Mathematica 教學：Project Euler problem 213

Q: A 30 x 30 grid of squares contains 900 fleas, initially one flea per square. When a bell is rung, each flea jumps to an adjacent square at random (usually 4 possibilities, except for fleas on the edge of the grid or at the corners).

What is the expected number of unoccupied squares after 50 rings of the bell? Give your answer rounded to six decimal places.

next[i_,j_]:=Block[{temp},
temp=Table[0,{x,30},{y,30}];
Which[
i==1&&j==1,temp[[1,2]]=temp[[2,1]]=1/2,
i==1&&2<=j<=29,temp[[1,j-1]]=temp[[1,j+1]]=temp[[2,j]]=1/3,
i==1&&j==30,temp[[1,29]]=temp[[2,30]]=1/2,
2<=i<=29&&2<=j<=29,temp[[i-1,j]]=temp[[i+1,j]]=temp[[i,j-1]]=temp[[i,j+1]]=1/4,
2<=i<=29&&j==1,temp[[i-1,1]]=temp[[i+1,1]]=temp[[i,2]]=1/3,
2<=i<=29&&j==30,temp[[i-1,30]]=temp[[i+1,30]]=temp[[i,29]]=1/3,
i==30&&j==1,temp[[29,1]]=temp[[30,2]]=1/2,
i==30&&2<=j<=29,temp[[30,j-1]]=temp[[30,j+1]]=temp[[29,j]]=1/3,
i==30&&j==30,temp[[29,30]]=temp[[30,29]]=1/2
];
N@Flatten[Table[temp[[x,y]],{x,30},{y,30}]]
]

(*轉移矩陣*)

tm=next@@@Flatten[Table[{i,j},{i,30},{j,30}],1];

ans213=Total[Times@@@(Table[1,{i,900},{j,900}]-Nest[#.tm&,tm,50])]


nextsquare[i_,j_]:=
Which[
i==1&&j==1,RandomChoice[{{1,2},{2,1}}],
i==1&&2<=j<=29,RandomChoice[{{1,j-1},{1,j+1},{2,j}}],
i==1&&j==30,RandomChoice[{{1,29},{2,30}}],
2<=i<=29&&2<=j<=29,RandomChoice[{{i-1,j},{i+1,j},{i,j-1},{i,j+1}}],
2<=i<=29&&j==1,RandomChoice[{{i-1,1},{i+1,1},{i,2}}],
2<=i<=29&&j==30,RandomChoice[{{i-1,30},{i+1,30},{i,29}}],
i==30&&j==1,RandomChoice[{{29,1},{30,2}}],
i==30&&2<=j<=29,RandomChoice[{{30,j-1},{30,j+1},{29,j}}],
i==30&&j==30,RandomChoice[{{29,30},{30,29}}]
]

(* 所有方格座標 *)

flea=Flatten[Table[{i,j},{i,30},{j,30}],1];

DistributeDefinitions[nextsquare,flea];

(* 計算每隻跳蚤在50響鐘聲之後的座標，之後取餘集，實驗1000次，最後求平均 *)

Mean@ParallelTable[Length[Complement[flea,
Table[Nest[nextsquare@@#&,z,50],{z,flea}]]],{100}]

Tags: Mathematica 教學

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