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Abelian sandpile model/Identity

From Rosetta Code
Task
Abelian sandpile model/Identity
You are encouraged to solve this task according to the task description, using any language you may know.

Our sandpiles are based on a 3 by 3 rectangular grid giving nine areas that contain a number from 0 to 3 inclusive. (The numbers are said to represent grains of sand in each area of the sandpile).

E.g. s1 =

    1 2 0
    2 1 1
    0 1 3

Or s2 =

    2 1 3
    1 0 1
    0 1 0

Addition on sandpiles is done by adding numbers in corresponding grid areas, so for the above:

              1 2 0     2 1 3     3 3 3
    s1 + s2 = 2 1 1  +  1 0 1  =  3 1 2
              0 1 3     0 1 0     0 2 3

If the addition would result in more than 3 "grains of sand" in any area then those areas cause the whole sandpile to become "unstable" and the sandpile areas are "toppled" in an "avalanche" until the "stable" result is obtained.

Any unstable area (with a number >= 4), is "toppled" by loosing one grain of sand to each of its four horizontal or vertical neighbours. Grains are lost at the edge of the grid, but otherwise increase the number in neighbouring cells by one, whilst decreasing the count in the toppled cell by four in each toppling.

A toppling may give an adjacent area more than four grains of sand leading to a chain of topplings called an "avalanche". E.g.

    4 3 3     0 4 3     1 0 4     1 1 0     2 1 0
    3 1 2 ==> 4 1 2 ==> 4 2 2 ==> 4 2 3 ==> 0 3 3
    0 2 3     0 2 3     0 2 3     0 2 3     1 2 3

The final result is the stable sandpile on the right.

Note: The order in which cells are toppled does not affect the final result.

Task
  • Create a class or datastructure and functions to represent and operate on sandpiles.
  • Confirm the result of the avalanche of topplings shown above
  • Confirm that s1 + s2 == s2 + s1 # Show the stable results
  • If s3 is the sandpile with number 3 in every grid area, and s3_id is the following sandpile:
    2 1 2  
    1 0 1  
    2 1 2
* Show that s3 + s3_id == s3
* Show that s3_id + s3_id == s3_id

Show confirming output here, with your example.

References


C++[edit]

#include <algorithm>
#include <array>
#include <cassert>
#include <initializer_list>
#include <iostream>
 
constexpr size_t sp_rows = 3;
constexpr size_t sp_columns = 3;
constexpr size_t sp_cells = sp_rows * sp_columns;
constexpr size_t sp_limit = 4;
 
class abelian_sandpile {
public:
abelian_sandpile();
explicit abelian_sandpile(std::initializer_list<int> init);
void stabilize();
bool is_stable() const;
void topple();
abelian_sandpile& operator+=(const abelian_sandpile& other);
void print(std::ostream&) const;
private:
int& cell_value(size_t row, size_t column) {
return cells_[cell_index(row, column)];
}
static size_t cell_index(size_t row, size_t column) {
return row * sp_columns + column;
}
static size_t row_index(size_t cell_index) {
return cell_index/sp_columns;
}
static size_t column_index(size_t cell_index) {
return cell_index % sp_columns;
}
friend bool operator==(const abelian_sandpile&, const abelian_sandpile&);
std::array<int, sp_cells> cells_;
};
 
abelian_sandpile::abelian_sandpile() {
cells_.fill(0);
}
 
abelian_sandpile::abelian_sandpile(std::initializer_list<int> init) {
assert(init.size() == sp_cells);
std::copy(init.begin(), init.end(), cells_.begin());
}
 
abelian_sandpile& abelian_sandpile::operator+=(const abelian_sandpile& other) {
for (size_t i = 0; i < sp_cells; ++i)
cells_[i] += other.cells_[i];
stabilize();
return *this;
}
 
bool abelian_sandpile::is_stable() const {
return std::none_of(cells_.begin(), cells_.end(),
[](int a) { return a >= sp_limit; });
}
 
void abelian_sandpile::topple() {
for (size_t i = 0; i < sp_cells; ++i) {
if (cells_[i] >= sp_limit) {
cells_[i] -= sp_limit;
size_t row = row_index(i);
size_t column = column_index(i);
if (row > 0)
++cell_value(row - 1, column);
if (row + 1 < sp_rows)
++cell_value(row + 1, column);
if (column > 0)
++cell_value(row, column - 1);
if (column + 1 < sp_columns)
++cell_value(row, column + 1);
break;
}
}
}
 
void abelian_sandpile::stabilize() {
while (!is_stable())
topple();
}
 
bool operator==(const abelian_sandpile& a, const abelian_sandpile& b) {
return a.cells_ == b.cells_;
}
 
abelian_sandpile operator+(const abelian_sandpile& a, const abelian_sandpile& b) {
abelian_sandpile c(a);
c += b;
return c;
}
 
void abelian_sandpile::print(std::ostream& out) const {
for (size_t i = 0; i < sp_cells; ++i) {
if (i > 0)
out << (column_index(i) == 0 ? '\n' : ' ');
out << cells_[i];
}
out << '\n';
}
 
int main() {
std::cout << std::boolalpha;
 
std::cout << "Avalanche:\n";
abelian_sandpile sp{4,3,3, 3,1,2, 0,2,3};
while (!sp.is_stable()) {
sp.print(std::cout);
std::cout << "stable? " << sp.is_stable() << "\n\n";
sp.topple();
}
sp.print(std::cout);
std::cout << "stable? " << sp.is_stable() << "\n\n";
 
std::cout << "Commutativity:\n";
abelian_sandpile s1{1,2,0, 2,1,1, 0,1,3};
abelian_sandpile s2{2,1,3, 1,0,1, 0,1,0};
abelian_sandpile sum1(s1 + s2);
abelian_sandpile sum2(s2 + s1);
std::cout << "s1 + s2 equals s2 + s1? " << (sum1 == sum2) << "\n\n";
std::cout << "s1 + s2 = \n";
sum1.print(std::cout);
std::cout << "\ns2 + s1 = \n";
sum2.print(std::cout);
std::cout << '\n';
 
