Abelian sandpile model/Identity
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).
You are encouraged to solve this task according to the task description, using any language you may know.
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
AArch64 Assembly
<lang AArch64 Assembly> /* 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" </lang>
- 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
<lang ARM Assembly> /* 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" </lang>
- 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
This Ada example works with Ada 2012 because of the use the aspect with Default_Component_Value.
The package specification for Abelian_Sandpile is: <lang Ada>-- 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; </lang> The package body for Abelian_Sandpile is <lang Ada>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; </lang> The main procedure performing the same tests as the C++ example is <lang Ada> 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; </lang>
- 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
C++
<lang cpp>#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;
}</lang>
- 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#) <lang fsharp> 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) </lang>
- 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.
<lang factor>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+</lang>
- 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
<lang go>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)
}</lang>
- 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
<lang 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</lang>
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
<lang 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 < [: >./ ,) </lang>
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│ └─────────┴─────────┘
Julia
<lang 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")
</lang>
- 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.. <lang Lua>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()</lang>
- 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
Nim
<lang 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) </lang>
- 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
Phix
<lang 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))})</lang>
- 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
<lang python>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") </lang>
- 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
<lang python>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 ------------------------
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()</lang>
- 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
Most of the logic is lifted straight from the Abelian sandpile model task.
<lang perl6>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, $asp.topple;
put %p<indent> ~ qww<identity 'test pile' toppled 'plus identity' toppled>».¢er($size * 4 + 1).join: %p<indent>;
.put for [Z~] @display».lines;
put ;
}</lang>
- 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 │ ╰───┴───┴───┴───┴───╯ ╰───┴───┴───┴───┴───╯ ╰───┴───┴───┴───┴───╯ ╰───┴───┴───┴───┴───╯ ╰───┴───┴───┴───┴───╯
REXX
<lang 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</lang>
- 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.
Wren
<lang ecmascript>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)</lang>
- 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
Rust
<lang 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]
);
} </lang>
- 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]