CGAC2022 Day 1: Let's build a chocolate pyramid!

Question

Following last year's event, we're doing Code Golf Advent Calendar 2022!

On each day from today (Dec 1) until Christmas (Dec 25), a Christmas-themed challenge will be posted, just like an Advent calendar. It is a free-for-all and just-have-fun-by-participation event, no leaderboards and no prizes for solving them fast or solving them in the shortest code. More details can be found in the link above.

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I've got an infinite supply of two kinds of weirdly shaped chocolate:

• White chocolate, a square pyramid of side lengths 1
• Dark chocolate, a regular tetrahedon of side lengths 1

To celebrate the upcoming Christmas, I want to assemble them into a giant chocolate pyramid. When the base of the pyramid is a rectangle of size \$R \times C\$, the process to build such a pyramid is as follows:

1. Fill the floor with \$RC\$ copies of White chocolate.
2. Fill the gaps between White chocolate with Dark chocolate.
3. Fill the holes between Dark chocolate with White chocolate. Now the top face is a rectangle of size \$(R-1) \times (C-1)\$.
4. Repeat 1-3 until the top face has the area of 0.

The diagram below shows the process for \$2 \times 3\$. It takes 8 White and 7 Dark chocolate to complete the first floor, and 10 White and 8 Dark for the entire pyramid.

Given the width and height of the base rectangle, how many White and Dark chocolate do I need to form the chocolate pyramid?

You may assume the width and height are positive integers. You may output two numbers in any order.

Standard code-golf rules apply. The shortest code in bytes wins.

Test cases

(width, height) -> (white, dark)
(2, 3) -> (10, 8)
(10, 10) -> (670, 660)
(10, 1) -> (10, 9)

Answer 1

Vyxal, 14 bytes

gʁ$vεƛΠε∑;:›"Ṡ

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The main observation here is that the amount of white chocolate \$w\$ and dark chocolate \$d\$ in a layer of size (a, b) is: $$ w = ab +(a-1)(b-1) = 2ab - a - b + 1\\ d = (a-1)b+(b-1)a = 2ab - a - b $$

In other words, the difference between white chocolate and dark chocolate for a layer is 1

 ʁ             # Range from 0 to...
g              # Minimum dimension of input
  $vε          # Subtract each from the input, creating a list of pairs of bases
     ƛ   ;     # Over each base pair [a, b]
      Π        # Product ab
       ε       # Take the absolute difference - [ab - a, ab - b]
        ∑      # Sum
          :›"  # Create an incremented pair for white
             Ṡ # Sum each into the final amounts

Answer 2

Desmos, 47 46 37 bytes

-1 byte thanks to emanresu A

-9 bytes (!!!) thanks to alephalpha

N=min(h,w)
f(w,h)=Nhw-N(NN+2)/3+[N,0]

\$f(w,h)\$ takes in the width \$w\$ and the height \$h\$ and returns a two-element list, with the number of white chocolates in the first element and the number of dark ones in the second element.

Try It On Desmos!

Made my own closed form formula because I figured it would be fun :)

For those who are really curious, here’s some of the scratch work for how I came up with this formula (warning: it’s completely unreadable :P): Scratch work

Answer 3

PARI/GP, 41 bytes

f(r,c)=if(r*c,[t=r*c+r--*c--,t-1]+f(r,c))

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PARI/GP, 41 bytes

f(r,c)=m=min(r,c);[t=1+3*r*c-m^2,t-3]*m/3

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f(r,c)=m=min(r,c);[t=r*c*m+m/3-m^3/3,t-m]

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Using the closed form formula.

Answer 4

R, 42 39 bytes

Edit: -3 bytes thanks to @Giuseppe.

\(r,c)-c(0,m<-min(r,c))+r*c*m+m/3-m^3/3

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Port of alephalpha's answer.

Answer 5

Ruby, 41 40 bytes

-1 byte thanks to Sisyphus

Based on alephalpha’s PARI/GP answer.