std::cout << "Identity:\n";
abelian_sandpile s3{3,3,3, 3,3,3, 3,3,3};
abelian_sandpile s3_id{2,1,2, 1,0,1, 2,1,2};
abelian_sandpile sum3(s3 + s3_id);
abelian_sandpile sum4(s3_id + s3_id);
std::cout << "s3 + s3_id equals s3? " << (sum3 == s3) << '\n';
std::cout << "s3_id + s3_id equals s3_id? " << (sum4 == s3_id) << "\n\n";
std::cout << "s3 + s3_id = \n";
sum3.print(std::cout);
std::cout << "\ns3_id + s3_id = \n";
sum4.print(std::cout);
 
return 0;
}
Output:
Avalanche:
4 3 3
3 1 2
0 2 3
stable? false

0 4 3
4 1 2
0 2 3
stable? false

1 0 4
4 2 2
0 2 3
stable? false

1 1 0
4 2 3
0 2 3
stable? false

2 1 0
0 3 3
1 2 3
stable? true

Commutativity:
s1 + s2 equals s2 + s1? true

s1 + s2 = 
3 3 3
3 1 2
0 2 3

s2 + s1 = 
3 3 3
3 1 2
0 2 3

Identity:
s3 + s3_id equals s3? true
s3_id + s3_id equals s3_id? true

s3 + s3_id = 
3 3 3
3 3 3
3 3 3

s3_id + s3_id = 
2 1 2
1 0 1
2 1 2

F#[edit]

This task uses Abelian Sandpile Model (F#)

 
let s1=Sandpile(3,3,[|1;2;0;2;1;1;0;1;3|])
let s2=Sandpile(3,3,[|2;1;3;1;0;1;0;1;0|])
printfn "%s\n" ((s1+s2).toS)
printfn "%s\n" ((s2+s1).toS);;
printfn "%s\n" ((s1+s1).toS)
printfn "%s\n" ((s2+s2).toS);;
printfn "%s\n" (Sandpile(3,3,[|4;3;3;3;1;2;0;2;3|])).toS;;
let s3=Sandpile(3,3,(Array.create 9 3))
let s3_id=Sandpile(3,3,[|2;1;2;1;0;1;2;1;2|])
printfn "%s\n" (s3+s3_id).toS
printfn "%s\n" (s3_id+s3_id).toS
//Add together 2 5x5 Sandpiles
let e1=Array.zeroCreate<int> 25 in e1.[12]<-6
let e2=Array.zeroCreate<int> 25 in e2.[12]<-16
printfn "%s\n" ((Sandpile(5,5,e1)+Sandpile(5,5,e2)).toS)
 
Output:
[[3; 3; 3]
 [3; 1; 2]
 [0; 2; 3]]

[[3; 3; 3]
 [3; 1; 2]
 [0; 2; 3]]

[[0; 2; 2]
 [2; 2; 1]
 [2; 1; 0]]

[[1; 0; 3]
 [3; 1; 3]
 [0; 2; 0]]

[[2; 1; 0]
 [0; 3; 3]
 [1; 2; 3]]

[[3; 3; 3]
 [3; 3; 3]
 [3; 3; 3]]

[[2; 1; 2]
 [1; 0; 1]
 [2; 1; 2]]

[[0; 0; 1; 0; 0]
 [0; 2; 2; 2; 0]
 [1; 2; 2; 2; 1]
 [0; 2; 2; 2; 0]
 [0; 0; 1; 0; 0]]

Factor[edit]

I wouldn't call it a translation, but the idea of storing sandpiles as flat arrays came from the Wren entry.

Works with: Factor version 0.99 2020-07-03
USING: arrays grouping io kernel math math.vectors prettyprint
qw sequences ;
 
CONSTANT: neighbors {
{ 1 3 } { 0 2 4 } { 1 5 } { 0 4 6 } { 1 3 5 7 }
{ 2 4 8 } { 3 7 } { 4 6 8 } { 5 7 }
}
 
! Sandpile words
: find-tall ( seq -- n ) [ 3 > ] find drop ;
: tall? ( seq -- ? ) find-tall >boolean ;
: distribute ( ind seq -- ) [ [ 1 + ] change-nth ] curry each ;
: adjacent ( n seq -- ) [ neighbors nth ] dip distribute ;
: shrink ( n seq -- ) [ 4 - ] change-nth ;
: (topple) ( n seq -- ) [ shrink ] [ adjacent ] 2bi ;
: topple ( seq -- seq' ) [ find-tall ] [ (topple) ] [ ] tri ;
: avalanche ( seq -- ) [ dup tall? ] [ topple ] while drop ;
: s+ ( seq1 seq2 -- seq3 ) v+ dup avalanche ;
 
! Output words
: mappend ( seq1 seq2 -- seq3 ) [ flip ] [email protected] append flip ;
: sym ( seq str -- seq ) 1array " " 1array tuck 3array mappend ;
: arrow ( seq -- new-seq ) ">" sym ;
: plus ( seq -- new-seq ) "+" sym ;
: eq ( seq -- new-seq ) "=" sym ;
: topple> ( seq seq -- seq seq ) arrow over topple 3 group mappend ;
: (.s+) ( seq seq seq -- seq ) [ plus ] [ mappend eq ] [ mappend ] tri* ;
: .s+ ( seq1 seq2 -- ) 2dup s+ [ 3 group ] [email protected] (.s+) simple-table. ;
 
! Task
CONSTANT: s1 { 1 2 0 2 1 1 0 1 3 }
CONSTANT: s2 { 2 1 3 1 0 1 0 1 0 }
CONSTANT: s3 { 3 3 3 3 3 3 3 3 3 }
CONSTANT: id { 2 1 2 1 0 1 2 1 2 }
 
"Avalanche:" print nl
{ 4 3 3 3 1 2 0 2 3 }
dup 3 group topple> topple> topple> topple> nip simple-table. nl
 
"s1 + s2 = s2 + s1" print nl
s1 s2 .s+ nl s2 s1 .s+ nl
 
"s3 + s3_id = s3" print nl
s3 id .s+ nl
 
"s3_id + s3_id = s3_id" print nl
id id .s+
Output:
Avalanche:

