# Visualize Nicomachus's Theorem

Nichomachus's Theorem relates the square of a sum to the sum of cubes:

and has a beautiful geometric visualization:

## Challenge: Create the 2d part of this visualization in ascii.

You will need to ensure that all visual demarcations are upheld by your diagram. This is simplest to do with four "colors," though it's possible to achieve with only three (see last example below for how). With four colors, you use two to distinguish between regions within a "strip" (ie, the different parts that make up a single cube), and two to distinguish between adjacent strips. You may also use more than four colors if you like. If any of this is confusing, the example output below should clarify.

## Input / Output

Input is a single integer greater than 0. Output is an ascii grid similar to the examples below, corresponding to the flattened grid for that input number in the image above. Leading and trailing whitespace are ok.

This is code golf, with standard rules.

## Sample outputs

N = 1

#

N = 2

#oo
o@@
o@@

N = 3

#oo+++
o@@+++
o@@+++
+++###
+++###
+++###

N = 4

#oo+++oooo
o@@+++oooo
o@@+++@@@@
+++###@@@@
+++###@@@@
+++###@@@@
oo@@@@oooo
oo@@@@oooo
oo@@@@oooo
oo@@@@oooo

N = 5

#oo+++oooo+++++
o@@+++oooo+++++
o@@+++@@@@+++++
+++###@@@@+++++
+++###@@@@+++++
+++###@@@@#####
oo@@@@oooo#####
oo@@@@oooo#####
oo@@@@oooo#####
oo@@@@oooo#####
+++++#####+++++
+++++#####+++++
+++++#####+++++
+++++#####+++++
+++++#####+++++

Three color version for N = 4, thanks to @BruceForte:

#oo+++oooo
o##+++oooo
o##+++####
+++ooo####
+++ooo####
+++ooo####
oo####++++
oo####++++
oo####++++
oo####++++
• Four-colour theorem :D Sep 19 '17 at 18:16
• Can you add the output for N=5 please? Sep 19 '17 at 19:31
• @Uriel Done. See my edit. Sep 19 '17 at 19:35
• Thanks! Also, can I switch the @ and o s only in the outer strip in N=4? Or must the output be a strict substitution of these texts with another charset? Sep 19 '17 at 19:37
• @Uriel switching is fine. All that matters is that adjacent colors don't conflict, so that the pattern is visible. Sep 19 '17 at 19:37

# MATL, 3028 27 bytes

t:P"@:s:@/Xk&+@+8MPt&(]30+c

Try it online!

Bonus features:

• For 26 bytes, the following modified version produces graphical output:

t:P"@:s:@/Xk&+@+8MPt&(]1YG

Try it at MATL Online!

• The image is begging for some colour, and it only costs 7 bytes:

t:P"@:s:@/Xk&+@+8MPt&(]1YG59Y02ZG

Try it at MATL Online!

• Or use a longer version (37 bytes) to see how the character matrix is gradually built:

t:P"@:s:@/Xk&+@+8MPt&(t30+cD9&Xx]30+c

Try it at MATL Online!

## Example outputs

For input is 8, the following shows the basic version, graphical output, and colour graphical output.

## Explanation

### General procedure

A numeric matrix is built from outer to inner layers in N steps, where N is the input. Each step overwrittes an inner (upper-left) part of the previous matrix. At the end, the numbers in the obtained matrix are changed to characters.

### Example

For input 4 the first matrix is

10 10  9  9  9  9  8  8  8  8
10 10  9  9  9  9  8  8  8  8
9  9  8  8  8  8  7  7  7  7
9  9  8  8  8  8  7  7  7  7
9  9  8  8  8  8  7  7  7  7
9  9  8  8  8  8  7  7  7  7
8  8  7  7  7  7  6  6  6  6
8  8  7  7  7  7  6  6  6  6
8  8  7  7  7  7  6  6  6  6
8  8  7  7  7  7  6  6  6  6

As a second step, the matrix

7 7 7 6 6 6
7 7 7 6 6 6
7 7 7 6 6 6
6 6 6 5 5 5
6 6 6 5 5 5
6 6 6 5 5 5

is overwritten into the upper half of the latter. Then the same is done with

6 5 5
5 4 4
5 4 4

and finally with

3

The resulting matrix is

3 5 5 6 6 6 8 8 8 8
5 4 4 6 6 6 8 8 8 8
5 4 4 6 6 6 7 7 7 7
6 6 6 5 5 5 7 7 7 7
6 6 6 5 5 5 7 7 7 7
6 6 6 5 5 5 7 7 7 7
8 8 7 7 7 7 6 6 6 6
8 8 7 7 7 7 6 6 6 6
8 8 7 7 7 7 6 6 6 6
8 8 7 7 7 7 6 6 6 6

Lastly, 30 is added to each entry and the resulting numbers are interpreted as codepoints and converted to characters (thus starting at 33, corresponding to !).

