# Abelian sandpile model/Identity

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
```

and `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 examples.

References

## 11l

Translation of: Python
```T Sandpile
DefaultDict[(Int, Int), Int] grid

F (gridtext)
V array = gridtext.split_py().map(x -> Int(x))
L(x) array
.grid[(L.index I/ 3, L.index % 3)] = x

Set[(Int, Int)] _border = Set(cart_product(-1 .< 4, -1 .< 4).filter((r, c) -> !(r C 0..2) | !(c C 0..2)))
_cell_coords = cart_product(0.<3, 0.<3)

F topple()
V& g = .grid
L(r, c) ._cell_coords
I g[(r, c)] >= 4
g[(r - 1, c)]++
g[(r + 1, c)]++
g[(r, c - 1)]++
g[(r, c + 1)]++
g[(r, c)] -= 4
R 1B
R 0B

F stabilise()
L .topple() {}

L(row_col) ._border.intersection(Set(.grid.keys()))
.grid.pop(row_col)

F ==(other)
R all(._cell_coords.map(row_col -> @.grid[row_col] == @other.grid[row_col]))

F +(other)
V ans = Sandpile(‘’)
L(row_col) ._cell_coords
ans.grid[row_col] = .grid[row_col] + other.grid[row_col]
ans.stabilise()
R ans

F String()
[String] txt
L(row) 3
txt.append((0.<3).map(col -> String(@.grid[(@row, col)])).join(‘ ’))
R txt.join("\n")

V unstable = Sandpile(‘4 3 3
3 1 2
0 2 3’)
V s1 = Sandpile(‘1 2 0
2 1 1
0 1 3’)
V s2 = Sandpile(‘2 1 3
1 0 1
0 1 0’)
V s3 = Sandpile(‘3 3 3  3 3 3  3 3 3’)
V s3_id = Sandpile(‘2 1 2  1 0 1  2 1 2’)

print(unstable)
print()
unstable.stabilise()
print(unstable)
print()
print(s1 + s2)
print()
print(s2 + s1)
print()
print(s1 + s2 == s2 + s1)
print()
print(s3 + s3_id)
print()
print(s3 + s3_id == s3)
print()
print(s3_id + s3_id)
print()
print(s3_id + s3_id == s3_id)```
Output:
```4 3 3
3 1 2
0 2 3

2 1 0
0 3 3
1 2 3

3 3 3
3 1 2
0 2 3

3 3 3
3 1 2
0 2 3

1B

3 3 3
3 3 3
3 3 3

1B

2 1 2
1 0 1
2 1 2

1B
```

## AArch64 Assembly

Works with: as version Raspberry Pi 3B version Buster 64 bits
or android 64 bits with application Termux
```/* ARM assembly AARCH64 Raspberry PI 3B or android 64 bits */
/*  program abelianSum64.s   */

/*******************************************/
/* Constantes file                         */
/*******************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"
.equ MAXI, 3

/*********************************/
/* Initialized data              */
/*********************************/
.data
szMessValue:        .asciz "@ "
szMessAdd1:         .asciz "Add sandpile 1 to sandpile 2  \n"
szMessAdd2:         .asciz "Add sandpile 2 to sandpile 1  \n"
szMessAdd2A:        .asciz "Add sandpile 2A to sandpile result  \n"
szMessAdd3:         .asciz "Add sandpile 3 to sandpile 3ID  \n"
szMessAdd3ID:       .asciz "Add sandpile 3ID to sandpile 3ID  \n"

szCarriageReturn:   .asciz "\n"

qSandPile1:    .quad 1,2,0
.quad 2,1,1
.quad 0,1,3

qSandPile2:    .quad 2,1,3
.quad 1,0,1
.quad 0,1,0

qSandPile2A:    .quad 1,0,0
.quad 0,0,0
.quad 0,0,0

qSandPile3:    .quad 3,3,3
.quad 3,3,3
.quad 3,3,3

qSandPile3ID:  .quad 2,1,2
.quad 1,0,1
.quad 2,1,2
/*********************************/
/* UnInitialized data            */
/*********************************/
.bss
sZoneConv:        .skip 24
qSandPilex1:      .skip 8 * MAXI * MAXI
qSandPilex2:      .skip 8 * MAXI * MAXI
/*********************************/
/*  code section                 */
/*********************************/
.text
.global main
main:                            // entry of program

ldr x0,qAdrqSandPile1        // sandpile1 address
ldr x1,qAdrqSandPile2        // sandpile2 address
ldr x2,qAdrqSandPilex1       // sandpile result address
bl addSandPile

ldr x0,qAdrszMessAdd1        // display message
bl affichageMess
ldr x0,qAdrqSandPilex1       // display sandpile
bl displaySandPile

ldr x0,qAdrqSandPile2        // sandpile2 address
ldr x1,qAdrqSandPile1        // sandpile1 address
ldr x2,qAdrqSandPilex1       // sandpile result address
bl addSandPile

ldr x0,qAdrszMessAdd2
bl affichageMess
ldr x0,qAdrqSandPilex1
bl displaySandPile

ldr x0,qAdrqSandPilex1        // sandpile1 address
ldr x1,qAdrqSandPile2A        // sandpile2A address
ldr x2,qAdrqSandPilex2        // sandpile result address
bl addSandPile

ldr x0,qAdrszMessAdd2A
bl affichageMess
ldr x0,qAdrqSandPilex2
bl displaySandPile

ldr x0,qAdrqSandPile3          // sandpile3 address
ldr x1,qAdrqSandPile3ID        // sandpile3ID address
ldr x2,qAdrqSandPilex2         // sandpile result address
bl addSandPile

ldr x0,qAdrszMessAdd3
bl affichageMess
ldr x0,qAdrqSandPilex2
bl displaySandPile

ldr x0,qAdrqSandPile3ID        // sandpile3 address
ldr x1,qAdrqSandPile3ID        // sandpile3ID address
ldr x2,qAdrqSandPilex2         // sandpile result address
bl addSandPile

ldr x0,qAdrszMessAdd3ID
bl affichageMess
ldr x0,qAdrqSandPilex2
bl displaySandPile
100:                               // standard end of the program
mov x0, #0                     // return code
mov x8, #EXIT                  // request to exit program
svc #0                         // perform the system call

qAdrszCarriageReturn:     .quad szCarriageReturn
qAdrsZoneConv:            .quad sZoneConv
qAdrszMessAdd1:           .quad szMessAdd1
qAdrszMessAdd2:           .quad szMessAdd2
qAdrszMessAdd2A:          .quad szMessAdd2A
qAdrszMessAdd3:           .quad szMessAdd3
qAdrszMessAdd3ID:         .quad szMessAdd3ID
qAdrqSandPile1:           .quad qSandPile1
qAdrqSandPilex1:          .quad qSandPilex1
qAdrqSandPilex2:          .quad qSandPilex2
qAdrqSandPile2:           .quad qSandPile2
qAdrqSandPile2A:          .quad qSandPile2A
qAdrqSandPile3:           .quad qSandPile3
qAdrqSandPile3ID:         .quad qSandPile3ID
/***************************************************/
/*     add two  sandpile               */
/***************************************************/
// x0 contains address to sandpile 1
// x1 contains address to sandpile 2
// x2 contains address to sandpile result
addSandPile:
stp x1,lr,[sp,-16]!       // save  registres
stp x2,x3,[sp,-16]!       // save  registres
stp x4,x5,[sp,-16]!       // save  registres
stp x6,x7,[sp,-16]!       // save  registres
mov x6,x1                 // save addresse sandpile2
mov x1,x2                 // and copy sandpile 1 to sandpile result
bl copySandPile
mov x0,x2                 // sanspile result
mov x2,#0                 // indice y
mov x4,#MAXI
1:
mov x1,#0                  // indice x
2:
madd x5,x2,x4,x1           // compute offset
ldr x7,[x0,x5,lsl #3]      // load value at pos x,y sanspile result
ldr x3,[x6,x5,lsl #3]      // load value at pos x,y sandpile 2
add x7,x7,x3
str x7,[x0,x5,lsl #3]      // store sum on sandpile result
bl avalancheRisk
add x1,x1,#1
cmp x1,#MAXI
blt 2b
add x2,x2,#1
cmp x2,#MAXI
blt 1b
100:
ldp x6,x7,[sp],16         // restaur des  2 registres
ldp x4,x5,[sp],16         // restaur des  2 registres
ldp x2,x3,[sp],16         // restaur des  2 registres
ldp x1,lr,[sp],16         // restaur des  2 registres
ret
/***************************************************/
/*     copy sandpile                               */
/***************************************************/
// x0 contains address to sandpile
// x1 contains address to sandpile result
copySandPile:
stp x1,lr,[sp,-16]!         // save  registres
stp x2,x3,[sp,-16]!         // save  registres
stp x4,x5,[sp,-16]!         // save  registres
stp x6,x7,[sp,-16]!         // save  registres
mov x2,#0                   // indice y
mov x3,#MAXI
1:
mov x4,#0                   // indice x
2:
madd x5,x2,x3,x4            // compute offset
ldr x6,[x0,x5,lsl #3]       // load value at pos x,y sanspile
str x6,[x1,x5,lsl #3]       // store value at pos x,y sandpile result
add x4,x4,#1
cmp x4,#MAXI
blt 2b
add x2,x2,#1
cmp x2,#MAXI
blt 1b
100:
ldp x6,x7,[sp],16          // restaur des  2 registres
ldp x4,x5,[sp],16          // restaur des  2 registres
ldp x2,x3,[sp],16          // restaur des  2 registres
ldp x1,lr,[sp],16          // restaur des  2 registres
ret
/***************************************************/
/*     display  sandpile               */
/***************************************************/
// x0 contains address to sandpile
displaySandPile:
stp x1,lr,[sp,-16]!         // save  registres
stp x2,x3,[sp,-16]!         // save  registres
stp x4,x5,[sp,-16]!         // save  registres
stp x6,x7,[sp,-16]!         // save  registres
mov x6,x0
mov x3,#0                   // indice y
mov x4,#MAXI
1:
mov x2,#0                   // indice x
2:
madd x5,x3,x4,x2            // compute offset
ldr x0,[x6,x5,lsl #3]       // load value at pos x,y
ldr x1,qAdrsZoneConv
bl conversion10             // call decimal conversion
add x1,x1,#1
mov x7,#0
strb w7,[x1,x0]
ldr x0,qAdrszMessValue
ldr x1,qAdrsZoneConv        // insert value conversion in message
bl strInsertAtCharInc
bl affichageMess
add x2,x2,#1
cmp x2,#MAXI
blt 2b
ldr x0,qAdrszCarriageReturn
bl affichageMess
add x3,x3,#1
cmp x3,#MAXI
blt 1b

100:
ldp x6,x7,[sp],16          // restaur des  2 registres
ldp x4,x5,[sp],16          // restaur des  2 registres
ldp x2,x3,[sp],16          // restaur des  2 registres
ldp x1,lr,[sp],16          // restaur des  2 registres
ret
qAdrszMessValue:       .quad szMessValue
/***************************************************/
/*     avalanche risk              */
/***************************************************/
// x0 contains address to sanspile
// x1 contains position x
// x2 contains position y
avalancheRisk:
stp x1,lr,[sp,-16]!         // save  registres
stp x2,x3,[sp,-16]!         // save  registres
stp x4,x5,[sp,-16]!         // save  registres
mov x3,#MAXI
madd x4,x3,x2,x1
ldr x5,[x0,x4,lsl #3]
1:
cmp x5,#4                   // 4 grains ?
blt 100f
sub x5,x5,#4                   // yes sustract
str x5,[x0,x4,lsl #3]
cmp x1,#MAXI-1              // right position ok ?
beq 2f
add x1,x1,#1                   // yes
bl add1Sand                 // add 1 grain
bl avalancheRisk                  // and compute new pile
sub x1,x1,#1
2:
cmp x1,#0                   // left position ok ?
beq 3f
sub x1,x1,#1
bl add1Sand
bl avalancheRisk
add x1,x1,#1
3:
cmp x2,#0                   // higt position ok ?
beq 4f
sub x2,x2,#1
bl add1Sand
bl avalancheRisk
add x2,x2,#1
4:
cmp x2,#MAXI-1               // low position ok ?
beq 5f
add x2,x2,#1
bl add1Sand
bl avalancheRisk
sub x2,x2,#1
5:
ldr x5,[x0,x4,lsl #3]       // reload value
b 1b                        // and loop
100:
ldp x4,x5,[sp],16          // restaur des  2 registres
ldp x2,x3,[sp],16          // restaur des  2 registres
ldp x1,lr,[sp],16          // restaur des  2 registres
ret
/***************************************************/
/*     add 1 grain of sand              */
/***************************************************/
// x0 contains address to sanspile
// x1 contains position x
// x2 contains position y
add1Sand:
stp x3,lr,[sp,-16]!         // save  registres
stp x4,x5,[sp,-16]!         // save  registres
mov x3,#MAXI
madd x4,x3,x2,x1
ldr x5,[x0,x4,lsl #3]     // load value at pos x,y
add x5,x5,#1
str x5,[x0,x4,lsl #3]     // and store
100:
ldp x4,x5,[sp],16          // restaur des  2 registres
ldp x3,lr,[sp],16          // restaur des  2 registres
ret
/********************************************************/
/*        File Include fonctions                        */
/********************************************************/
/* for this file see task include a file in language AArch64 assembly */
.include "../includeARM64.inc"```
Output:
```~/.../rosetta/asm1 \$ abelianSum64
Add sandpile 1 to sandpile 2
3 3 3
3 1 2
0 2 3
Add sandpile 2 to sandpile 1
3 3 3
3 1 2
0 2 3
Add sandpile 2A to sandpile result
2 1 0
0 3 3
1 2 3
Add sandpile 3 to sandpile 3ID
3 3 3
3 3 3
3 3 3
Add sandpile 3ID to sandpile 3ID
2 1 2
1 0 1
2 1 2
```

## ARM Assembly

Works with: as version Raspberry Pi
or android 32 bits with application Termux
```/* ARM assembly Raspberry PI  or android 32 bits */
/*  program abelianSum.s   */