4 3 3   0 4 3   1 0 4   1 1 0   2 1 0
3 1 2 > 4 1 2 > 4 2 2 > 4 2 3 > 0 3 3
0 2 3   0 2 3   0 2 3   0 2 3   1 2 3

s1 + s2 = s2 + s1

1 2 0   2 1 3   3 3 3
2 1 1 + 1 0 1 = 3 1 2
0 1 3   0 1 0   0 2 3

2 1 3   1 2 0   3 3 3
1 0 1 + 2 1 1 = 3 1 2
0 1 0   0 1 3   0 2 3

s3 + s3_id = s3

3 3 3   2 1 2   3 3 3
3 3 3 + 1 0 1 = 3 3 3
3 3 3   2 1 2   3 3 3

s3_id + s3_id = s3_id

2 1 2   2 1 2   2 1 2
1 0 1 + 1 0 1 = 1 0 1
2 1 2   2 1 2   2 1 2

Go[edit]

Translation of: Wren
package main
 
import (
"fmt"
"strconv"
"strings"
)
 
type sandpile struct{ a [9]int }
 
var neighbors = [][]int{
{1, 3}, {0, 2, 4}, {1, 5}, {0, 4, 6}, {1, 3, 5, 7}, {2, 4, 8}, {3, 7}, {4, 6, 8}, {5, 7},
}
 
// 'a' is in row order
func newSandpile(a [9]int) *sandpile { return &sandpile{a} }
 
func (s *sandpile) plus(other *sandpile) *sandpile {
b := [9]int{}
for i := 0; i < 9; i++ {
b[i] = s.a[i] + other.a[i]
}
return &sandpile{b}
}
 
func (s *sandpile) isStable() bool {
for _, e := range s.a {
if e > 3 {
return false
}
}
return true
}
 
// just topples once so we can observe intermediate results
func (s *sandpile) topple() {
for i := 0; i < 9; i++ {
if s.a[i] > 3 {
s.a[i] -= 4
for _, j := range neighbors[i] {
s.a[j]++
}
return
}
}
}
 
func (s *sandpile) String() string {
var sb strings.Builder
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
sb.WriteString(strconv.Itoa(s.a[3*i+j]) + " ")
}
sb.WriteString("\n")
}
return sb.String()
}
 
func main() {
fmt.Println("Avalanche of topplings:\n")
s4 := newSandpile([9]int{4, 3, 3, 3, 1, 2, 0, 2, 3})
fmt.Println(s4)
for !s4.isStable() {
s4.topple()
fmt.Println(s4)
}
 
fmt.Println("Commutative additions:\n")
s1 := newSandpile([9]int{1, 2, 0, 2, 1, 1, 0, 1, 3})
s2 := newSandpile([9]int{2, 1, 3, 1, 0, 1, 0, 1, 0})
s3_a := s1.plus(s2)
for !s3_a.isStable() {
s3_a.topple()
}
s3_b := s2.plus(s1)
for !s3_b.isStable() {
s3_b.topple()
}
fmt.Printf("%s\nplus\n\n%s\nequals\n\n%s\n", s1, s2, s3_a)
fmt.Printf("and\n\n%s\nplus\n\n%s\nalso equals\n\n%s\n", s2, s1, s3_b)
 
fmt.Println("Addition of identity sandpile:\n")
s3 := newSandpile([9]int{3, 3, 3, 3, 3, 3, 3, 3, 3})
s3_id := newSandpile([9]int{2, 1, 2, 1, 0, 1, 2, 1, 2})
s4 = s3.plus(s3_id)
for !s4.isStable() {
s4.topple()
}
fmt.Printf("%s\nplus\n\n%s\nequals\n\n%s\n", s3, s3_id, s4)
 
fmt.Println("Addition of identities:\n")
s5 := s3_id.plus(s3_id)
for !s5.isStable() {
s5.topple()
}
fmt.Printf("%s\nplus\n\n%s\nequals\n\n%s", s3_id, s3_id, s5)
}
Output:
Avalanche of topplings:

4 3 3 
3 1 2 
0 2 3 

0 4 3 
4 1 2 
0 2 3 

1 0 4 
4 2 2 
0 2 3 

1 1 0 
4 2 3 
0 2 3 

2 1 0 
0 3 3 
1 2 3 

Commutative additions:

1 2 0 
2 1 1 
0 1 3 

plus

2 1 3 
1 0 1 
0 1 0 

equals

3 3 3 
3 1 2 
0 2 3 

and

2 1 3 
1 0 1 
0 1 0 

plus

1 2 0 
2 1 1 
0 1 3 

also equals

3 3 3 
3 1 2 
0 2 3 

Addition of identity sandpile:

3 3 3 
3 3 3 
3 3 3 

plus

2 1 2 
1 0 1 
2 1 2 

equals

3 3 3 
3 3 3 
3 3 3 

Addition of identities:

2 1 2 
1 0 1 
2 1 2 

plus

2 1 2 
1 0 1 
2 1 2 

equals

2 1 2 
1 0 1 
2 1 2 

Haskell[edit]

{-# LANGUAGE TupleSections #-}
 
import Data.List (findIndex, transpose)
import Data.List.Split (chunksOf)
 
--------------------------- TEST ---------------------------
main :: IO ()
main = do
let s0 = [[4, 3, 3], [3, 1, 2], [0, 2, 3]]
s1 = [[1, 2, 0], [2, 1, 1], [0, 1, 3]]
s2 = [[2, 1, 3], [1, 0, 1], [0, 1, 0]]
s3_id = [[2, 1, 2], [1, 0, 1], [2, 1, 2]]
s3 = replicate 3 (replicate 3 3)
x:xs = reverse $ cascade s0
mapM_
putStrLn
[ "Cascade:"
, showCascade $ ([], x) : fmap ("->", ) xs
 
, "s1 + s2 == s2 + s1 -> " <> show (addSand s1 s2 == addSand s2 s1)
, showCascade [([], s1), (" +", s2), (" =", addSand s1 s2)]
, showCascade [([], s2), (" +", s1), (" =", addSand s2 s1)]
 
, "s3 + s3_id == s3 -> " <> show (addSand s3 s3_id == s3)
, showCascade [([], s3), (" +", s3_id), (" =", addSand s3 s3_id)]
 
, "s3_id + s3_id == s3_id -> " <> show (addSand s3_id s3_id == s3_id)
, showCascade [([], s3_id), (" +", s3_id), (" =", addSand s3_id s3_id)]
]
 
