又一个由算24点引出的问题

manthanein

先规定：只允许使用正整数范围内进行的四则运算，分步计算中只能出现正整数。开局可用的数字是1~13。
问：
（1）能凑成3×8算出来的情形有多少？
（2）能凑成4×6算出来的情形有多少？
（3）能凑成2×12算出来的情形有多少？
（4）能凑成10+14算出来的情形有多少？
（5）能凑成12+12算出来的情形有多少？
（6）能算出来但不能通过以上五种方式算出来的情形有多少？

mathematica

三个字:穷举法!
四个数,24重排列,5种运算
13^4*24*5=3427320
也就340万种情况,完全可以穷举法!
穷举法万岁!

manthanein

我知道穷举行得通，但我不会敲代码啊

zeroieme

2019-01-03 12:00
旧版没有Groupings，自己写了个递归；
2019-01-03 18:00
然后不满意默认的ToString不会调整负号为减法，又自己写了个递归。
2019-01-04 12:30
修改了等价判断依据

1. Array[Subscript[a, #]&,4]//If[Length[#]==1,{#},Table[Table[{j+k,j-k,j k,j/k},{j,#0[#[[1;;i]]]},{k,#0[#[[i+1;;-1]]]}],{i,Length[#]-1}]//Flatten//Union]&//#/.(Range[4]//Permutations//MapIndexed[Subscript[a, #2[[1]]]->Subscript[a, #1]&,#]&/@#&)&//Flatten//Union//(#~Function[{ArithmeticPermutations,DataCombinations},Parallelize[Function[{OneSingleData},MapIndexed[Subscript[a, #2[[1]]]->#1&,OneSingleData]//{#,GatherBy[#,#[[2]]&]//(#[[All,1]]//Table[\[Sigma]->#[[1]],{\[Sigma],#[[2;;-1]]}]&)&/@#&//Flatten}&//Function[{RulesTypeData},Select[ArithmeticPermutations,Quiet[#/.RulesTypeData[[1]]]==24&]//Select[#,(Level[#,{0,\[Infinity]}]//Which[#===a,Null,Variables[#]=={},Null,#[[0]]===Times\[And]#[[1]]===-1,-#,#[[0]]===Power\[And]#[[2]]===-1,#^-1,True,#]&/@#&//Complement[#,{Null}]&//#/.RulesTypeData[[1]]&//(#\[Element]Integers\[And]#>0)&/@#&//And@@#&)&]&//GatherBy[#,{#[[0]],#/.RulesTypeData[[2]]//Factor}&]&//(SortBy[#,Length]//{#[[-1]],RulesTypeData[[1]]}&)&/@#&]]/@DataCombinations]//Join@@#&]~(Range[13]//Tuples[#,4]&//Sort/@#&//Union))&//(Print["自然数范围内计算的"<>ToString[Length[#]]<>"条"];#)&//Parallelize[Append[({#[[1,0]],Subsets[#[[1]],{0,\[Infinity]}]/.#[[2]]}//If[#[[1]]===Times,Table[MemberQ[#[[2]],x],{x,{3,4,2}}],{False,False,False}]~Join~If[#[[1]]===Plus,Table[MemberQ[#[[2]],y],{y,{10,12}}],{False,False}]&//Append[#,Or@@#//Not]/.{True->1,False->0}&),(#//{#[[1]]//Which[#[[0]]===Subscript,ToString[#,InputForm],#[[0]]===Plus,(#0/@(Select[#,Not[#[[0]]===Times\[And]#[[1]]===-1]&]//If[#[[0]]===Plus,List@@#,{#}]&)//Riffle[#,"+"]&//StringJoin)<>(Which[#1===0,"",#1[[0]]===Plus,"-("<>#2[#1]<>")",True,"-"<>#2[#1]]&[-Select[#,#[[0]]===Times\[And]#[[1]]===-1&],#0]),#[[0]]===Times,(Table[#0[v]//If[v[[0]]===Plus,"("<>#<>")",#]&,{v,Select[#,Not[#[[0]]===Power\[And]#[[2]]===-1]&]//If[#[[0]]===Times,List@@#,{#}]&}]//Riffle[#,"*"]&//StringJoin)<>(Which[#1===1,"",#1[[0]]===Subscript,"/"<>#2[#1],True,"/("<>#2[#1]<>")"]&[Select[#,#[[0]]===Power\[And]#[[2]]===-1&]^-1,#0])]&,ToString[#[[1]],InputForm]->ToString[#[[2]]]&/@#[[2]]}&//StringReplace[#[[1]],#[[2]]]&//StringReplace[#," "->""]&)]&/@#]&//Transpose//(Do[Total[#[[i]]]//ToString//{"（1）能凑成3*8算出来的情形有","（2）能凑成4*6算出来的情形有","（3）能凑成2*12算出来的情形有","（4）能凑成10+14算出来的情形有","（5）能凑成12+12算出来的情形有","（6）能算出来但不能通过以上五种方式算出来的情形有"}[[i]]<>#&//Print,{i,6}];#[[-1]])&

复制代码

自然数范围内计算的3858条
（1）能凑成3*8算出来的情形有686
（2）能凑成4*6算出来的情形有729
（3）能凑成2*12算出来的情形有716
（4）能凑成10+14算出来的情形有249
（5）能凑成12+12算出来的情形有311
（6）能算出来但不能通过以上五种方式算出来的情形有1483

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