Wind me a number snake!

Question

Given an input integer n, draw a number snake, that is, a grid measuring n x n consisting of the numbers 1 through n^2 that are wound around each other in the following fashion:

Input n = 3:

7 8 9
6 1 2
5 4 3

Input n = 4:

 7  8  9 10
 6  1  2 11
 5  4  3 12
16 15 14 13

Input n = 5:

21 22 23 24 25
20  7  8  9 10
19  6  1  2 11
18  5  4  3 12
17 16 15 14 13

(Inspired by this problem from Project Euler.)

This is code-golf, the shortest answer in bytes wins!

Answer 1

MATL, 3 bytes

1YL

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Explanation

Built-in... ¯\_(ツ)_/¯

Answer 2

C#, 203 202 196 193 178 bytes

n=>{var r=new int[n,n];for(int o=n-2+n%2>>1,i=r[o,o]=1,c=2,w=o,h=o,b=1-2*(i%2),j;n>i++;){r[h,w+=b]=c++;for(j=0;j<i-1;++j)r[h+=b,w]=c++;for(j=0;j<i-1;++j)r[h,w-=b]=c++;}return r;}

Saved a byte thanks to @StefanDelport.
Saved 22 bytes thanks to @FelipeNardiBatista.

This works by the following observation of how the squares are built up:

As you can see each bit is added on to the previous square. For even numbers we go right of where we were, down till were one lower than where the square was and then left to the end. Odd numbers are essentially the opposite, we go left one, up till were one above the current height and then right to the end.

Full/Formatted version:

using System;
using System.Linq;

class P
{
    static void Main()
    {
        Func<int, int[,]> f = n =>
        {
            var r = new int[n, n];
            for (int o = n - 2 + n % 2 >> 1, i = r[o, o] = 1, c = 2, w = o, h = o, b = 1 - 2 * (i % 2), j; n > i++;)
            {
                r[h, w += b] = c++;

                for (j = 0; j < i - 1; ++j)
                    r[h += b, w] = c++;

                for (j = 0; j < i - 1; ++j)
                    r[h, w -= b] = c++;
            }

            return r;
        };

        Console.WriteLine(String.Join("\n", f(3).ToJagged().Select(line => String.Join(" ", line.Select(l => (l + "").PadLeft(2))))) + "\n");
        Console.WriteLine(String.Join("\n", f(4).ToJagged().Select(line => String.Join(" ", line.Select(l => (l + "").PadLeft(2))))) + "\n");
        Console.WriteLine(String.Join("\n", f(5).ToJagged().Select(line => String.Join(" ", line.Select(l => (l + "").PadLeft(2))))) + "\n");

        Console.ReadLine();
    }
}

public static class ArrayExtensions
{
    public static T[][] ToJagged<T>(this T[,] value)
    {
        T[][] result = new T[value.GetLength(0)][];

        for (int i = 0; i < value.GetLength(0); ++i)
            result[i] = new T[value.GetLength(1)];

        for (int i = 0; i < value.GetLength(0); ++i)
            for (int j = 0; j < value.GetLength(1); ++j)
                result[i][j] = value[i, j];

        return result;
    }
}

Answer 3

Dyalog APL, 70 56 45 41 bytes

,⍨⍴∘(⍋+\)×⍨↑(⌈2÷⍨×⍨),(+⍨⍴1,⊢,¯1,-)(/⍨)2/⍳

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How?

(+⍨⍴1,⊢,¯1,-)(/⍨)2/⍳

calculates the differences between the indices; 1 and ¯1 for right and left, ¯⍵ and ⍵ for up and down.

1,⊢,¯1,- comes as 1 ⍵ ¯1 ¯⍵, +⍨⍴ stretches this array to the length of ⍵×2, so the final 2/⍳ can repeat each of them, with a repetition count increasing every second element:

      (1,⊢,¯1,-) 4
1 4 ¯1 ¯4
      (+⍨⍴1,⊢,¯1,-) 4
1 4 ¯1 ¯4 1 4 ¯1 ¯4
      (2/⍳) 4
1 1 2 2 3 3 4 4
      ((+⍨⍴1,⊢,¯1,-)(/⍨)2/⍳) 4
1 4 ¯1 ¯1 ¯4 ¯4 1 1 1 4 4 4 ¯1 ¯1 ¯1 ¯1 ¯4 ¯4 ¯4 ¯4

then,

(⌈2÷⍨×⍨),

prepends the top-left element of the spiral,

×⍨↑

limit the first ⍵² elements of this distances list,

+\

performs cumulative sum,

⍋

grades up the indices (⍵[i] = ⍵[⍵[i]]), to translate the original matrix with the indices of every element, and finally

,⍨⍴

shapes as a ⍵×⍵ matrix.