------------------------ SAND PILES ------------------------
addSand :: [[Int]] -> [[Int]] -> [[Int]]
addSand xs ys =
(head . cascade . chunksOf (length xs)) $ zipWith (+) (concat xs) (concat ys)
 
cascade :: [[Int]] -> [[[Int]]]
cascade xs = chunksOf w <$> convergence (==) (iterate (tumble w) (concat xs))
where
w = length xs
 
convergence :: (a -> a -> Bool) -> [a] -> [a]
convergence p = go
where
go (x:ys@(y:_))
| p x y = [x]
| otherwise = go ys <> [x]
 
tumble :: Int -> [Int] -> [Int]
tumble w xs = maybe xs go $ findIndex (w <) xs
where
go i = zipWith f [0 ..] xs
where
neighbours = indexNeighbours w i
f j x
| j `elem` neighbours = succ x
| i == j = x - succ w
| otherwise = x
 
indexNeighbours :: Int -> Int -> [Int]
indexNeighbours w = go
where
go i =
concat
[ [ j
| j <- [i - w, i + w]
, -1 < j
, wSqr > j ]
, [ pred i
| 0 /= col ]
, [ succ i
| pred w /= col ]
]
where
wSqr = w * w
col = rem i w
 
------------------------- DISPLAY --------------------------
showCascade :: [(String, [[Int]])] -> String
showCascade pairs =
unlines $
fmap unwords $
transpose $
fmap
(\(pfx, xs) ->
unwords <$> transpose (centered pfx : transpose (fmap (fmap show) xs)))
pairs
 
centered :: String -> [String]
centered s = [pad, s, pad <> replicate r ' ']
where
lng = length s
pad = replicate lng ' '
(q, r) = quotRem (2 + lng) 2
Cascade:
 4 3 3    0 4 3    1 0 4    1 1 0    2 1 0
 3 1 2 -> 4 1 2 -> 4 2 2 -> 4 2 3 -> 0 3 3
 0 2 3    0 2 3    0 2 3    0 2 3    1 2 3

s1 + s2 == s2 + s1 -> True
 1 2 0    2 1 3    3 3 3
 2 1 1  + 1 0 1  = 3 1 2
 0 1 3    0 1 0    0 2 3

 2 1 3    1 2 0    3 3 3
 1 0 1  + 2 1 1  = 3 1 2
 0 1 0    0 1 3    0 2 3

s3 + s3_id == s3 -> True
 3 3 3    2 1 2    3 3 3
 3 3 3  + 1 0 1  = 3 3 3
 3 3 3    2 1 2    3 3 3

s3_id + s3_id == s3_id -> True
 2 1 2    2 1 2    2 1 2
 1 0 1  + 1 0 1  = 1 0 1
 2 1 2    2 1 2    2 1 2

J[edit]

 
While=:2 :'u^:([email protected]:-:v)^:_'
index_of_maximum=: $ #: (i. >./)@:,
 
frame=: ({.~ [email protected]:>:@:$)@:({.~ >:@:$) :. ([;.0~ (1,:_2+$))
NEIGHBORS=: _2]\_1 0 0 _1 0 0 0 1 1 0
AVALANCHE =: 1 1 _4 1 1
 
avalanche=: (AVALANCHE + {)`[`]}~ ([: <"1 NEIGHBORS +"1 index_of_maximum)
erode=: avalanche&.:frame While(3 < [: >./ ,)
 
   NB. common ways to construct a matrix in j from directly entered vectors
   s3_id=: >2 1 2;1 0 1;2 1 2 NB. 3 3$2 1 2 1 0 1 2 1 2 NB. _3]\2 1 2 1 0 1 2 1 2 NB. 2 1 2,1 0 1,:2 1 2
   s3=: 3 3 $ 3 NB. ($~,~)3 NB. 3"0 i.3 3

   matches =: -:
   Commutes=: adverb def '(u matches u~)~'

   NB. demonstrate Commutes adbverb
   4 - Commutes 3
0
   4 + Commutes 3
1

   NB. confirmation
   <"2 A , ] avalanche&.:[email protected]:([ 3 :'A=:A,y') While(3 < [: >./ ,) 10#.inv 433 312 023 [ A=:0 3 3$0
┌─────┬─────┬─────┬─────┬─────┐
│4 3 3│0 4 3│1 0 4│1 1 0│2 1 0│
│3 1 2│4 1 2│4 2 2│4 2 3│0 3 3│
│0 2 3│0 2 3│0 2 3│0 2 3│1 2 3│
└─────┴─────┴─────┴─────┴─────┘

   NB. matrix addition commutes
   's1 s2'=: 120 211 013 ;&:(10&#.inv) 213 101 010
   s1 + Commutes s2
1
   erode s1 + s2
3 3 3
3 1 2
0 2 3

   NB. use: IDENTITY verify_identity MATRIX
   verify_identity=: ([email protected]:+ matches ]) erode

   raku_id verify_identity raku
1

   (; erode) raku
┌─────────┬─────────┐
│4 1 0 5 1│1 3 2 1 0│
│9 3 6 1 0│2 2 3 3 1│
│8 1 2 5 3│1 1 2 0 3│
│3 0 1 7 5│2 0 3 2 0│
│4 2 2 4 0│3 2 3 2 1│
└─────────┴─────────┘

Julia[edit]

import Base.+, Base.print
 
struct Sandpile
pile::Matrix{UInt8}
end
 
function Sandpile(s::String)
arr = [parse(UInt8, x.match) for x in eachmatch(r"\d+", s)]
siz = isqrt(length(arr))
return Sandpile(reshape(UInt8.(arr), siz, siz)')
end
 
const HMAX = 3
 
function avalanche!(s::Sandpile, lim=HMAX)
nrows, ncols = size(s.pile)
while any(x -> x > lim, s.pile)
for j in 1:ncols, i in 1:nrows
if s.pile[i, j] > lim
i > 1 && (s.pile[i - 1, j] += 1)
i < nrows && (s.pile[i + 1, j] += 1)
j > 1 && (s.pile[i, j - 1] += 1)
j < ncols && (s.pile[i, j + 1] += 1)
s.pile[i, j] -= 4
end
end
end
s
end
 