### Construction of the intermediate matrices

For input N, consider decreasing values of k from N to 1. For each k, a vector of integers from 1 to k*(k+1) is generated, and then each entry is divided by k and rounded up. As an example, for k=4 this gives (all blocks have size k except the last):

1 1 1 1 2 2 2 2 3 3

whereas for k=3 the result would be (all blocks have size k):

1 1 1 2 2 2

This vector is added, element-wise with broadcast, to a transposed copy of itself; and then k is added to each entry. For k=4 this gives

6  6  6  6  7  7  7  7  8  8
6  6  6  6  7  7  7  7  8  8
6  6  6  6  7  7  7  7  8  8
6  6  6  6  7  7  7  7  8  8
7  7  7  7  8  8  8  8  9  9
7  7  7  7  8  8  8  8  9  9
7  7  7  7  8  8  8  8  9  9
7  7  7  7  8  8  8  8  9  9
8  8  8  8  9  9  9  9 10 10
8  8  8  8  9  9  9  9 10 10

This is one of the intermediate matrices shown above, except that it is flipped horizontally and vertically. So all that remains is to flip this matrix and write it into the upper-left corner of the "accumulated" matrix so far, initiallized to an empty matrix for the first (k=N) step.

### Code

t       % Implicitly input N. Duplicate. The first copy of N serves as the
% initial state of the "accumulated" matrix (size 1×1). This will be
% extended to size N*(N+1)/2 × N*(N+1)/2 in the first iteration
:P     % Range and flip: generates vector [N, N-1, ..., 1]
"       % For each k in that vector
@:    %   Push vector [1, 2, ..., k]
s     %   Sum of this vector. This gives 1+2+···+k = k*(k+1)/2
:     %   Range: gives vector [1, 2, ..., k*(k+1)/2]
@/    %   Divide each entry by k
Xk    %   Round up
%   a square matrix of size k*(k+1)/2 × k*(k+1)/2
@+    %   Add k to each entry of the this matrix. This is the flipped
%   intermediate matrix
8M    %   Push vector [1, 2, ..., k*(k+1)/2] again
Pt    %   Flip and duplicate. The two resulting, equal vectors are the row and
%   column indices where the generated matrix will be written. Note that
%   flipping the indices has the same effect as flipping the matrix
%   horizontally and vertically (but it's shorter)
&(    %   Write the (flipped) intermediate matrix into the upper-left
%   corner of the accumulated matrix, as given by the two (flipped)
%   index vectors
]       % End
30+     % Add 30 to each entry of the final accumulated matrix
c       % Convert to char. Implicitly display
• I don't know MATL at all, but could you save any bytes by taking mod10 rather than adding 30 and converting to character? Sep 20 '17 at 10:13
• Or even mod4... Sep 20 '17 at 10:19
• @user2390246 Do you mean to keep them as numbers with a single digit and avoid converting to char? That wouldn't work, because the numeric matrix would be printed with spaces between the numbers. But thanks for the idea anyway :-) Sep 20 '17 at 13:42
• Fair enough. What happens with n > 226? Won't that go outside the range of valid characters? (Unsurprisingly, times out on TIO, so I couldn't check) Sep 20 '17 at 16:03
• @user2390246 Yes, for high input numbers it goes outside. And if we consider ASCII chars the maximum code point is 127, so it goes outside even sooner. But as you noticed it runs out of memory before that happens (the resulting char matrix is too large). Anyway, working up to a given input size only due to memory or data type limitations is usually allowed Sep 20 '17 at 20:33

# Python 2, 187178164162 152 bytes

-8 bytes thanks to Mr.Xcoder
-1 byte thanks to Stephen
-10 bytes thanks to Jonathan Frech

g=lambda y:y>1and[l+y*f(y,i)for i,l in enumerate(g(y-1))]+y*[''.join(f(y,i)for i in range(y*-~y/2))]or['#']
f=lambda y,i:'0@+#'[(y*~-y/2%y+i)/y%2+y%2*2]

Try it online!