/* REMARK 1 : this program use routines in a include file
see task Include a file language arm assembly
for the routine affichageMess conversion10
see at end of this program the instruction include */
/* for constantes see task include a file in arm assembly */
/************************************/
/* Constantes                       */
/************************************/
.include "../constantes.inc"
.equ MAXI, 3

/*********************************/
/* Initialized data              */
/*********************************/
.data
szMessValue:        .asciz "@ "
szMessAdd1:         .asciz "Add sandpile 1 to sandpile 2  \n"
szMessAdd2:         .asciz "Add sandpile 2 to sandpile 1  \n"
szMessAdd2A:        .asciz "Add sandpile 2A to sandpile result  \n"
szMessAdd3:         .asciz "Add sandpile 3 to sandpile 3ID  \n"
szMessAdd3ID:       .asciz "Add sandpile 3ID to sandpile 3ID  \n"

szMessFin:          .asciz "End display :\n"
szCarriageReturn:   .asciz "\n"

iSandPile1:    .int 1,2,0
.int 2,1,1
.int 0,1,3

iSandPile2:    .int 2,1,3
.int 1,0,1
.int 0,1,0

iSandPile2A:    .int 1,0,0
.int 0,0,0
.int 0,0,0

iSandPile3:    .int 3,3,3
.int 3,3,3
.int 3,3,3

iSandPile3ID:  .int 2,1,2
.int 1,0,1
.int 2,1,2
/*********************************/
/* UnInitialized data            */
/*********************************/
.bss
sZoneConv:        .skip 24
iSandPileR1:      .skip 4 * MAXI * MAXI
iSandPileR2:      .skip 4 * MAXI * MAXI
/*********************************/
/*  code section                 */
/*********************************/
.text
.global main
main:                            @ entry of program

ldr r0,iAdriSandPile1        @ sandpile1 address
ldr r1,iAdriSandPile2        @ sandpile2 address
ldr r2,iAdriSandPileR1       @ sandpile result address
bl addSandPile

ldr r0,iAdrszMessAdd1        @ display message
bl affichageMess
ldr r0,iAdriSandPileR1       @ display sandpile
bl displaySandPile

ldr r0,iAdriSandPile2        @ sandpile2 address
ldr r1,iAdriSandPile1        @ sandpile1 address
ldr r2,iAdriSandPileR1       @ sandpile result address
bl addSandPile

ldr r0,iAdrszMessAdd2
bl affichageMess
ldr r0,iAdriSandPileR1
bl displaySandPile

ldr r0,iAdriSandPileR1        @ sandpile1 address
ldr r1,iAdriSandPile2A        @ sandpile2A address
ldr r2,iAdriSandPileR2        @ sandpile result address
bl addSandPile

ldr r0,iAdrszMessAdd2A
bl affichageMess
ldr r0,iAdriSandPileR2
bl displaySandPile

ldr r0,iAdriSandPile3          @ sandpile3 address
ldr r1,iAdriSandPile3ID        @ sandpile3ID address
ldr r2,iAdriSandPileR2         @ sandpile result address
bl addSandPile

ldr r0,iAdrszMessAdd3
bl affichageMess
ldr r0,iAdriSandPileR2
bl displaySandPile

ldr r0,iAdriSandPile3ID        @ sandpile3 address
ldr r1,iAdriSandPile3ID        @ sandpile3ID address
ldr r2,iAdriSandPileR2         @ sandpile result address
bl addSandPile

ldr r0,iAdrszMessAdd3ID
bl affichageMess
ldr r0,iAdriSandPileR2
bl displaySandPile
100:                               @ standard end of the program
mov r0, #0                     @ return code
mov r7, #EXIT                  @ request to exit program
svc #0                         @ perform the system call