+(s1::Sandpile, s2::Sandpile) = avalanche!(Sandpile((s1.pile + s2.pile)))
 
function print(io::IO, s::Sandpile)
for row in 1:size(s.pile)[1]
for col in 1:size(s.pile)[2]
print(io, lpad(s.pile[row, col], 4))
end
println()
end
end
 
const s1 = Sandpile("""
1 2 0
2 1 1
0 1 3""")
 
const s2 = Sandpile("""
2 1 3
1 0 1
0 1 0""")
 
const s3 = Sandpile("""
3 3 3
3 3 3
3 3 3""")
 
const s3_id = Sandpile("""
2 1 2
1 0 1
2 1 2""")
 
const s3a = Sandpile("""
4 3 3
3 1 2
0 2 3""")
 
println("Avalanche reduction to group:\n", s3a, " =>")
println(avalanche!(s3a), "\n")
 
println("Commutative Property:\ns1 + s2 =\n", s1 + s2, "\ns2 + s1 =\n", s2 + s1, "\n")
 
println("Addition:\n", s3, " +\n", s3_id, " =\n", s3 + s3_id, "\n")
println(s3_id, " +\n", s3_id, " =\n", s3_id + s3_id, "\n")
 
 
Output:
Avalanche reduction to group:
   4   3   3
   3   1   2
   0   2   3
   =>
   2   1   0
   0   3   3
   1   2   3


Commutative Property:
s1 + s2 =
   3   3   3
   3   1   2
   0   2   3

s2 + s1 =
   3   3   3
   3   1   2
   0   2   3


Addition:
   3   3   3
   3   3   3
   3   3   3
   +
   2   1   2
   1   0   1
   2   1   2
   =
   3   3   3
   3   3   3
   3   3   3


   2   1   2
   1   0   1
   2   1   2
   +
   2   1   2
   1   0   1
   2   1   2
   =
   2   1   2
   1   0   1
   2   1   2

Phix[edit]

constant s1 = {"1 2 0",
"2 1 1",
"0 1 3"},
 
s2 = {"2 1 3",
"1 0 1",
"0 1 0"},
 
s3 = {"3 3 3",
"3 3 3",
"3 3 3"},
 
s3_id = {"2 1 2",
"1 0 1",
"2 1 2"},
 
s4 = {"4 3 3",
"3 1 2",
"0 2 3"}
 
function add(sequence s, t)
for i=1 to 3 do
for j=1 to 5 by 2 do
s[i][j] += t[i][j]-'0'
end for
end for
return s
end function
 
function topple(sequence s, integer one=0)
for i=1 to 3 do
for j=1 to 5 by 2 do
if s[i][j]>'3' then
s[i][j] -= 4
if i>1 then s[i-1][j] += 1 end if
if i<3 then s[i+1][j] += 1 end if
if j>1 then s[i][j-2] += 1 end if
if j<5 then s[i][j+2] += 1 end if
if one=1 then return s end if
one = -1
end if
end for
end for
return iff(one=1?{}:iff(one=-1?topple(s):s))
end function
 
procedure shout(sequence s)
sequence r = repeat("",5)
for i=1 to length(s) do
sequence si = s[i]
if string(si) then
string ti = repeat(' ',length(si))
r[1] &= ti
r[2] &= si
r[3] &= ti
else
for j=1 to 3 do
r[j] &= si[j]
end for
end if
end for
puts(1,join(r,"\n"))
end procedure
 
puts(1,"1. Show avalanche\n\n")
sequence s = s4,
res = {" ",s}
while true do
s = topple(s,1)
if s={} then exit end if
res &= {" ==> ",s}
end while
shout(res)
 
puts(1,"2. Prove s1 + s2 = s2 + s1\n\n")
shout({" ",s1," + ",s2," = ",topple(add(s1,s2))})
shout({" ",s2," + ",s1," = ",topple(add(s2,s1))})
 
puts(1,"3. Show that s3 + s3_id == s3\n\n")
shout({" ",s3," + ",s3_id," = ",topple(add(s3,s3_id))})
 
puts(1,"4. Show that s3_id + s3_id == s3_id\n\n")
shout({" ",s3_id," + ",s3_id," = ",topple(add(s3_id,s3_id))})
Output:
1. Show avalanche

    4 3 3     0 4 3     1 0 4     1 1 0     2 1 0
    3 1 2 ==> 4 1 2 ==> 4 2 2 ==> 4 2 3 ==> 0 3 3
    0 2 3     0 2 3     0 2 3     0 2 3     1 2 3

2. Prove s1 + s2 = s2 + s1

    1 2 0     2 1 3     3 3 3
    2 1 1  +  1 0 1  =  3 1 2
    0 1 3     0 1 0     0 2 3

    2 1 3     1 2 0     3 3 3
    1 0 1  +  2 1 1  =  3 1 2
    0 1 0     0 1 3     0 2 3

3. Show that s3 + s3_id == s3

    3 3 3     2 1 2     3 3 3
    3 3 3  +  1 0 1  =  3 3 3
    3 3 3     2 1 2     3 3 3

4. Show that s3_id + s3_id == s3_id

    2 1 2     2 1 2     2 1 2
    1 0 1  +  1 0 1  =  1 0 1
    2 1 2     2 1 2     2 1 2

Python[edit]

Object Oriented[edit]

from itertools import product
from collections import defaultdict
 
 
class Sandpile():
def __init__(self, gridtext):
array = [int(x) for x in gridtext.strip().split()]
self.grid = defaultdict(int,
{(i //3, i % 3): x
for i, x in enumerate(array)})
 
_border = set((r, c)
for r, c in product(range(-1, 4), repeat=2)
if not 0 <= r <= 2 or not 0 <= c <= 2
)
_cell_coords = list(product(range(3), repeat=2))
 
def topple(self):
g = self.grid
for r, c in self._cell_coords:
if g[(r, c)] >= 4:
g[(r - 1, c)] += 1
g[(r + 1, c)] += 1
g[(r, c - 1)] += 1
g[(r, c + 1)] += 1
g[(r, c)] -= 4
return True
return False
 
def stabilise(self):
while self.topple():
pass
# Remove extraneous grid border
g = self.grid
for row_col in self._border.intersection(g.keys()):
del g[row_col]
return self
 