• For when you get home, 179 bytes. Sep 19 '17 at 20:01
• @Mr.Xcoder 178 bytes Sep 19 '17 at 21:09
• Is it allowed to not include your lambda function's name byte count when you are using it recursively, i.e. using its name in the rest of the code? Sep 20 '17 at 1:36
• sum(range(y))%y -> y*~-y/2%y Sep 20 '17 at 1:41
• @JonathanFrech yeah, when it's recursive it must be there.
– Rod
Sep 20 '17 at 3:11

# Charcoal, 50 46 bytes

Ｆ⮌…·¹Ｎ«≔⊘×ι⊕ιθＦ⊕⊘ι«Ｆ§#+@⁺ικ«ＵＯ⁻θ×ικθλＵＯθ⁻θ×ικλ

Try it online! Link is to verbose version of code. Previous 50-byte version with explanation: Try it online!

Ｆ⮌…·¹Ｎ«≔÷×ι⁺¹ι²θＦ⁺¹÷ι²«Ｆ§#+@⁺ικ«ＵＯ⁻θ×ικθλＵＯθ⁻θ×ικλ

Ｆ     «     Loop over
…·¹       Inclusive range from 1 to
Ｎ      Input as a number
⮌          Reversed

ι⁺¹        Add 1 to current index
×   ι       Multiply by current index
÷     ²      Divide by 2
≔       θ     Assign to q

Ｆ     «      Loop over
Implicit range from 0 to
÷ι²       Half the current index
⁺¹          Plus 1

Ｆ       «    Loop over
#+@        Literal string
§           Circularly indexed by
⁺ικ     Sum of outer and inner index

×ικ     Multiply outer and inner index
⁻θ        Subtract from q
ＵＯ     θλ   Draw an oblong (q-ik, q) using that character

ＵＯθ⁻θ×ικλ   Draw an oblong (q, q-ik) using that character

Note: I loop over the character rather than trying to assign the character directly to l because you can't directly assign the result of indexing a string to a variable as it's an ambiguous construct in Charcoal. Fortunately the byte count is the same.

• Technically you can, with an ASCII variable since its argument order is reversed (note that it needs an operator to access so it's still less golfy) Sep 20 '17 at 7:02

# C (gcc), 135128 120 bytes

f(n,m,i,x,y,k){for(m=n*-~n/2,i=m*m;i--;printf("\n%d"+!!(~i%m),(x/k+y/k+k)%3))for(x=i%m,y=i/m,k=n;x>=k&y>=k;x-=k--)y-=k;}

Try it online!

Uses only three colors.

Conceptually, works on a grid rotated by 180 degrees:

000111
000111
000111
111220
111220
111001

And computes colors according to the formula:

c(x,y,n) = c(x-n,y-n,n-1)                   if x >= n and y >= n
= (x div n + y div n + n) mod 3    otherwise
• 123 bytes. Sep 20 '17 at 1:30
• @JonathanFrech This isn't valid C and breaks with gcc -O2. Sep 20 '17 at 1:53
• Fair enough; is it possible that the second code only works for three colors because of the modulus three (g(i%m,i/m,n)%3)? Sep 20 '17 at 2:11
• Suggest x/k&&y/k instead of x>=k&y>=k Jun 19 '18 at 1:39

# R, 131126 123 bytes

3 bytes saved thanks to @Giuseppe

function(n){l=w=sum(1:n)
m=matrix(,l,l)
for(i in n:1){m[l:1,l:1]=outer(x<-(1:l-1)%/%i,x,+)+i
l=l-i}
write(m%%4,"",w,,"")}

Try it online!

This uses the same algorithm as @LuisMendo 's MATL answer. The only difference is that rather than converting to characters, the matrix is output with all of the values mod4 to ensure that each element is a single ascii character.

• 123 bytes! I brought the for loop back for -1 byte :) Sep 20 '17 at 13:51

# Python 2, 176 175 bytes

n=input()
R,J=range,''.join;r=[]
for i in R(n+1):
S=sum(R(i));c='AxBo'[i%2::2]
for j in R(S):r[~j]+=c[j/i%2]*i
r+=[J(c[-j/i%2]for j in R(S+i,0,-1))]*i
for l in r:print J(l)

Try it online!

• If you define J="".join; (+10 bytes) and replace both "".joins (-2*7=-14 bytes) with J (+2 bytes), you can save a byte (as there has to be an additional space after the print; +1 byte). Sep 20 '17 at 13:32