iAdrszCarriageReturn:     .int szCarriageReturn
iAdrsZoneConv:            .int sZoneConv
iAdrszMessFin:            .int szMessFin
iAdrszMessAdd1:           .int szMessAdd1
iAdrszMessAdd2:           .int szMessAdd2
iAdrszMessAdd2A:          .int szMessAdd2A
iAdrszMessAdd3:           .int szMessAdd3
iAdrszMessAdd3ID:         .int szMessAdd3ID
iAdriSandPile1:           .int iSandPile1
iAdriSandPileR1:          .int iSandPileR1
iAdriSandPileR2:          .int iSandPileR2
iAdriSandPile2:           .int iSandPile2
iAdriSandPile2A:          .int iSandPile2A
iAdriSandPile3:           .int iSandPile3
iAdriSandPile3ID:         .int iSandPile3ID
/***************************************************/
/*     add two  sandpile               */
/***************************************************/
// r0 contains address to sandpile 1
// r1 contains address to sandpile 2
// r2 contains address to sandpile result
addSandPile:
push {r1-r7,lr}           @ save  registers
mov r6,r1                 @ save addresse sandpile2
mov r1,r2                 @ and copy sandpile 1 to sandpile result
bl copySandPile
mov r0,r2                 @ sanspile result
mov r2,#0                 @ indice y
mov r4,#MAXI
1:
mov r1,#0                  @ indice x
2:
mla r5,r2,r4,r1            @ compute offset
ldr r7,[r0,r5,lsl #2]      @ load value at pos x,y sanspile result
ldr r3,[r6,r5,lsl #2]      @ load value at pos x,y sandpile 2
add r7,r3
str r7,[r0,r5,lsl #2]      @ store sum on sandpile result
bl avalancheRisk
add r1,r1,#1
cmp r1,#MAXI
blt 2b
add r2,r2,#1
cmp r2,#MAXI
blt 1b
100:
pop {r1-r7,lr}             @ restaur registers
bx lr                      @ return
/***************************************************/
/*     copy sandpile                               */
/***************************************************/
// r0 contains address to sandpile
// r1 contains address to sandpile result
copySandPile:
push {r1-r6,lr}           @ save  registers
mov r2,#0                 @ indice y
mov r3,#MAXI
1:
mov r4,#0                   @ indice x
2:
mla r5,r2,r3,r4            @ compute offset
ldr r6,[r0,r5,lsl #2]      @ load value at pos x,y sanspile
str r6,[r1,r5,lsl #2]      @ store value at pos x,y sandpile result
add r4,r4,#1
cmp r4,#MAXI
blt 2b
add r2,r2,#1
cmp r2,#MAXI
blt 1b
100:
pop {r1-r6,lr}             @ restaur registers
bx lr                      @ return
/***************************************************/
/*     display  sandpile               */
/***************************************************/
// r0 contains address to sandpile
displaySandPile:
push {r1-r6,lr}             @ save  registers
mov r6,r0
mov r3,#0                   @ indice y
mov r4,#MAXI
1:
mov r2,#0                   @ indice x
2:
mul r5,r3,r4
add r5,r2                   @ compute offset
ldr r0,[r6,r5,lsl #2]       @ load value at pos x,y
ldr r1,iAdrsZoneConv
bl conversion10             @ call decimal conversion
add r1,#1
mov r7,#0
strb r7,[r1,r0]
ldr r0,iAdrszMessValue
ldr r1,iAdrsZoneConv        @ insert value conversion in message
bl strInsertAtCharInc
bl affichageMess
add r2,#1
cmp r2,#MAXI
blt 2b
ldr r0,iAdrszCarriageReturn
bl affichageMess
add r3,#1
cmp r3,#MAXI
blt 1b

100:
pop {r1-r6,lr}             @ restaur registers
bx lr                      @ return
iAdrszMessValue:       .int szMessValue
/***************************************************/
/*     avalanche risk              */
/***************************************************/
// r0 contains address to sanspile
// r1 contains position x
// r2 contains position y
avalancheRisk:
push {r1-r5,lr}             @ save  registers
mov r3,#MAXI
mul r4,r3,r2
add r4,r1
ldr r5,[r0,r4,lsl #2]
1:
cmp r5,#4                   @ 4 grains ?
blt 100f
sub r5,#4                   @ yes sustract
str r5,[r0,r4,lsl #2]
cmp r1,#MAXI-1              @ right position ok ?
beq 2f
add r1,#1                   @ yes
bl add1Sand                 @ add 1 grain
bl avalancheRisk                  @ and compute new pile
sub r1,#1
2:
cmp r1,#0                   @ left position ok ?
beq 3f
sub r1,#1
bl add1Sand
bl avalancheRisk
add r1,#1
3:
cmp r2,#0                   @ higt position ok ?
beq 4f
sub r2,#1
bl add1Sand
bl avalancheRisk
add r2,#1
4:
cmp r2,#MAXI-1               @ low position ok ?
beq 5f
add r2,#1
bl add1Sand
bl avalancheRisk
sub r2,#1
5:
ldr r5,[r0,r4,lsl #2]       @ reload value
b 1b                        @ and loop
100:
pop {r1-r5,lr}             @ restaur registers
bx lr                      @ return
/***************************************************/
/*     add 1 grain of sand              */
/***************************************************/
// r0 contains address to sanspile
// r1 contains position x
// r2 contains position y
add1Sand:
push {r3-r5,lr}           @ save  registers
mov r3,#MAXI
mul r4,r3,r2
add r4,r1                 @ compute offset
ldr r5,[r0,r4,lsl #2]     @ load value at pos x,y
add r5,#1
str r5,[r0,r4,lsl #2]     @ and store
100:
pop {r3-r5,lr}            @ restaur registers
bx lr                     @ return
/***************************************************/
/*      ROUTINES INCLUDE                           */
/***************************************************/
.include "../affichage.inc"```
Output:
```Add sandpile 1 to sandpile 2
3  3  3
3  1  2
0  2  3
Add sandpile 2 to sandpile 1
3  3  3
3  1  2
0  2  3
Add sandpile 2A to sandpile result
2  1  0
0  3  3
1  2  3
Add sandpile 3 to sandpile 3ID
3  3  3
3  3  3
3  3  3
Add sandpile 3ID to sandpile 3ID
2  1  2
1  0  1
2  1  2
```

## Ada

Translation of: C++

This Ada example works with Ada 2012 because of the use the aspect with Default_Component_Value.

The package specification for Abelian_Sandpile is:

```-- Works with Ada 2012

package Abelian_Sandpile is
Limit : constant Integer := 4;

type Sandpile is array (0 .. 2, 0 .. 2) of Natural with
Default_Component_Value => 0;

procedure Stabalize (Pile : in out Sandpile);
function Is_Stable (Pile : in Sandpile) return Boolean;
procedure Topple (Pile : in out Sandpile);
function "+" (Left, Right : Sandpile) return Sandpile;
procedure Print(PIle : in Sandpile);

end Abelian_Sandpile;
```

The package body for Abelian_Sandpile is

```with Ada.Text_Io; use Ada.Text_IO;

package body Abelian_Sandpile is

---------------
-- Stabalize --
---------------

procedure Stabalize (Pile : in out Sandpile) is
begin
while not Is_Stable(Pile) loop
Topple(Pile);
end loop;
end Stabalize;

---------------
-- Is_Stable --
---------------

function Is_Stable (Pile : in Sandpile) return Boolean is
begin
return (for all E of Pile => E < Limit);
end Is_Stable;

------------
-- Topple --
------------

procedure Topple (Pile : in out Sandpile) is
begin
outer:
for Row in Pile'Range(1) loop
for Col in Pile'Range(2) loop
if Pile(Row, Col) >= Limit then
Pile(Row, Col) := Pile(Row, Col) - Limit;
if Row > 0 then
Pile(Row - 1, Col) := Pile(Row -1, Col) + 1;
end if;
if Row < Pile'Last(1) then
Pile(Row + 1, Col) := Pile(Row + 1, Col) + 1;
end if;
if Col > 0 then
Pile(Row, Col - 1) := Pile(Row, Col - 1) + 1;
end if;
if Col < Pile'Last(2) then
Pile(Row, Col + 1) := Pile(Row, Col + 1) + 1;
end if;

exit outer;
end if;
end loop;
end loop outer;
end Topple;

---------
-- "+" --
---------

function "+" (Left, Right : Sandpile) return Sandpile is
Result : Sandpile;
begin
for I in Sandpile'Range(1) loop
for J in Sandpile'Range(2) loop
Result(I, J) := Left(I, J) + Right(I, J);
end loop;
end loop;
Stabalize(Result);
return Result;
end "+";

-----------
-- Print --
-----------

procedure Print(Pile : in Sandpile) is
begin
for row in Pile'Range(1) loop
for col in Pile'Range(2) loop
Put(Integer'Image(Pile(row, col)));
end loop;
New_Line;
end loop;
New_Line;
end Print;
end Abelian_Sandpile;
```

The main procedure performing the same tests as the C++ example is

```with Ada.Text_IO;      use Ada.Text_IO;
with Abelian_Sandpile; use Abelian_Sandpile;

procedure Main is
sp    : 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));
sum1  : Sandpile := s1 + s2;
sum2  : Sandpile := s2 + s1;
sum3  : Sandpile := s3 + s3_id;
sum4  : Sandpile := s3_id + s3_id;

begin
Put_Line ("Avalanche:");
while not Is_Stable (sp) loop
Print (sp);
Put_Line ("stable? " & Boolean'Image (Is_Stable (sp)));
New_Line;
Topple (sp);
end loop;
Print (sp);
Put_Line ("stable? " & Boolean'Image (Is_Stable (sp)));
New_Line;

Put_Line ("Commutivity:");
Put_Line ("s1 + s2 equals s2 + s2? " & Boolean'Image (sum1 = sum2));
New_Line;
Put_Line ("S1 : s2 =");
Print (sum1);
New_Line;
Put_Line ("s2 + s1 =");
Print (sum2);
New_Line;

Put_Line ("Identity:");
Put_Line ("s3 + s3_id equals s3? " & Boolean'Image (sum3 = s3));
Put_Line ("s3_id + s3_id equals s3_id? " & Boolean'Image (sum4 = s3_id));
New_Line;
Put_Line ("s3 + s3_id =");
Print (sum3);
Put_Line ("s3_id + s3_id =");
Print (sum4);

end Main;
```
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