->a{r,c=a.sort
[t=r/3+c*r*r-r**3/3,t-r]}

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Answer 6

C (GCC), 76 72 70 bytes

• -4 bytes thanks to Conor O'Brien
• -2 bytes thanks to Keven Cruijssen

#define C(n,c)n(a,b){a=a*b?c*--b+n(a,b):0;}
C(f,a*b+--a)C(g,~-a*b+a--)

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f(a,b) returns the number of black chocolates while g(a,b) the number of white chocolates.

Answer 7

Charcoal, 20 19 bytes

Ｉ⁺ΣＥ⌊θΣ⁻Π⁻θι⁻θι⟦⌊θ⁰

Try it online! Link is to verbose version of code. Takes input as a tuple or list of (rows, columns). Explanation:

     θ              Input tuple
    ⌊               Minimum
   Ｅ                Map over implicit range
          θ         Input tuple
         ⁻          Vectorised minus
           ι        Current value
        Π           Take the product
       ⁻            Minus
             θ      Input tuple
            ⁻       Vectorised minus
              ι     Current value
      Σ             Take the sum
  Σ                 Take the sum
 ⁺                  Vectorised add
               ⟦    List of
                 θ  Input tuple
                ⌊   Minimum
                  ⁰ Literal integer `0`
Ｉ                   Cast to string
                    Implicitly print

Edit: Saved 1 byte by porting @Ausername's formula. The best closed form formula I could find was 22 bytes, but adapting one of @alephalpha's versions saves 1 byte:

Ｉ⁻⁻×Πθ⌊θ÷⁻Ｘ⌊θ³⌊θ³⟦⁰⌊θ

Try it online! Link is to verbose version of code. Takes input as a tuple or list of (rows, columns). Explanation:

     θ                  Input tuple
    Π                   Product
   ×                    Multiplied by
       θ                Input tuple
      ⌊                 Minimum
  ⁻                     Subtract
            θ           Input tuple
           ⌊            Minimum
          Ｘ             Raised to power
             ³          Literal integer `3`
         ⁻              Subtract
               θ        Input tuple
              ⌊         Minimum
        ÷               (Integer) Divide by
                 ³      Literal integer `3`
 ⁻                      Vectorised subtract
                 ⟦      List of
                  ⁰     Literal integer `0`
                    θ   Input tuple
                   ⌊    Minimum
Ｉ                       Cast to string
                        Implicitly print

Answer 8

Jelly, 14 bytes

_ⱮṂ‘PḤ_SƲ€S;ṂÄ

A monadic Link that accepts a list of two integers, the base dimensions, and yields a pair of integers, [dark, white].

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16 bytes

 ^{Probably  beatable with a constructive method, but I like it.}

Ṃð‘c3×4+²×IAʋ;⁸Ä

A monadic Link that accepts a list of two integers, the base dimensions, and yields a pair of integers, [dark, white].

Try it online!

How?

Ṃð‘c3×4+²×IAʋ;⁸Ä - Link: Dimensions
Ṃ                - minimum (Dimensions)
 ð               - start a new dyadic chain - f(M=that, Dimensions)
  ‘              - increment (M)
   c3            - choose three -> (M-1)th triangular pyramidal number
     ×4          - times four (call this Q)
            ʋ    - last four links as a dyad - f(M, Dimensions):
        ²        -   square (M)
          I      -   forward differences (Dimensions) -> [Height-Width]
         ×       -   multiply -> [M*M*(Height-Width)]
           A     -   absolute value -> [M*M*|Height-Width|]
       +         - (Q) add (that) -> [dark piece count]
              ⁸  - chain's left argument -> M
             ;   - (dark piece count) concatenate (M) -> [dark piece count, M]
               Ä - cumulative sums -> [dark piece count, white piece count]

Answer 9

05AB1E, 13 bytes

Wݨ€-DPαOD>‚O

Port of @emanresuA's Vyxal answer, so make sure to upvote him/her as well!

Input as a pair \$[width,height]\$; output as a pair \$[dark,white]\$.

Try it online or verify all test cases.

