2\$, and the square of the lengths of all the diagonals being \$2a^2, 3a^2,\dots, na^2\$.


Also checks that the given vertices are all equidistant from the centre of mass.

  • \$\begingroup\$ I feel like this should work and not give false positives, but I don't see how to prove it. The multiset of distances doesn't always determine the set of points. In one dimension, I found that [0, 1, 3, 3, 7, 8] and [0, 1, 1, 4, 6, 8] have the same multiset of pairwise distances even though they are not translations or reflections of each other. But it feels like the distances of vertices in a cube constraints it in a more structured way. \$\endgroup\$
    – xnor
    May 12, 2020 at 20:57
  • 2
    \$\begingroup\$ I think the points \$\{(0,0,0), (1,0,0), (\frac 12, \frac{\sqrt{3}}{2}, 0), (\frac 12, \frac{\sqrt{3}}{2}, 1)\}\$ meet this criteria but don't form a square. I'll see if I can rotate and scale this to make it fit on the integer lattice. \$\endgroup\$ May 13, 2020 at 2:06
  • 3
    \$\begingroup\$ @PeterKagey So after rotation it's [(1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0),(0,0,1,1,1)] \$\endgroup\$
    – l4m2
    May 13, 2020 at 3:42
  • \$\begingroup\$ @l4m2, thanks! I added a new test case to reflect this. \$\endgroup\$ May 13, 2020 at 5:04
  • \$\begingroup\$ @PeterKagey Thanks - have added a correction. \$\endgroup\$
    – Noodle9
    May 13, 2020 at 8:40

Python 3, 339 338

lambda P:1==L(P)or P in map(g,permutations(P))
from itertools import*
D=lambda a,b:sum(x*y for x,y in Z(a,b))
def g(Q):B=[[x-y for x,y in Z(p,Q[0])]for p in Q[3-L(bin(L(Q))):]];return any(D(a,b)or D(a,a)-D(b,b)for a,b in combinations(B,2))or{tuple(x+sum(y)for x,y in Z(Q[0],Z(*C)))for C in product(*[(p,(0,)*L(p))for p in B])}

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Takes a set of points as input.

Pseudocode explanation:

def f(points):
    let n = log_2(|points|)
    for each permutation Q of the points:
        let q be the first point in Q
        let B be the following n points, with q subtracted from each
        if all pairs of points in B are orthogonal and have equal magnitude:
            let S be the set of points which can be obtained by summing q and any subset of B
            if S == points: return True
    return False

Can definitely be golfed further but it's bedtime.


JavaScript (Node.js), 258 bytes


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Similar to Noodle9's answer, but generate another square to compare rather than use formula and add midpoint like normal ones

  • \$\begingroup\$ Error like Noodle9's answer \$\endgroup\$
    – l4m2
    May 13, 2020 at 3:43
  • \$\begingroup\$ I added a new test case to reflect this. \$\endgroup\$ May 13, 2020 at 5:04

JavaScript (Node.js), 182 bytes


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Check sums of length squares of A-B-C, where A,B,C can be same


Haskell, 139 bytes

import Data.List
import Data.List.Ordered
c p=has[(2^n,n*2^n)|n<-[0..]](length p,length$group(sort[sum$(^2)<$>zipWith(-)x y|x<-p,y<-p])!!1)

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