Commutivity:
s1 + s2 equals s2 + s2? 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
```

## ALGOL 68

```BEGIN # model Abelian sandpiles #
# represents a sandpile #
INT elements = 3;
MODE SANDPILE = [ 1 : elements, 1 : elements ]INT;
# returns TRUE if the sandpiles a and b have the same values, FALSE otherwise #
OP = = ( SANDPILE a, b )BOOL:
BEGIN
BOOL result := TRUE;
FOR i TO elements WHILE result DO
FOR j TO elements WHILE ( result := a[ i, j ] = b[ i, j ] ) DO SKIP OD
OD;
result
END # = # ;
# returns TRUE if the sandpile s is stable, FALSE otherwise #
OP STABLE = ( SANDPILE s )BOOL:
BEGIN
BOOL result := TRUE;
FOR i TO elements WHILE result DO
FOR j TO elements WHILE result := s[ i, j ] < 4 DO SKIP OD
OD;
result
END # STABLE # ;
# returns the sandpile s after avalanches #
OP AVALANCHE = ( SANDPILE s )SANDPILE:
BEGIN
SANDPILE result := s;
WHILE BOOL had avalanche := FALSE;
FOR i TO elements DO
FOR j TO elements DO
IF result[ i, j ] >= 4 THEN
# unstable pile #
had avalanche := TRUE;
result[ i, j ] -:= 4;
IF i > 1        THEN result[ i - 1, j     ] +:= 1 FI;
IF i < elements THEN result[ i + 1, j     ] +:= 1 FI;
IF j > 1        THEN result[ i,     j - 1 ] +:= 1 FI;
IF j < elements THEN result[ i,     j + 1 ] +:= 1 FI
FI
OD
OD;
had avalanche
DO SKIP OD;
result
END # AVALANCHE # ;
# returns the result of adding the sandpile b to a, handling avalanches #
OP + = ( SANDPILE a, b )SANDPILE:
BEGIN
SANDPILE result;
FOR i TO elements DO
FOR j TO elements DO result[ i, j ] := a[ i, j ] + b[ i, j ] OD
OD;
# handle avalanches #
AVALANCHE result
END # + # ;
# prints the sandpile s #
PROC show sandpile = ( STRING title, SANDPILE s )VOID:
BEGIN
print( ( title, newline ) );
FOR i TO elements DO
FOR j TO elements DO
print( ( " ", whole( s[ i, j ], 0 ) ) )
OD;
print( ( newline ) )
OD
END # show sandpile # ;
# task test cases #
SANDPILE us    = ( ( 4, 3, 3 )
, ( 3, 1, 2 )
, ( 0, 2, 3 )
);
SANDPILE s1    = ( ( 1, 2, 0 )
, ( 2, 1, 1 )
, ( 0, 1, 3 )
);
SANDPILE s2    = ( ( 2, 1, 3 )
, ( 1, 0, 1 )
, ( 0, 1, 0 )
);
SANDPILE s3    = ( ( 3, 3, 3 )
, ( 3, 3, 3 )
, ( 3, 3, 3 )
);
SANDPILE s3_id = ( ( 2, 1, 2 )
, ( 1, 0, 1 )
, ( 2, 1, 2 )
);
SANDPILE t := us;
WHILE NOT STABLE t DO
show sandpile( "unstable:", t );
t := AVALANCHE t
OD;
show sandpile( "stable: ", t );
print( ( newline ) );
show sandpile( "s1:", s1 );
show sandpile( "s2:", s2 );
show sandpile( "s1 + s2:", s1 + s2 );
show sandpile( "s2 + s1:", s2 + s1 );
print( ( newline ) );
show sandpile( "s3:", s3 );
show sandpile( "s3_id:", s3_id );
print( ( newline ) );
print( ( "s3 + s3_id = s3 is ", IF s3 + s3_id = s3 THEN "TRUE" ELSE "FALSE" FI, newline ) );
show sandpile( "s3 + s3_id", s3 + s3_id );
print( ( "s3_id + s3 = s3 is ", IF s3 + s3_id = s3 THEN "TRUE" ELSE "FALSE" FI, newline ) );
show sandpile( "s3_id + s3", s3_id + s3 );
show sandpile( "s3_id + s3_id:", s3_id + s3_id )

END```
Output:
```unstable:
4 3 3
3 1 2
0 2 3
stable:
2 1 0
0 3 3
1 2 3

s1:
1 2 0
2 1 1
0 1 3
s2:
2 1 3
1 0 1
0 1 0
s1 + s2:
3 3 3
3 1 2
0 2 3
s2 + s1:
3 3 3
3 1 2
0 2 3

s3:
3 3 3
3 3 3
3 3 3
s3_id:
2 1 2
1 0 1
2 1 2

s3 + s3_id = s3 is TRUE
s3 + s3_id
3 3 3
3 3 3
3 3 3
s3_id + s3 = s3 is TRUE
s3_id + s3
3 3 3
3 3 3
3 3 3
s3_id + s3_id:
2 1 2
1 0 1
2 1 2
```

## C++

```#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 int sp_limit = 4;

class abelian_sandpile {
friend std::ostream& operator<<(std::ostream&, const 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&);
bool operator==(const abelian_sandpile&);

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;
}

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::operator==(const abelian_sandpile& other) {
return cells_ == other.cells_;
}

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();
}

abelian_sandpile operator+(const abelian_sandpile& a, const abelian_sandpile& b) {
abelian_sandpile c(a);
c += b;
return c;
}

std::ostream& operator<<(std::ostream& out, const abelian_sandpile& as) {
for (size_t i = 0; i < sp_cells; ++i) {
if (i > 0)
out << (as.column_index(i) == 0 ? '\n' : ' ');
out << as.cells_[i];
}
return 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()) {
std::cout << sp << "stable? " << sp.is_stable() << "\n\n";
sp.topple();
}
std::cout << sp << "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;
std::cout << "\ns2 + s1 = \n" << sum2;
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;
std::cout << "\ns3_id + s3_id = \n" << sum4;

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#

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

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 ] bi@ 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 ] tri@ (.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
```

## FutureBasic

```void local fn SandpilePrint( s(2,2) as ^long )
long r, c
for r = 0 to 2
for c = 0 to 2
printf @"%ld\t",s(r,c)
next
print
next
print
end fn

void local fn SandpileTopple( s(2,2) as ^long )
BOOL stable = NO
long r, c, value

while ( stable == NO )
stable = YES
for r = 0 to 2
for c = 0 to 2
value = s(r,c)
if ( value > 3 )
s(r,c) -= 4
if ( r > 0 ) then s(r-1,c)++
if ( r < 2 ) then s(r+1,c)++
if ( c > 0 ) then s(r,c-1)++
if ( c < 2 ) then s(r,c+1)++
print @"⇣ ⇣ ⇣ ⇣ ⇣"
print
fn SandpilePrint( s(0,0) )
stable = NO : break
end if
next
if ( stable == NO ) then break
next
wend
end fn

void local fn SandpileLoad( s(2,2) as ^long, values as CFStringRef )
long r, c, i = 0
for r = 0 to 2
for c = 0 to 2
s(r,c) = intval(mid(values,i,1))
i++
next
next
end fn

void local fn DoIt
long r, c, s(2,2), s1(2,2), s2(2,2), s3(2,2), s3_id(2,2)

// s
text @"Menlo-Bold" : print @"avalanche"
text @"Menlo" : print @"----------"
fn SandpileLoad( s(0,0), @"433312023" )
fn SandpilePrint( s(0,0) )
fn SandpileTopple( s(0,0) )

// s1
fn SandpileLoad( s1(0,0), @"120211013" )

// s2
fn SandpileLoad( s2(0,0), @"213101010" )

// s1 + s2
for r = 0 to 2
for c = 0 to 2
s(r,c) = s1(r,c) + s2(r,c)
next
next
text @"Menlo-Bold" : print @"s1 + s2"
text @"Menlo" : print @"----------"
fn SandpileTopple( s(0,0) )
fn SandpilePrint( s(0,0) )

// s2 + s1
for r = 0 to 2
for c = 0 to 2
s(r,c) = s2(r,c) + s1(r,c)
next
next
text @"Menlo-Bold" : print @"s2 + s1"
text @"Menlo" : print @"----------"
fn SandpileTopple( s(0,0) )
fn SandpilePrint( s(0,0) )

// s3
fn SandpileLoad( s3(0,0), @"333333333" )
text @"Menlo-Bold" : print @"s3"
text @"Menlo" : print @"----------"
fn SandpilePrint( s3(0,0) )

// s3_id
fn SandpileLoad( s3_id(0,0), @"212101212" )
text @"Menlo-Bold" : print @"s3_id"
text @"Menlo" : print @"----------"
fn SandpilePrint( s3_id(0,0) )

// s3 + s3_id
for r = 0 to 2
for c = 0 to 2
s(r,c) = s3(r,c) + s3_id(r,c)
next
next
text @"Menlo-Bold" : print @"s3+s3_id"
text @"Menlo" : print @"----------"
fn SandpilePrint( s(0,0) )
fn SandpileTopple( s(0,0) )

// s3_id + s3_id
for r = 0 to 2
for c = 0 to 2
s(r,c) = s3_id(r,c) + s3_id(r,c)
next
next
text @"Menlo-Bold" : print @"s3_id+s3_id"
text @"Menlo" : print @"-----------"
fn SandpilePrint( s(0,0) )
fn SandpileTopple( s(0,0) )

end fn

fn DoIt

HandleEvents```
Output:
```avalanche
----------
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