__pos__ = stabilise # +s == s.stabilise()
 
def __eq__(self, other):
g = self.grid
return all(g[row_col] == other.grid[row_col]
for row_col in self._cell_coords)
 
def __add__(self, other):
g = self.grid
ans = Sandpile("")
for row_col in self._cell_coords:
ans.grid[row_col] = g[row_col] + other.grid[row_col]
return ans.stabilise()
 
def __str__(self):
g, txt = self.grid, []
for row in range(3):
txt.append(' '.join(str(g[(row, col)])
for col in range(3)))
return '\n'.join(txt)
 
def __repr__(self):
return f'{self.__class__.__name__}(""""\n{self.__str__()}""")'
 
 
unstable = Sandpile("""
4 3 3
3 1 2
0 2 3"""
)
s1 = Sandpile("""
1 2 0
2 1 1
0 1 3
"""
)
s2 = Sandpile("""
2 1 3
1 0 1
0 1 0
"""
)
s3 = Sandpile("3 3 3 3 3 3 3 3 3")
s3_id = Sandpile("2 1 2 1 0 1 2 1 2")
 
Command line session to complete task.
In [2]: unstable
Out[2]: 
Sandpile(""""
4 3 3
3 1 2
0 2 3""")

In [3]: unstable.stabilise()
Out[3]: 
Sandpile(""""
2 1 0
0 3 3
1 2 3""")

In [4]: s1 + s2
Out[4]: 
Sandpile(""""
3 3 3
3 1 2
0 2 3""")

In [5]: s2 + s1
Out[5]: 
Sandpile(""""
3 3 3
3 1 2
0 2 3""")

In [6]: s1 + s2 == s2 + s1
Out[6]: True

In [7]: s3
Out[7]: 
Sandpile(""""
3 3 3
3 3 3
3 3 3""")

In [8]: s3_id
Out[8]: 
Sandpile(""""
2 1 2
1 0 1
2 1 2""")

In [9]: s3 + s3_id
Out[9]: 
Sandpile(""""
3 3 3
3 3 3
3 3 3""")

In [10]: s3 + s3_id == s3
Out[10]: True

In [11]: s3_id + s3_id
Out[11]: 
Sandpile(""""
2 1 2
1 0 1
2 1 2""")

In [12]: s3_id + s3_id == s3_id
Out[12]: True

In [13]: 

Functional[edit]

'''Abelian Sandpile – Identity'''
 
from operator import add, eq
 
 
# -------------------------- TEST --------------------------
# main :: IO ()
def main():
'''Tests of cascades and additions'''
s0 = [[4, 3, 3], [3, 1, 2], [0, 2, 3]]
s1 = [[1, 2, 0], [2, 1, 1], [0, 1, 3]]
s2 = [[2, 1, 3], [1, 0, 1], [0, 1, 0]]
s3 = [[3, 3, 3], [3, 3, 3], [3, 3, 3]]
s3_id = [[2, 1, 2], [1, 0, 1], [2, 1, 2]]
 
series = list(cascadeSeries(s0))
for expr in [
'Cascade:',
showSandPiles(
[(' ', series[0])] + [
(':', xs) for xs in series[1:]
]
),
'',
f's1 + s2 == s2 + s1 -> {addSand(s1)(s2) == addSand(s2)(s1)}',
showSandPiles([
(' ', s1),
('+', s2),
('=', addSand(s1)(s2))
]),
'',
showSandPiles([
(' ', s2),
('+', s1),
('=', addSand(s2)(s1))
]),
'',
f's3 + s3_id == s3 -> {addSand(s3)(s3_id) == s3}',
showSandPiles([
(' ', s3),
('+', s3_id),
('=', addSand(s3)(s3_id))
]),
'',
f's3_id + s3_id == s3_id -> {addSand(s3_id)(s3_id) == s3_id}',
showSandPiles([
(' ', s3_id),
('+', s3_id),
('=', addSand(s3_id)(s3_id))
]),
 
]:
print(expr)
 
 
# ----------------------- SANDPILES ------------------------
 
# addSand :: [[Int]] -> [[Int]] -> [[Int]]
def addSand(xs):
'''The stabilised sum of two sandpiles.
'''

def go(ys):
return cascadeSeries(
chunksOf(len(xs))(
map(
add,
concat(xs),
concat(ys)
)
)
)[-1]
return go
 
 
# cascadeSeries :: [[Int]] -> [[[Int]]]
def cascadeSeries(rows):
'''The sequence of states from a given
sand pile to a stable condition.
'''

xs = list(rows)
w = len(xs)
return [
list(chunksOf(w)(x)) for x
in convergence(eq)(
iterate(nextState(w))(
concat(xs)
)
)
]
 
 
# convergence :: (a -> a -> Bool) -> [a] -> [a]
def convergence(p):
'''All items of xs to the point where the binary
p returns True over two successive values.
'''

def go(xs):
def conv(prev, ys):
y = next(ys)
return [prev] + (
[] if p(prev, y) else conv(y, ys)
)
return conv(next(xs), xs)
return go
 
 
# nextState Int -> Int -> [Int] -> [Int]
def nextState(w):
'''The next state of a (potentially unstable)
flattened sand-pile matrix of row length w.
'''

def go(xs):
def tumble(i):
neighbours = indexNeighbours(w)(i)
return [
1 + k if j in neighbours else (
k - (1 + w) if j == i else k
) for (j, k) in enumerate(xs)
]
return maybe(xs)(tumble)(
findIndex(lambda x: w < x)(xs)
)
return go
 
 
# indexNeighbours :: Int -> Int -> [Int]
def indexNeighbours(w):
'''Indices vertically and horizontally adjoining the
given index in a flattened matrix of dimension w.
'''

def go(i):
lastCol = w - 1
iSqr = (w * w)
col = i % w
return [
j for j in [i - w, i + w]
if -1 < j < iSqr
] + ([i - 1] if 0 != col else []) + (
[1 + i] if lastCol != col else []
)
return go
 
 
# ------------------------ DISPLAY -------------------------
 
# showSandPiles :: [(String, [[Int]])] -> String
def showSandPiles(pairs):
'''Indented multi-line representation
of a sequence of matrices, delimited
by preceding operators or indents.
'''

return '\n'.join([
' '.join([' '.join(map(str, seq)) for seq in tpl])
for tpl in zip(*[
zip(
*[list(str(pfx).center(len(rows)))]
+ list(zip(*rows))
)
for (pfx, rows) in pairs
])
])
 
 
# ------------------------ GENERIC -------------------------
 
# chunksOf :: Int -> [a] -> [[a]]
def chunksOf(n):
'''A series of lists of length n, subdividing the
contents of xs. Where the length of xs is not evenly
divible, the final list will be shorter than n.
'''

def go(xs):
ys = list(xs)
return (
ys[i:n + i] for i in range(0, len(ys), n)
) if 0 < n else None
return go
 