Explanation:

W             # Get the minimum of the (implicit) input-pair
 Ý            # Pop and push a list in the range [0,min]
  ¨           # Remove the last item to make the range [0,min)
   €-         # Subtract each from the (implicit) input-pair
     D        # Duplicate this list of pairs
      P       # Get the product of each inner pairs
       α      # Get the absolute difference with the values in the pairs
        O     # Sum each inner pair
         D    # Duplicate this list
          >   # Increase each inner value by 1
           ‚  # Pair the two lists together
            O # Sum each inner list
              # (after which this pair is output implicitly as result [dark,white])

Answer 10

Pyth, 20 bytes

_-h=SQ
sm+*FQ*F=tMQh

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Takes (width, height), outputs (white, dark)

Explanation

   =SQ            sort the input, set to Q
_-h               output dark - min(Q)

s                 sum of
 m          hQ    run min(Q) times:
  +                add
   *FQ             product of elements of Q
        =tMQ       decrement each element of Q
      *F           product of elements of Q

Answer 11

Python 3, 83 bytes:

f=lambda r,c,w=0,d=0:f(r-1,c-1,w+r*c+(r-1)*(c-1),d+r*(c-1)+c*(r-1))if r*c else(w,d)

Try it online!

Answer 12

JavaScript (ES7), 40 bytes

Uses the closed form formula from alephalpha's answer.

Expects (w)(h).

w=>h=>[n=w*h*(w<h?w:w=h)+(w-w**3)/3,n-w]

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Answer 13

Python, 50 bytes

lambda p,q:(y:=(x:=min(p,q))*(3*p*q+1-x*x)//3,y-x)

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Same closed formula (@alephalpha's) everybody is using.

Answer 14

Lil, 59 bytes

on f x do
r:min x
t:(r/3)+(r*r*max x)-(r^3)/3
t,t-r
end

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Same as alephalpha.

Answer 15

APL(Dyalog Unicode), 15 bytes ^SBCS

⌊××+3÷⍨1 ¯2-⌊×⌊

Try it on APLgolf!

Closed form formula from alephalpha's PARI/GP answer.

Answer 16

Perl 5, 68 + 3 (-pla option) = 71 bytes

($i,$j)=@F;{$d+=$k=$i*$j*2-$i---$j--;$w+=$k+1;$i*$j&&redo}$_="$w $d"

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Answer 17

Pip -p, 26 bytes

Y Ngy*a*b-y*(y*y+2)/3+[y0]

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Port of my Desmos answer.

Answer 18

Raku, 49 bytes

{my (\r,\c)=.sort;my \t=-r**3/3+r/3+c*r*r;t,-r+t}

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Answer 19

x86-64 machine code, 26 bytes

92 39 F0 73 01 96 6A 03 59 F7 E1 29 F0 F7 E6 F7 E6 01 F0 F7 F1 AB 29 F0 AB C3

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Following the standard calling convention for Unix-like systems (from the System V AMD64 ABI), this takes in RDI an address at which two 32-bit integers will be consecutively placed, for the white number and the dark number, in that order; and takes the width and height in ESI and EDX.

In assembly:

f:  xchg edx, eax    # Switch one argument into EAX.
    cmp eax, esi     # Compare the two arguments.
    jae s            # Jump if EAX holds the higher value.
    xchg esi, eax    # Otherwise, exchange the values.
s:                   # (Now, EAX holds the higher value (which we will call m)
                     #   and ESI holds the lower value (which we will call n).)
    push 3; pop rcx  # Set ECX to 3.
    mul ecx          # Multiply EAX by ECX (3), producing 3m.
                     # (Also, the high half of the product goes in EDX.)
    sub eax, esi     # Subtract ESI (n) from EAX, producing -n+3m.
    mul esi          # Multiply EAX by ESI (n), producing -n²+3mn.
    mul esi          # Multiply EAX by ESI (n), producing -n³+3mn².
    add eax, esi     # Add ESI (n) to EAX, producing -n³+3mn²+n.
    div ecx          # Divide EDX:EAX by 3, producing (-n³+3mn²+n)/3.
    stosd            # Write that value to the address EDI, advancing the pointer.
    sub eax, esi     # Subtract ESI (n) from EAX, producing (-n³+3mn²-2n)/3.
    stosd            # Write that value to the address EDI, advancing the pointer.
    ret              # Return.