s1 + s2
----------
3   3   3
3   1   2
0   2   3

s2 + s1
----------
3   3   3
3   1   2
0   2   3

s3
----------
3   3   3
3   3   3
3   3   3

s3_id
----------
2   1   2
1   0   1
2   1   2

s3+s3_id
----------
5   4   5
4   3   4
5   4   5

⇣ ⇣ ⇣ ⇣ ⇣

1   5   5
5   3   4
5   4   5

⇣ ⇣ ⇣ ⇣ ⇣

2   1   6
5   4   4
5   4   5

⇣ ⇣ ⇣ ⇣ ⇣

2   2   2
5   4   5
5   4   5

⇣ ⇣ ⇣ ⇣ ⇣

3   2   2
1   5   5
6   4   5

⇣ ⇣ ⇣ ⇣ ⇣

3   3   2
2   1   6
6   5   5

⇣ ⇣ ⇣ ⇣ ⇣

3   3   3
2   2   2
6   5   6

⇣ ⇣ ⇣ ⇣ ⇣

3   3   3
3   2   2
2   6   6

⇣ ⇣ ⇣ ⇣ ⇣

3   3   3
3   3   2
3   2   7

⇣ ⇣ ⇣ ⇣ ⇣

3   3   3
3   3   3
3   3   3

s3_id+s3_id
----------
4   2   4
2   0   2
4   2   4

⇣ ⇣ ⇣ ⇣ ⇣

0   3   4
3   0   2
4   2   4

⇣ ⇣ ⇣ ⇣ ⇣

0   4   0
3   0   3
4   2   4

⇣ ⇣ ⇣ ⇣ ⇣

1   0   1
3   1   3
4   2   4

⇣ ⇣ ⇣ ⇣ ⇣

1   0   1
4   1   3
0   3   4

⇣ ⇣ ⇣ ⇣ ⇣

2   0   1
0   2   3
1   3   4

⇣ ⇣ ⇣ ⇣ ⇣

2   0   1
0   2   4
1   4   0

⇣ ⇣ ⇣ ⇣ ⇣

2   0   2
0   3   0
1   4   1

⇣ ⇣ ⇣ ⇣ ⇣

2   0   2
0   4   0
2   0   2

⇣ ⇣ ⇣ ⇣ ⇣

2   1   2
1   0   1
2   1   2
```

## Go

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

```{-# 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

```While=:2 :'u^:(0-.@:-:v)^:_'
index_of_maximum=: \$ #: (i. >./)@:,

frame=: ({.~ -@:>:@:\$)@:({.~ >:@:\$) :. ([;.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&.:frame@:([ 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=: (erode@:+ 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│
└─────────┴─────────┘
```

## Java

```import java.util.ArrayList;
import java.util.List;

public final class AbelianSandpileModel {

public static void main(String[] aArgs) {
Sandpile avalanche = new Sandpile(List.of( 4, 3, 3, 3, 1, 2, 0, 2, 3 ));
System.out.println("Avalanche reduction to stable state:");
avalanche.display();
System.out.println(" ==> ");
avalanche.stabilise();
avalanche.display();

Sandpile s1 = new Sandpile(List.of( 1, 2, 0, 2, 1, 1, 0, 1, 3 ));
Sandpile s2 = new Sandpile(List.of( 2, 1, 3, 1, 0, 1, 0, 1, 0 ));
Sandpile sum1 = s1.add(s2);
Sandpile sum2 = s2.add(s1);
System.out.println(System.lineSeparator() + "Commutativity of addition" + System.lineSeparator());
System.out.println("Sandpile1 + Sandpile2:");
sum1.display();
System.out.println("Sandpile2 + Sandpile1:");
sum2.display();
System.out.println("Sandpile1 + Sandpile2 = Sandpile2 + Sandpile1: " + sum1.equals(sum2));

Sandpile s3 = new Sandpile(List.of( 3, 3, 3, 3, 3, 3, 3, 3, 3 ));
Sandpile s3_id = new Sandpile(List.of( 2, 1, 2, 1, 0, 1, 2, 1, 2 ));
Sandpile sum3 = s3.add(s3_id);
Sandpile sum4 = s3_id.add(s3_id);
System.out.println(System.lineSeparator() + "Identity Sandpile" + System.lineSeparator());
System.out.println("Sandpile3 + Sandpile3_id:");
sum3.display();
System.out.println("Sandpile3_id + Sandpile3_id:");
sum4.display();
}

}

final class Sandpile {

public Sandpile(List<Integer> aList) {
if ( aList.size() != CELL_COUNT ) {
throw new IllegalArgumentException("Initialiser list must contain " + CELL_COUNT + " elements");
}
cells = new ArrayList<Integer>(aList);
}

public void stabilise() {
while ( ! isStable() ) {
topple();
}
}

public boolean isStable() {
return cells.stream().noneMatch( i -> i >= CELL_LIMIT );
}

public void topple() {
for ( int i = 0; i < CELL_COUNT; i++ ) {
if ( cells.get(i) >= CELL_LIMIT ) {
cells.set(i, cells.get(i) - CELL_LIMIT);
final int row = rowIndex(i);
final int col = colIndex(i);
if ( row > 0 ) {
increment(row - 1, col);
}
if ( row + 1 < ROW_COUNT ) {
increment(row + 1, col);
}
if ( col > 0 ) {
increment(row, col - 1);
}
if ( col + 1 < COL_COUNT ) {
increment(row, col + 1);
}
}
}
}

public Sandpile add(Sandpile aOther) {
List<Integer> list = new ArrayList<Integer>();
for ( int i = 0; i < CELL_COUNT; i++ ) {
list.add(cells.get(i) + aOther.cells.get(i));
}
Sandpile result = new Sandpile(list);
result.stabilise();
return result;
}

public boolean equals(Sandpile aOther) {
return cells.equals(aOther.cells);
}

public void display() {
for ( int i = 0; i < CELL_COUNT; i++ ) {
System.out.print(cells.get(i));
System.out.print( ( colIndex(i + 1) == 0 ) ? System.lineSeparator() : " ");
}
}

private void increment(int aRow, int aCol) {
final int index = cellIndex(aRow, aCol);
cells.set(index, cells.get(index) + 1);
}

private static int cellIndex(int aRow, int aCol) {
return aRow * COL_COUNT + aCol;
}

private static int rowIndex(int aCellIndex) {
return aCellIndex / COL_COUNT;
}

private static int colIndex(int aCellIndex) {
return aCellIndex % COL_COUNT;
}

private List<Integer> cells;

private static final int ROW_COUNT = 3;
private static final int COL_COUNT = 3;
private static final int CELL_COUNT = ROW_COUNT * COL_COUNT;
private static final int CELL_LIMIT = 4;

}
```
Output:
```Avalanche reduction to stable state:
4 3 3
3 1 2
0 2 3
==>
2 1 0
0 3 3
1 2 3

Commutativity of addition

Sandpile1 + Sandpile2:
3 3 3
3 1 2
0 2 3
Sandpile2 + Sandpile1:
3 3 3
3 1 2
0 2 3
Sandpile1 + Sandpile2 = Sandpile2 + Sandpile1: true

Identity Sandpile

Sandpile3 + Sandpile3_id:
3 3 3
3 3 3
3 3 3
Sandpile3_id + Sandpile3_id:
2 1 2
1 0 1
2 1 2
```

## jq

Adapted from Wren

Works with jq and gojq, the C and Go implementations of jq

```# `whilst/2` is like `while/2` but emits the final term rather than the first one
def whilst(cond; update):
def _whilst:
if cond then update | (., _whilst) else empty end;
_whilst;

# module Sandpile

def new(\$a): {\$a};

def 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]
];

def add(\$other):
. as \$in
| reduce range(0; .a|length) as \$i (\$in; .a[\$i] += \$other.a[\$i] );

def isStable:
all(.a[]; . <= 3);

# just topple once so we can observe intermediate results
def topple:
last(
label \$out
| foreach range(0; .a|length) as \$i (.;
if .a[\$i] > 3
then .a[\$i] += -4
| reduce neighbors[\$i][] as \$j (.; .a[\$j] += 1)
| ., break \$out
else .
end ) );

def tos:
. as \$in
| reduce range(0;3) as \$i ("";
reduce range(0;3) as \$j (.;
. + " \(\$in.a[3*\$i + \$j])" )
| . +"\n" );

# Some sandpiles:
def s1: new([1, 2, 0, 2, 1, 1, 0, 1, 3]);
def s2: new([2, 1, 3, 1, 0, 1, 0, 1, 0]);
def s3: new([range(0;9)|3]);
def s4: new([4, 3, 3, 3, 1, 2, 0, 2, 3]);

def s3_id: new([2, 1, 2, 1, 0, 1, 2, 1, 2]);

# For brevity
def report_add(\$s1; \$s2):
"\(\$s1|tos)\nplus\n\n\(\$s2|tos)\nequals\n\n\(\$s1 | add(\$s2) | until(isStable; topple) | tos)";

def task1:
"Avalanche of topplings:\n",
(s4
| (., whilst(isStable|not; topple))
| tos ) ;

def task2:
def s3_a: s1 | add(s2);
def s3_b: s2 | add(s1);

"Commutative additions:\n",
( (s3_b | until(isStable; topple)) as \$s3_b
| report_add(s1; s2),
"and\n\n\(s2|tos)\nplus\n\n\(s1|tos)\nalso equals\n\n\(\$s3_b|tos)" ) ;

def task3:
"Addition of identity sandpile:\n",
report_add(s3; s3_id);

def task4:
"Addition of identities:\n",
report_add(s3_id; s3_id);

task1, task2, task3, task4```
Output:

As for Wren.

## Julia

```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
```

## Lua

Uses Abelian sandpile model here, then extends..