Answer 4

C, 321 307 295 284 283 282 bytes

Thanks to both @Zachary T and @Jonathan Frech for golfing a byte!

#define F for(b=a;b--;)l
i,j,k,a,b,m;**l;f(n){n*=n;l=calloc(a=m=3*n,4);F[b]=calloc(m,4);for(l[i=j=n][j]=a=k=1;n>k;++a){F[i][++j]=++k;F[++i][j]=++k;++a;F[i][--j]=++k;F[--i][j]=++k;}for(i=0;i<m;++i,k&&puts(""))for(j=k=0;j<m;)(a=l[i][j++])>0&&a<=n&&printf("%*d ",(int)log10(n)+1,k=a);}

Allocates a two-dimensional array of zeroes, then starts filling it from somewhere in the middle. Lastly the values that are larger than zero but smaller than or equal to the square of the input are printed.

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Formatted:

#define F for(b=a; b--;)l
i, j, k, a, b, m; **l;
f(n)
{
    n *= n;
    l = calloc(a=m=3*n, 4);

    F[b] = calloc(m, 4);

    for(l[i=j=n][j]=a=k=1; n>k; ++a)
    {
        F[i][++j] = ++k;
        F[++i][j] = ++k;
        ++a;

        F[i][--j] = ++k;
        F[--i][j] = ++k;
    }

    for(i=0; i<m; ++i, k&&puts(""))
        for(j=k=0; j<m;)
            (a=l[i][j++])>0 && a<=n && printf("%*d ", (int)log10(n)+1, k=a);
}

Answer 5

APL (Dyalog Classic), 32 29 bytes

1+×⍨-{⊖∘⌽⍣⍵⌽{⌽⍉,⌸⍵+≢⍵}⍣2⍣⍵⍪⍬}

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Uses ⎕io←1. Starts with a 0-by-1 matrix (⍪⍬). 2N times (⍣2⍣⍵) adds the height of the matrix (≢⍵) to each of its elements, puts 1 2...height on its right (,⌸), and rotates (⌽⍉). When that's finished, corrects the orientation of the result (⊖∘⌽⍣⍵⌽) and reverses the numbers by subtracting them from N²+1 (1+×⍨-).

Answer 6

PHP, 192 bytes

for($l=strlen($q=($a=$argn)**2)+$d=1,$x=$y=$a/2^$w=0;$i++<$q;${yx[$w%2]}+=$d&1?:-1,$i%$d?:$d+=$w++&1)$e[$x-!($a&1)][$y]=sprintf("%$l".d,$i);for(;$k<$a;print join($o)."\n")ksort($o=&$e[+$k++]);

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Same way build a string instead of an array

PHP, 217 bytes

for($l=strlen($q=($a=$argn)**2)+$d=1,$x=$y=($a/2^$w=0)-!($a&1),$s=str_pad(_,$q*$l);$i++<$q;${yx[$w%2]}+=$d&1?:-1,$i%$d?:$d+=$w++&1)$s=substr_replace($s,sprintf("%$l".d,$i),($x*$a+$y)*$l,$l);echo chunk_split($s,$a*$l);

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Answer 7

PHP, 185 176 174 bytes

for(;$n++<$argn**2;${xy[$m&1]}+=$m&2?-1:1,$k++<$p?:$p+=$m++%2+$k=0)$r[+$y][+$x]=$n;ksort($r);foreach($r as$o){ksort($o);foreach($o as$i)printf(" %".strlen($n).d,$i);echo"
";}

Run as pipe with -nR or test it online.