 
# concat :: [[a]] -> [a]
def concat(xs):
'''The concatenation of all
elements in a list.
'''

return [x for lst in xs for x in lst]
 
 
# findIndex :: (a -> Bool) -> [a] -> Maybe Int
def findIndex(p):
'''Just the first index at which an
element in xs matches p,
or Nothing if no elements match.
'''

def go(xs):
return next(
(i for (i, x) in enumerate(xs) if p(x)),
None
)
return go
 
 
# iterate :: (a -> a) -> a -> Gen [a]
def iterate(f):
'''An infinite list of repeated
applications of f to x.
'''

def go(x):
v = x
while True:
yield v
v = f(v)
return go
 
 
# maybe :: b -> (a -> b) -> Maybe a -> b
def maybe(v):
'''Either the default value v, if x is None,
or the application of f to x.
'''

def go(f):
def g(x):
return v if None is x else f(x)
return g
return go
 
 
# MAIN ---
if __name__ == '__main__':
main()
Output:
Cascade:
  4 3 3   0 4 3   1 0 4   1 1 0   2 1 0
  3 1 2 : 4 1 2 : 4 2 2 : 4 2 3 : 0 3 3
  0 2 3   0 2 3   0 2 3   0 2 3   1 2 3

s1 + s2 == s2 + s1 -> True
  1 2 0   2 1 3   3 3 3
  2 1 1 + 1 0 1 = 3 1 2
  0 1 3   0 1 0   0 2 3

  2 1 3   1 2 0   3 3 3
  1 0 1 + 2 1 1 = 3 1 2
  0 1 0   0 1 3   0 2 3

s3 + s3_id == s3 -> True
  3 3 3   2 1 2   3 3 3
  3 3 3 + 1 0 1 = 3 3 3
  3 3 3   2 1 2   3 3 3

s3_id + s3_id == s3_id -> True
  2 1 2   2 1 2   2 1 2
  1 0 1 + 1 0 1 = 1 0 1
  2 1 2   2 1 2   2 1 2

Raku[edit]

Works with: Rakudo version 2020.05

Most of the logic is lifted straight from the Abelian sandpile model task.

class ASP {
has $.h = 3;
has $.w = 3;
has @.pile = 0 xx $!w * $!h;
 
method topple {
my $buf = $!w * $!h;
my $done;
repeat {
$done = True;
loop (my int $row; $row < $!h; $row = $row + 1) {
my int $rs = $row * $!w; # row start
my int $re = $rs + $!w; # row end
loop (my int $idx = $rs; $idx < $re; $idx = $idx + 1) {
if self.pile[$idx] >= 4 {
my $grain = self.pile[$idx] div 4;
self.pile[ $idx - $!w ] += $grain if $row > 0;
self.pile[ $idx - 1 ] += $grain if $idx - 1 >= $rs;
self.pile[ $idx + $!w ] += $grain if $row < $!h - 1;
self.pile[ $idx + 1 ] += $grain if $idx + 1 < $re;
self.pile[ $idx ] %= 4;
$done = False;
}
}
}
} until $done;
self.pile;
}
}
 
# some handy display layout modules
use Terminal::Boxer:ver<0.2+>;
use Text::Center;
 
for 3, (4,3,3,3,1,2,0,2,3), (2,1,2,1,0,1,2,1,2), # 3 square task example
3, (2,1,2,1,0,1,2,1,2), (2,1,2,1,0,1,2,1,2), # 3 square identity
5, (4,1,0,5,1,9,3,6,1,0,8,1,2,5,3,3,0,1,7,5,4,2,2,4,0), (2,3,2,3,2,3,2,1,2,3,2,1,0,1,2,3,2,1,2,3,2,3,2,3,2) # 5 square test
-> $size, $pile, $identity {
 
my $asp = ASP.new(:h($size), :w($size));
 
$asp.pile = |$pile;
 
my @display;
 
my %p = :col($size), :3cw, :indent("\t");
 
@display.push: rs-box |%p, |$identity;
 
@display.push: rs-box |%p, $asp.pile;
 
@display.push: rs-box |%p, $asp.topple;
 
$asp.pile Z+= $identity.list;
 
@display.push: rs-box |%p, $asp.pile;
 
@display.push: rs-box |%p, $asp.topple;
 
put %p<indent> ~ qww<identity 'test pile' toppled 'plus identity' toppled>».&center($size * 4 + 1).join: %p<indent>;
 
.put for [Z~] @display».lines;
 
put '';
}
Output:
	   identity  	  test pile  	   toppled   	plus identity	   toppled   
	╭───┬───┬───╮	╭───┬───┬───╮	╭───┬───┬───╮	╭───┬───┬───╮	╭───┬───┬───╮
	│ 2 │ 1 │ 2 │	│ 4 │ 3 │ 3 │	│ 2 │ 1 │ 0 │	│ 4 │ 2 │ 2 │	│ 2 │ 1 │ 0 │
	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤
	│ 1 │ 0 │ 1 │	│ 3 │ 1 │ 2 │	│ 0 │ 3 │ 3 │	│ 1 │ 3 │ 4 │	│ 0 │ 3 │ 3 │
	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤
	│ 2 │ 1 │ 2 │	│ 0 │ 2 │ 3 │	│ 1 │ 2 │ 3 │	│ 3 │ 3 │ 5 │	│ 1 │ 2 │ 3 │
	╰───┴───┴───╯	╰───┴───┴───╯	╰───┴───┴───╯	╰───┴───┴───╯	╰───┴───┴───╯

	   identity  	  test pile  	   toppled   	plus identity	   toppled   
	╭───┬───┬───╮	╭───┬───┬───╮	╭───┬───┬───╮	╭───┬───┬───╮	╭───┬───┬───╮
	│ 2 │ 1 │ 2 │	│ 2 │ 1 │ 2 │	│ 2 │ 1 │ 2 │	│ 4 │ 2 │ 4 │	│ 2 │ 1 │ 2 │
	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤
	│ 1 │ 0 │ 1 │	│ 1 │ 0 │ 1 │	│ 1 │ 0 │ 1 │	│ 2 │ 0 │ 2 │	│ 1 │ 0 │ 1 │
	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤	├───┼───┼───┤
	│ 2 │ 1 │ 2 │	│ 2 │ 1 │ 2 │	│ 2 │ 1 │ 2 │	│ 4 │ 2 │ 4 │	│ 2 │ 1 │ 2 │
	╰───┴───┴───╯	╰───┴───┴───╯	╰───┴───┴───╯	╰───┴───┴───╯	╰───┴───┴───╯