```sandpile.__index = sandpile
sandpile.new = function(self, vals)
local inst = setmetatable({},sandpile)
inst.cell, inst.dim = {}, #vals
for r = 1, inst.dim do
inst.cell[r] = {}
for c = 1, inst.dim do
inst.cell[r][c] = vals[r][c]
end
end
return inst
end
sandpile.add = function(self, other)
local vals = {}
for r = 1, self.dim do
vals[r] = {}
for c = 1, self.dim do
vals[r][c] = self.cell[r][c] + other.cell[r][c]
end
end
local inst = sandpile:new(vals)
inst:iter()
return inst
end

local s1 = sandpile:new{{1,2,0},{2,1,1},{0,1,3}}
local s2 = sandpile:new{{2,1,3},{1,0,1},{0,1,0}}
print("s1 =")  s1:draw()
print("\ns2 =")  s2:draw()
local s1ps2 = s1:add(s2)
print("\ns1 + s2 =")  s1ps2:draw()
local s2ps1 = s2:add(s1)
print("\ns2 + s1 =")  s2ps1:draw()
local topple = sandpile:new{{4,3,3},{3,1,2},{0,2,3}}
print("\ntopple, before =")  topple:draw()
topple:iter()
print("\ntopple, after =")  topple:draw()
local s3 = sandpile:new{{3,3,3},{3,3,3},{3,3,3}}
print("\ns3 =")  s3:draw()
local s3_id = sandpile:new{{2,1,2},{1,0,1},{2,1,2}}
print("\ns3_id =") s3_id:draw()
local s3ps3_id = s3:add(s3_id)
print("\ns3 + s3_id =")  s3ps3_id:draw()
local s3_idps3_id = s3_id:add(s3_id)
print("\ns3_id + s3_id =")  s3_idps3_id:draw()
```
Output:
```s1 =
1 2 0
2 1 1
0 1 3

s2 =
2 1 3
1 0 1
0 1 0

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

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

topple, before =
4 3 3
3 1 2
0 2 3

topple, after =
2 1 0
0 3 3
1 2 3

s3 =
3 3 3
3 3 3
3 3 3

s3_id =
2 1 2
1 0 1
2 1 2

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```

## Mathematica/Wolfram Language

```ClearAll[sp]
sp[s_List] + sp[n_Integer] ^:= sp[s] + sp[ConstantArray[n, Dimensions[s]]]
sp[s_List] + sp[t_List] ^:= Module[{dim, r, tmp, neighbours}, dim = Dimensions[s];
r = s + t;
While[Max[r] > 3,
r = ArrayPad[r, 1, 0];
tmp = Quotient[r, 4];
r -= 4 tmp;
r += RotateLeft[tmp, {0, 1}] + RotateLeft[tmp, {1, 0}] +
RotateLeft[tmp, {0, -1}] + RotateLeft[tmp, {-1, 0}];
r = ArrayPad[r, -1];
];
sp[r]
]
Format[x_sp] := Grid[x[[1]]]

s1 = sp[{{1, 2, 0}, {2, 1, 1}, {0, 1, 3}}];
s2 = sp[{{2, 1, 3}, {1, 0, 1}, {0, 1, 0}}];
s3 = sp[ConstantArray[3, {3, 3}]];
s3id = sp[{{2, 1, 2}, {1, 0, 1}, {2, 1, 2}}];

s1 + s2
s2 + s1
sp[{{4, 3, 3}, {3, 1, 2}, {0, 2, 3}}] + sp[0]
s3 + s3id === s3
s3id + s3id === s3id
```
Output:
```3	3	3
3	1	2
0	2	3

3	3	3
3	1	2
0	2	3

2	1	0
0	3	3
1	2	3

True

True```

## Nim

```import sequtils
import strutils

type SandPile = array[3, array[3, int]]

#---------------------------------------------------------------------------------------------------

iterator neighbors(i, j: int): tuple[a, b: int] =
## Yield the indexes of the neighbours of cell at indexes (i, j).
if i > 0:
yield (i - 1, j)
if i < 2:
yield (i + 1, j)
if j > 0:
yield (i, j - 1)
if j < 2:
yield (i, j + 1)

#---------------------------------------------------------------------------------------------------

proc print(s: openArray[SandPile]) =
## Print a list of sandpiles.
for i in 0..2:
for n, sp in s:
if n != 0:
stdout.write(if i == 1: " ⇨ " else: "   ")
stdout.write(sp[i].join(" "))
stdout.write('\n')

#---------------------------------------------------------------------------------------------------

proc printSum(s1, s2, s3: SandPile) =
## Print "s1 + s2 = s3".
for i in 0..2:
stdout.write(s1[i].join(" "))
stdout.write(if i == 1: " + " else: "   ", s2[i].join(" "))
stdout.write(if i == 1: " = " else: "   ", s3[i].join(" "))
stdout.write('\n')

#---------------------------------------------------------------------------------------------------

func isStable(sandPile: SandPile): bool =
## Return true if the sandpile is stable, else false.
result = true
for row in sandPile:
if row.anyit(it > 3):
return false

#---------------------------------------------------------------------------------------------------

proc topple(sandPile: var SandPile) =
## Eliminate one value > 3, propagating a grain to each neighbor.
for i, row in sandPile:
for j, val in row:
if val > 3:
dec sandPile[i][j], 4
for (i, j) in neighbors(i, j):
inc sandPile[i][j]
return

#---------------------------------------------------------------------------------------------------

proc stabilize(sandPile: var SandPile) =
## Stabilize a sandpile.
while not sandPile.isStable():
sandPile.topple()

#---------------------------------------------------------------------------------------------------

proc `+`(s1, s2: SandPile): SandPile =
## Add two sandpiles, stabilizing the result.
for row in 0..2:
for col in 0..2:
result[row][col] = s1[row][col] + s2[row][col]
result.stabilize()

#---------------------------------------------------------------------------------------------------

const Separator = "\n-----\n"

echo "Avalanche\n"
var s: SandPile = [[4, 3, 3], [3, 1, 2], [0, 2, 3]]
var list = @[s]
while not s.isStable():
s.topple()
list.add(s)
list.print()
echo Separator

echo "s1 + s2 == s2 + s1\n"
let s1 = [[1, 2, 0], [2, 1, 1], [0, 1, 3]]
let s2 = [[2, 1, 3], [1, 0, 1], [0, 1, 0]]
printSum(s1, s2, s1 + s2)
echo ""
printSum(s2, s1, s2 + s1)
echo Separator

echo "s3 + s3_id == s3\n"
let s3 = [[3, 3, 3], [3, 3, 3], [3, 3, 3]]
let s3_id = [[2, 1, 2], [1, 0, 1], [2, 1, 2]]
printSum(s3, s3_id, s3 + s3_id)
echo Separator

echo "s3_id + s3_id = s3_id\n"
printSum(s3_id, s3_id, s3_id + s3_id)
```
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
```