breakdown

for(;$n++<$argn**2;     # loop $n from 1 to N squared
    ${xy[$m&1]}+=$m&2?-1:1, # 2. move cursor
    $k++<$p?:               # 3. if $p+1 numbers have been printed in that direction:
        $p+=$m++%2+             # increment direction $m, every two directions increment $p
        $k=0                    # reset $k
)$r[+$y][+$x]=$n;           # 1. paint current number at current coordinates

ksort($r);              # sort grid by indexes
foreach($r as$o){       # and loop through it
    ksort($o);              # sort row by indexes
    foreach($o as$i)        # and loop through it
        printf(" %".strlen($n).d,$i);   # print formatted number
    echo"\n";               # print newline
}

Answer 8

Mathematica, 177 bytes

(n=#;i=j=Floor[(n+1)/2];c=1;d=0;v={{1,0},{0,-1},{-1,0},{0,1}};a=Table[j+n(i-1),{i,n},{j,n}];Do[Do[Do[a[[j,i]]=c++;{i,j}+=v[[d+1]], {k,l}];d=Mod[d+1,4],{p,0,1}],{l,n-1}];Grid@a)&

Answer 9

C++, 245 228 bytes

void f(){for(int i=0,j=-1,v,x,y,a,b;i<n;i++,j=-1,cout<<endl)while(++j<n){x=(a=n%2)?j:n-j-1;y=a?i:n-i-1;v=(b=y<n-x)?n-1-2*(x<y?x:y):2*(x>y?x:y)-n;v=v*v+(b?n-y-(y>x?x:y*2-x):y+1-n+(x>y?x:2*y-x));cout<<setw(log10(n*n)+1)<<v<<' ';}}

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The function calculates and prints the value of each number of the matrix depending on its x, y position by applying this logic:

Formatted version:

#include <iostream>
#include <iomanip>
#include <math.h>

using namespace std;

void f(int n)
{
    for (int i = 0; i < n; i++)
    {
        for (int j = 0; j < n; j++)
        {
            int value = 0;

            // Invert x and y when n is even
            int x = n % 2 == 0 ? n - j - 1 : j;
            int y = n % 2 == 0 ? n - i - 1 : i;
            if (y < (n - x))
            {
                // Left-top part of the matrix
                int padding = x < y ? x : y;
                value = n - 1 - padding * 2;
                value *= value;
                value += y >= x ? n - x - y : n + x - y - (y * 2);
            }
            else
            {
                // Right-bottom part of the matrix
                int padding = x > y ? n - x : n - y;
                value = n - padding * 2;
                value *= value;
                value += x > y ? y - padding + 1 : n + y - x - (padding * 2) + 1;
            }

            cout << setw(log10(n * n) + 1);
            cout << value << ' ';
        }

        cout << endl;
    }
}

int main()
{
    int n;
    while (cin >> n && n > 0)
    {
        f(n);
        cout << endl;
    }
}

Answer 10

Python 3, 249 247 bytes

I initialize an 2D array and find the starting point, which is the center for odd n or offset (-1,-1) for even n, then scale the fill/cursor pattern with the current 'ring' number. I feel like I'm missing a trick for interpreting the directions but I haven't come up with anything cheaper.

def f(n):
 M=[n*[0]for a in range(n)]
 x=y=n//2-1+n%2
 M[x][y]=i=s=1
 while 1:
  t=s*2
  for d in'R'+'D'*(t-1)+'L'*t+'U'*t+'R'*t:
   if i==n*n:print(*M,sep='\n');return
   v=[1,-1][d in'LU']
   if d in'UD':x+=v
   else:y+=v
   M[x][y]=i=i+1
  s+=1

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-2 thanks to Zachary T!

Answer 11

Wolfram Language (Mathematica), (...) 83 bytes

Byte measured in UTF8, \[LeftFloor] (⌊) and \[RightFloor] (⌋) cost 3 bytes each. Mathematica doesn't have any special byte character set.

Table[Max[4x^2-Max[x+y,3x-y],4y
y-{x+y,3y-x}]+1,{y,b+1-#,b=⌊#/2⌋},{x,b+1-#,b}]&

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---

Uses the closed form for each of the 4 cases, then takes the maximum carefully to get the desired result.

Returns a 2D array of integers. I'm not sure if this is allowed, and although it has been asked in the comments, the OP didn't reply.