	       identity      	      test pile      	       toppled       	    plus identity    	       toppled       
	╭───┬───┬───┬───┬───╮	╭───┬───┬───┬───┬───╮	╭───┬───┬───┬───┬───╮	╭───┬───┬───┬───┬───╮	╭───┬───┬───┬───┬───╮
	│ 2 │ 3 │ 2 │ 3 │ 2 │	│ 4 │ 1 │ 0 │ 5 │ 1 │	│ 1 │ 3 │ 2 │ 1 │ 0 │	│ 3 │ 6 │ 4 │ 4 │ 2 │	│ 1 │ 3 │ 2 │ 1 │ 0 │
	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤
	│ 3 │ 2 │ 1 │ 2 │ 3 │	│ 9 │ 3 │ 6 │ 1 │ 0 │	│ 2 │ 2 │ 3 │ 3 │ 1 │	│ 5 │ 4 │ 4 │ 5 │ 4 │	│ 2 │ 2 │ 3 │ 3 │ 1 │
	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤
	│ 2 │ 1 │ 0 │ 1 │ 2 │	│ 8 │ 1 │ 2 │ 5 │ 3 │	│ 1 │ 1 │ 2 │ 0 │ 3 │	│ 3 │ 2 │ 2 │ 1 │ 5 │	│ 1 │ 1 │ 2 │ 0 │ 3 │
	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤
	│ 3 │ 2 │ 1 │ 2 │ 3 │	│ 3 │ 0 │ 1 │ 7 │ 5 │	│ 2 │ 0 │ 3 │ 2 │ 0 │	│ 5 │ 2 │ 4 │ 4 │ 3 │	│ 2 │ 0 │ 3 │ 2 │ 0 │
	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤	├───┼───┼───┼───┼───┤
	│ 2 │ 3 │ 2 │ 3 │ 2 │	│ 4 │ 2 │ 2 │ 4 │ 0 │	│ 3 │ 2 │ 3 │ 2 │ 1 │	│ 5 │ 5 │ 5 │ 5 │ 3 │	│ 3 │ 2 │ 3 │ 2 │ 1 │
	╰───┴───┴───┴───┴───╯	╰───┴───┴───┴───┴───╯	╰───┴───┴───┴───┴───╯	╰───┴───┴───┴───┴───╯	╰───┴───┴───┴───┴───╯

Wren[edit]

Library: Wren-fmt
import "/fmt" for Fmt
 
class Sandpile {
static init() {
__neighbors = [
[1, 3], [0, 2, 4], [1, 5], [0, 4, 6], [1, 3, 5, 7], [2, 4, 8], [3, 7], [4, 6, 8], [5, 7]
]
}
 
// 'a' is a list of 9 integers in row order
construct new(a) {
_a = a
}
 
a { _a }
 
+(other) {
var b = List.filled(9, 0)
for (i in 0..8) b[i] = _a[i] + other.a[i]
return Sandpile.new(b)
}
 
isStable { _a.all { |i| i <= 3 } }
 
// just topples once so we can observe intermediate results
topple() {
for (i in 0..8) {
if (_a[i] > 3) {
_a[i] = _a[i] - 4
for (j in __neighbors[i]) _a[j] = _a[j] + 1
return
}
}
}
 
toString {
var s = ""
for (i in 0..2) {
for (j in 0..2) s = s + "%(a[3*i + j]) "
s = s + "\n"
}
return s
}
}
 
Sandpile.init()
System.print("Avalanche of topplings:\n")
var s4 = Sandpile.new([4, 3, 3, 3, 1, 2, 0, 2, 3])
System.print(s4)
while (!s4.isStable) {
s4.topple()
System.print(s4)
}
 
System.print("Commutative additions:\n")
var s1 = Sandpile.new([1, 2, 0, 2, 1, 1, 0, 1, 3])
var s2 = Sandpile.new([2, 1, 3, 1, 0, 1, 0, 1, 0])
var s3_a = s1 + s2
while (!s3_a.isStable) s3_a.topple()
var s3_b = s2 + s1
while (!s3_b.isStable) s3_b.topple()
Fmt.print("$s\nplus\n\n$s\nequals\n\n$s", s1, s2, s3_a)
Fmt.print("and\n\n$s\nplus\n\n$s\nalso equals\n\n$s", s2, s1, s3_b)
 
System.print("Addition of identity sandpile:\n")
var s3 = Sandpile.new(List.filled(9, 3))
var s3_id = Sandpile.new([2, 1, 2, 1, 0, 1, 2, 1, 2])
s4 = s3 + s3_id
while (!s4.isStable) s4.topple()
Fmt.print("$s\nplus\n\n$s\nequals\n\n$s", s3, s3_id, s4)
 
System.print("Addition of identities:\n")
var s5 = s3_id + s3_id
while (!s5.isStable) s5.topple()
Fmt.write("$s\nplus\n\n$s\nequals\n\n$s", s3_id, s3_id, s5)
Output:
Avalanche of topplings:

4 3 3 
3 1 2 
0 2 3 

0 4 3 
4 1 2 
0 2 3 

1 0 4 
4 2 2 
0 2 3 

1 1 0 
4 2 3 
0 2 3 

2 1 0 
0 3 3 
1 2 3 

Commutative additions:

1 2 0 
2 1 1 
0 1 3 

plus

2 1 3 
1 0 1 
0 1 0 

equals

3 3 3 
3 1 2 
0 2 3 

and

2 1 3 
1 0 1 
0 1 0 

plus

1 2 0 
2 1 1 
0 1 3 

also equals

3 3 3 
3 1 2 
0 2 3 

Addition of identity sandpile:

3 3 3 
3 3 3 
3 3 3 

plus

2 1 2 
1 0 1 
2 1 2 

equals

3 3 3 
3 3 3 
3 3 3 

Addition of identities:

2 1 2 
1 0 1 
2 1 2 

plus

2 1 2 
1 0 1 
2 1 2 

equals

2 1 2 
1 0 1 
2 1 2