## OCaml

```(* https://en.wikipedia.org/wiki/Abelian_sandpile_model *)

module Make =
functor (M : sig val m : int val n : int end)
-> struct

type t = { grid : int array array ; unstable : ((int*int),unit) Hashtbl.t }

let make () = { grid = Array.init M.m (fun _ -> Array.make M.n 0); unstable = Hashtbl.create 10 }

let print {grid=grid} =
for i = 0 to M.m - 1
do for j = 0 to M.n - 1
do Printf.printf "%d " grid.(i).(j)
done
; print_newline ()
done

let add_grain {grid=grid;unstable=unstable} x y
= grid.(x).(y) <- grid.(x).(y) + 1
; if grid.(x).(y) >= 4 then
Hashtbl.replace unstable (x,y) () (* Use Hashtbl.replace for uniqueness *)

let topple ({grid=grid;unstable=unstable} as s) x y
= grid.(x).(y) <- grid.(x).(y) - 4
; if grid.(x).(y) < 4
then Hashtbl.remove unstable (x,y)
; let add_grain = add_grain s in match (x,y) with
(* corners *)
| (0,0) -> add_grain 1 0
; add_grain 0 1
| (0,n) when n = M.n - 1
-> add_grain 1 n
; add_grain 0 (n-1)
| (m,0) when m = M.m - 1
-> add_grain m 1
; add_grain (m-1) 0
| (m,n) when m = M.m - 1 && n = M.n - 1
-> add_grain ( m ) (n-1)
; add_grain (m-1) ( n )
(* sides *)
| (0,y) -> add_grain 1 y
; add_grain 0 (y+1)
; add_grain 0 (y-1)
| (m,y) when m = M.m - 1
-> add_grain ( m ) (y-1)
; add_grain ( m ) (y+1)
; add_grain (m-1) ( y )
| (x,0) -> add_grain (x+1) 0
; add_grain (x-1) 0
; add_grain ( x ) 1
| (x,n) when n = M.n - 1
-> add_grain (x-1) ( n )
; add_grain (x+1) ( n )
; add_grain ( x ) (n-1)
(* else *)
| (x,y) -> add_grain ( x ) (y+1)
; add_grain ( x ) (y-1)
; add_grain (x+1) ( y )
; add_grain (x-1) ( y )

let add_sand s n x y
= for i = 1 to n
do add_grain s x y
done

let avalanche ?(avalanche_print=fun _ -> ()) ({grid=grid;unstable=unstable} as s)
= while Hashtbl.length unstable > 0
do
let unstable' = Hashtbl.fold  (fun (x,y) () r -> (x,y) :: r) unstable []
in List.iter (fun (x,y) -> topple s x y; avalanche_print s ) unstable'
done

let init ?(avalanche_print=fun _ -> ()) f
= let s = { grid = Array.init M.m (fun x -> Array.init M.n (fun y -> f x y)) ; unstable = Hashtbl.create 10 }
in Array.iteri (fun x -> Array.iteri (fun y e -> if e >= 4 then Hashtbl.replace s.unstable (x,y) ())) s.grid
; avalanche_print s
; avalanche ~avalanche_print s
; s

let sandpile n
= let s = make ()
in add_sand s n (M.m/2) (M.n/2)
; avalanche s
; s

let (+.) {grid=a} {grid=b}
= let c = init (fun x y -> a.(x).(y) + b.(x).(y))
in avalanche c
; c
end

(* testing *)

let ()
= let module S = Make (struct let m = 3 let n = 3 end)
in let open S
in print_endline "Avalanche example"
; begin
let s0 = init ~avalanche_print:(fun s -> print s
; print_endline "  ↓")
(fun x y -> [| [| 4 ; 3 ; 3 |]
; [| 3 ; 1 ; 2 |]
; [| 0 ; 2 ; 3 |]
|].(x).(y))
in print s0
; print_endline "---------------"
end
; print_endline "Addition example"
; begin
let  s1 = init (fun x y -> [| [| 1 ; 2 ; 0 |]
; [| 2 ; 1 ; 1 |]
; [| 0 ; 1 ; 3 |]
|].(x).(y))
and s2 = init (fun x y -> [| [| 2 ; 1 ; 3 |]
; [| 1 ; 0 ; 1 |]
; [| 0 ; 1 ; 0|]
|].(x).(y))
and s3 = init (fun _ _ -> 3)
and s3_id = init (fun x y -> match (x,y) with
| ((0,0)|(2,0)|(0,2)|(2,2)) -> 2
| ((1,0)|(1,2)|(0,1)|(2,1)) -> 1
| _ -> 0)
in print s1
; print_endline "  +"
; print s2
; print_endline "  ="
; print (s1 +. s2)
; print_endline "------ Identity examples -----"
; print s3
; print_endline "  +"
; print s3_id
; print_endline "  ="
; print (s3 +. s3_id)
; print_endline "-----"
; print s3_id
; print_endline "  +"
; print s3_id
; print_endline "  ="
; print (s3_id +. s3_id)
end
```
Output:
```Avalanche example
4 3 3
3 1 2
0 2 3
↓
0 4 3
4 1 2
0 2 3
↓
1 4 3
0 2 2
1 2 3
↓
2 0 4
0 3 2
1 2 3
↓
2 1 0
0 3 3
1 2 3
↓
2 1 0
0 3 3
1 2 3
---------------
Addition example
1 2 0
2 1 1
0 1 3
+
2 1 3
1 0 1
0 1 0
=
3 3 3
3 1 2
0 2 3
------ Identity examples -----
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

```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

### Object Oriented

```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

```'''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

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 │
╰───┴───┴───┴───┴───╯	╰───┴───┴───┴───┴───╯	╰───┴───┴───┴───┴───╯	╰───┴───┴───┴───┴───╯	╰───┴───┴───┴───┴───╯
```

## Red

```Red [Purpose: "implement Abelian sandpile model"]

sadd: make object! [
comb: function [pile1 [series!] pile2 [series!]] [
repeat r 3 [
repeat c 3 [
pile2/:r/:c: pile2/:r/:c + pile1/:r/:c
]
]
check pile2
]
check: func [pile [series!]] [
stable: true row: col: none
repeat r 3[
repeat c 3[
if pile/:r/:c >= 4 [
stable: false
pile/:r/:c: pile/:r/:c - 4
row: r col: c
break]
]
if stable = false [break]
]
unless stable = false [print trim/with mold/only pile "[]" exit]
spill pile row col
]
spill: func [pile [series!] r [integer!] c [integer!]] [
neigh: reduce [
right: reduce [r c - 1] up: reduce [r + 1 c]
left: reduce [r c + 1] down: reduce [r - 1 c]
]
foreach n neigh [
unless any [(pile/(n/1) = none) (pile/(n/1)/(n/2) = none)] [
pile/(n/1)/(n/2): pile/(n/1)/(n/2) + 1
]
]
check pile
]
]

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]
]

ex: [
[4 3 3]
[3 1 2]
[0 2 3]
]

sadd/check copy/deep ex
sadd/comb copy/deep s1 copy/deep s2
sadd/comb copy/deep s2 copy/deep s1
sadd/comb copy/deep s3 copy/deep s3_id
sadd/comb copy/deep s3_id copy/deep s3_id
```
Output:
```
2 1 0
0 3 3
1 2 3

3 3 3
3 1 2
0 2 3

3 3 3
3 1 2
0 2 3

3 3 3
3 3 3
3 3 3

2 1 2
1 0 1
2 1 2

```

## REXX

```/*REXX program demonstrates a 3x3 sandpile model by addition with toppling & avalanches.*/
@.= 0;   size= 3                                 /*assign 0 to all grid cells; grid size*/
call init   1,      1 2 0    2 1 1    0 1 3      /*   "   grains of sand──► sandpile 1. */
call init   2,      2 1 3    1 0 1    0 1 0      /*   "      "    "   "  "     "     2  */
call init   3,      3 3 3    3 3 3    3 3 3      /*   "      "    "   "  "     "     3  */
call init 's3_id',  2 1 2    1 0 1    2 1 2      /*   "      "    "   "  "     "    3_id*/
call show   1                                    /*display sandpile  1  to the terminal.*/
call show   2                                    /*   "        "     2   "  "      "    */
call  add   1, 2, 'sum1',           'adding sandpile  s1  and  s2  yields:'
call show 'sum1'
call  add   2, 1, 'sum2',           'adding sandpile  s2  and  s1  yields:'
call show 'sum2'
call  eq? 'sum1', 'sum2'                         /*is  sum1  the same as  sum2 ?        */
call show   3
call show 's3_id'
call  add   3,     's3_id', 'sum3', 'adding sandpile  s3  and  s3_id  yields:'
call show 'sum3'
call  add 's3_id', 's3_id', 'sum4', 'adding sandpile  s3_id  and  s3_id  yields:'
call show 'sum4'
call  eq? 'sum4',  's3_id'                       /*is  sum4  the same as  s3_id ?       */
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
@get: procedure expose @.; parse arg grid,r,c    ;                      return @.grid.r.c
@set: procedure expose @.; parse arg grid,r,c,val;  @.grid.r.c= val;    return
tran: procedure; parse arg a;  if datatype(a,'W')  then a='s'a;         return a
/*──────────────────────────────────────────────────────────────────────────────────────*/
add:  parse arg x, y, t;    if t==''  then t= 'sum';  xx= tran(x);  yy= tran(y)
do r=1  for size;   do c=1  for size;   @.t.r.c= @.xx.r.c  +  @.yy.r.c
end   /*c*/
end   /*r*/;    say arg(4);             call norm t;            return
/*──────────────────────────────────────────────────────────────────────────────────────*/
eq?:  parse arg x, y;   xx= tran(x);      yy= tran(y);      ?= 1
do r=1  for size;         do c=1  for size;         ?= ?  &  (@.xx.r.c==@.yy.r.c)
end   /*c*/
end   /*r*/
if ?  then say  'comparison of '      xx       " and "       yy':  same.'
else say  'comparison of '      xx       " and "       yy':  not the same.'
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
init: parse arg x, \$;   xx= tran(x);   #= 0;      pad= left('', 8);   ind= left('', 45)
do r=1  for size;   do c=1  for size;     #= # + 1;           @.xx.r.c= word(\$, #)
end   /*c*/
end   /*r*/;                              shows= 0;             return
/*──────────────────────────────────────────────────────────────────────────────────────*/
norm: procedure expose @. size; parse arg x;        xx= tran(x);            recurse= 0
do r=1  for size;     do c=1  for size;     if @.xx.r.c<=size  then iterate
recurse= 1;              @.xx.r.c= @.xx.r.c - 4
call @set  xx, r-1, c  , @get(xx, r-1, c  )   +   1
call @set  xx, r+1, c  , @get(xx, r+1, c  )   +   1
call @set  xx, r  , c-1, @get(xx, r  , c-1)   +   1
call @set  xx, r  , c+1, @get(xx, r  , c+1)   +   1
end   /*c*/
end   /*r*/;    if recurse  then call norm xx;                  return
/*──────────────────────────────────────────────────────────────────────────────────────*/
show: parse arg x;  xx= tran(x);  say ind center("sandpile" xx,25,'─')  /*show the title*/
do r=1  for size;  \$=;      do c=1  for size;   \$= \$  @.xx.r.c  /*build a row.  */
end   /*c*/
say ind pad \$                                                   /*display a row.*/
end   /*r*/;     shows= shows + 1;     if shows==1  then say;   return
```
output   when using the default inputs:

(Shown at three-quarter size.)