Answer 12

Clojure, 206 bytes

(defmacro F[s]`(if(~'r(+ ~'p ~'v ~s))~'v ~s))
#(loop[i 1 p(*(quot(dec %)2)(inc %))v 1 r{}](if(<(* % %)i)(partition %(map r(range i)))(recur(inc i)(+ p v)({1(F %)-1(F(- %))%(F -1)(- %)(F 1)}v)(assoc r p i))))

I guess this is a decent start, builds the board in sequence to a hash-map and then partitions it into n x n lists. That defmacro ended up being quite long, but the code is still shorter with it than without. Is there a more succint syntax to describe it?

Bulk of bytes calculate the starting point, and build the look-up logic of the next velocity v. Perhaps a nested vec would be better, but then you've got two indexes and velocities to keep track of.

Answer 13

J, 41 bytes

(]|.@|:@[&0](|.@|:@,.*/@$+#\)@]^:[1:)2*<:

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Does the same thing as ngn’s APL submission but starts with a 1-by-1 matrix and repeats 2×N−2 times.

Answer 14

Python 165 (or 144)

from pylab import *
def S(n):
 a=r_[[[1]]];r=rot90;i=2
 while any(array(a.shape)<n):
  q=a.shape[0];a=vstack([range(i,i+q),r(a)]);i+=q
 if n%2==0:a=r(r(a))
 print(a)

This creates a numpy array, then rotates it and adds a side until the correct size is reached. The question didn't specify if the same start point needs to be used for even and odd numbers, if that's not the case the line if n%2==0:a=r(r(a)) can be removed, saving 21 bytes.

Answer 15

J, 41 bytes

,~$[:/:[:+/\_1|.1&,(]##@]$[,-@[)2}:@#1+i.

standard formatting

,~ $ [: /: [: +/\ _1 |. 1&, (] # #@] $ [ , -@[) 2 }:@# 1 + i.

This approach is based off At Play With J Volutes (Uriel's APL uses a similar technique).

It's unexpected and elegant enough to justify a 2nd J answer, I thought.

Essentially, we don't do anything procedural or even geometric. Instead, we arithmetically create a simple sequence that, when scan summed and graded up, gives the correct order of the spiraled number from left to right, top to bottom. We then shape that into a matrix and are done.

I'll add a more detailed explanation when time permits, but the linked article explains it in depth.

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Answer 16

K (ngn/k), 38 bytes

{(x,x)#<0,+\(,/-:\1,x)4!-1_&1+-1_&x#2}

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Answer 17

Python 3 (Stackless), 192 188 179 150 bytes

lambda n:[list(map(v,list(range(t-n,t)),[y]*n))for t in[1+n//2]for y in range(n-t,-t,-1)]
v=lambda x,y,r=0:y>=abs(x)and(3-2*r+4*y)*y+x+1or v(y,-x,r+1)

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The algorithm here is to form a phasor for each coordinate in the grid, and then rotate it 90-degrees clockwise until the phasor lies between the upper diagonals. A simple formula can be used to calculate the value based on the coordinates and the number of clockwise rotations: $$ (2y+1)^2-(y-x)-2yr $$

Saved 4-bytes since 90-degree phasor rotation is easily done without complex numbers

Answer 18

R, 183 bytes

x=scan()
b=t(d<-1)
while(2*x-1-d){m=max(b)
y=(m+1):(m+sum(1:dim(b)[2]|1))
z=d%%4
if(z==1)b=cbind(b,y)
if(z==2)b=rbind(b,rev(y))
if(z==3)b=cbind(rev(y),b)
if(z==0)b=rbind(y,b)
d=d+1}
b

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Output is a matrix snake (or snake matrix, whatever). It's probably not the most efficient method, and it could likely be golfed, but I thought it was worth showing. I'm rather proud of this actually!

The method builds the matrix from the inside out, always appending an additional number of integers equal to the number of columns in the matrix before appending. The pattern that follows is either binding by columns or by rows, while also reversing some values so they are appended in the correct order.

193 bytes

Exact same as above, but final b is

matrix(b,x)

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which gives slightly cleaner output, but I didn't see any special criteria for output, so the first answer should work if I'm not mistaken.