```                                              ──────sandpile s1  ──────
0 0 0
0 0 0
0 0 0

───────sandpile s2───────
2 1 3
1 0 1
0 1 0
adding sandpile  s1  and  s2  yields:
──────sandpile sum1──────
3 3 3
3 1 2
0 2 3
adding sandpile  s2  and  s1  yields:
──────sandpile sum2──────
3 3 3
3 1 2
0 2 3
comparison of  sum1  and  sum2:  same.
───────sandpile s3───────
3 3 3
3 3 3
3 3 3
─────sandpile s3_id──────
2 1 2
1 0 1
2 1 2
adding sandpile  s3  and  s3_id  yields:
──────sandpile sum3──────
3 3 3
3 3 3
3 3 3
adding sandpile  s3_id  and  s3_id  yields:
──────sandpile sum4──────
2 1 2
1 0 1
2 1 2
comparison of  sum4  and  s3_id:  same.
```

## Ruby

```class Sandpile

def initialize(ar) = @grid = ar

def to_a = @grid.dup

def + (other)
res = self.to_a.zip(other.to_a).map{|row1, row2| row1.zip(row2).map(&:sum) }
Sandpile.new(res)
end

def stable? = @grid.flatten.none?{|v| v > 3}

def avalanche
topple until stable?
self
end

def == (other) = self.avalanche.to_a == other.avalanche.to_a

def topple
a = @grid
a.each_index do |row|
a[row].each_index do |col|
next if a[row][col] < 4
a[row+1][col] += 1 unless row == a.size-1
a[row-1][col] += 1 if row > 0
a[row][col+1] += 1 unless col == a.size-1
a[row][col-1] += 1 if col > 0
a[row][col]   -= 4
end
end
self
end

def to_s = "\n" + @grid.map {|row| row.join(" ") }.join("\n")

end

puts "Sandpile:"
puts demo = Sandpile.new( [[4,3,3], [3,1,2],[0,2,3]] )
puts "\nAfter the avalanche:"
puts demo.avalanche
puts "_" * 30,""

s1 = Sandpile.new([[1, 2, 0], [2, 1, 1], [0, 1, 3]] )
puts "s1: #{s1}"
s2 = Sandpile.new([[2, 1, 3], [1, 0, 1], [0, 1, 0]] )
puts "\ns2: #{s2}"
puts "\ns1 + s2 == s2 + s1: #{s1 + s2 == s2 + s1}"
puts "_" * 30,""

s3    = Sandpile.new([[3, 3, 3], [3, 3, 3], [3, 3, 3]] )
s3_id = Sandpile.new([[2, 1, 2], [1, 0, 1], [2, 1, 2]] )
puts "s3 + s3_id == s3: #{s3 + s3_id == s3}"
puts "s3_id + s3_id == s3_id: #{s3_id + s3_id == s3_id}"
```
Output:
```4 3 3
3 1 2
0 2 3

After the avalanche:

2 1 0
0 3 3
1 2 3
______________________________

s1:
1 2 0
2 1 1
0 1 3

s2:
2 1 3
1 0 1
0 1 0

s1 + s2 == s2 + s1: true
______________________________

s3 + s3_id == s3: true
s3_id + s3_id == s3_id: true
```

## Rust

```#[derive(Clone)]
struct Box {
piles: [[u8; 3]; 3],
}

impl Box {
fn init(piles: [[u8; 3]; 3]) -> Box {
let a = Box { piles };

if a.piles.iter().any(|&row| row.iter().any(|&pile| pile >= 4)) {
return a.avalanche();
} else {
return a;
}
}

fn avalanche(&self) -> Box {
let mut a = self.clone();
for (i, row) in self.piles.iter().enumerate() {
for (j, pile) in row.iter().enumerate() {
if *pile >= 4u8 {
if i > 0 {
a.piles[i - 1][j] += 1u8
}
if i < 2 {
a.piles[i + 1][j] += 1u8
}
if j > 0 {
a.piles[i][j - 1] += 1u8
}
if j < 2 {
a.piles[i][j + 1] += 1u8
}
a.piles[i][j] -= 4;
}
}
}
Box::init(a.piles)
}

fn add(&self, a: &Box) -> Box {
let mut b = Box {
piles: [[0u8; 3]; 3],
};
for (row, columns) in b.piles.iter_mut().enumerate() {
for (col, pile) in columns.iter_mut().enumerate() {
*pile = self.piles[row][col] + a.piles[row][col]
}
}
Box::init(b.piles)
}
}

fn main() {
println!(
"The piles demonstration avalanche starts as:\n{:?}\n{:?}\n{:?}",
[4, 3, 3],
[3, 1, 2],
[0, 2, 3]
);
let s0 = Box::init([[4u8, 3u8, 3u8], [3u8, 1u8, 2u8], [0u8, 2u8, 3u8]]);
println!(
"And ends as:\n{:?}\n{:?}\n{:?}",
s0.piles[0], s0.piles[1], s0.piles[2]
);
let s1 = Box::init([[1u8, 2u8, 0u8], [2u8, 1u8, 1u8], [0u8, 1u8, 3u8]]);
let s2 = Box::init([[2u8, 1u8, 3u8], [1u8, 0u8, 1u8], [0u8, 1u8, 0u8]]);
let s1_2 = s1.add(&s2);
let s2_1 = s2.add(&s1);
println!(
"The piles in s1 + s2 are:\n{:?}\n{:?}\n{:?}",
s1_2.piles[0], s1_2.piles[1], s1_2.piles[2]
);
println!(
"The piles in s2 + s1 are:\n{:?}\n{:?}\n{:?}",
s2_1.piles[0], s2_1.piles[1], s2_1.piles[2]
);
let s3 = Box::init([[3u8; 3]; 3]);
let s3_id = Box::init([[2u8, 1u8, 2u8], [1u8, 0u8, 1u8], [2u8, 1u8, 2u8]]);
let s4 = s3.add(&s3_id);
println!(
"The piles in s3 + s3_id are:\n{:?}\n{:?}\n{:?}",
s4.piles[0], s4.piles[1], s4.piles[2]
);
let s5 = s3_id.add(&s3_id);
println!(
"The piles in s3_id + s3_id are:\n{:?}\n{:?}\n{:?}",
s5.piles[0], s5.piles[1], s5.piles[2]
);
}
```
Output:
```The piles demonstration avalanche starts as:
[4, 3, 3]
[3, 1, 2]
[0, 2, 3]
And ends as:
[2, 1, 0]
[0, 3, 3]
[1, 2, 3]
The piles in s1 + s2 are:
[3, 3, 3]
[3, 1, 2]
[0, 2, 3]
The piles in s2 + s1 are:
[3, 3, 3]
[3, 1, 2]
[0, 2, 3]
The piles in s3 + s3_id are:
[3, 3, 3]
[3, 3, 3]
[3, 3, 3]
The piles in s3_id + s3_id are:
[2, 1, 2]
[1, 0, 1]
[2, 1, 2]
```

## V (Vlang)

Translation of: Go
```import strings

struct Sandpile {
mut:
a [9]int
}

const (
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 in row order
fn new_sandpile(a [9]int) Sandpile { return Sandpile{a} }

fn (s &Sandpile) plus(other &Sandpile) Sandpile {
mut b := [9]int{}
for i in 0..9 {
b[i] = s.a[i] + other.a[i]
}
return Sandpile{b}
}

fn (s &Sandpile) is_stable() bool {
for e in s.a {
if e > 3 {
return false
}
}
return true
}

// just topples once so we can observe intermediate results
fn (mut s Sandpile) topple() {
for i in 0..9 {
if s.a[i] > 3 {
s.a[i] -= 4
for j in neighbors[i] {
s.a[j]++
}
return
}
}
}

fn (s Sandpile) str() string {
mut sb := strings.new_builder(64)
for i in 0..3 {
for j in 0..3 {
sb.write_string("\${u8(s.a[3*i+j])} ")
}
sb.write_string("\n")
}
return sb.str()
}

fn main() {
println("Avalanche of topplings:\n")
mut s4 := new_sandpile([4, 3, 3, 3, 1, 2, 0, 2, 3]!)
println(s4)
for !s4.is_stable() {
s4.topple()
println(s4)
}

println("Commutative additions:\n")
s1 := new_sandpile([1, 2, 0, 2, 1, 1, 0, 1, 3]!)
s2 := new_sandpile([2, 1, 3, 1, 0, 1, 0, 1, 0]!)
mut s3_a := s1.plus(s2)
for !s3_a.is_stable() {
s3_a.topple()
}
mut s3_b := s2.plus(s1)
for !s3_b.is_stable() {
s3_b.topple()
}
println("\$s1\nplus\n\n\$s2\nequals\n\n\$s3_a")
println("and\n\n\$s2\nplus\n\n\$s1\nalso equals\n\n\$s3_b")

println("Addition of identity sandpile:\n")
s3 := new_sandpile([3, 3, 3, 3, 3, 3, 3, 3, 3]!)
s3_id := new_sandpile([2, 1, 2, 1, 0, 1, 2, 1, 2]!)
s4 = s3.plus(s3_id)
for !s4.is_stable() {
s4.topple()
}
println("\$s3\nplus\n\n\$s3_id\nequals\n\n\$s4")

println("Addition of identities:\n")
mut s5 := s3_id.plus(s3_id)
for !s5.is_stable() {
s5.topple()
}
print("\$s3_id\nplus\n\n\$s3_id\nequals\n\n\$s5")
}```
Output:
```Same as Go entry
```

## Wren

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
```