Euler Project question 68th: Magic 5-gon Ring

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The sum of three numbers on any line is equal to 9, clockwise, starting from the smallest outer ring, and the resulting sequence is: 432621213

and sequence

A 9-bit string; The maximum string that a triangular ring can form is 432621513.

Using the numbers 1 through 10, you can get 16-bit or 17-bit strings through different arrangements. What is the largest 16-bit string that can be formed with a five-point ring?

16-bit character, 10 in outer ring

17-bit character, 10 inner ring

Find the largest string

6,7,8,9,10 should be in outer circle, 1,2,3,4,5 inner circle

(1+2+3+4+5) +6+7+8+9+10 = 70, every three digits and 14

The following can actually be done by hand

The smallest number of outside laps begins: 6, in 1-5 find two number so that three number of the and is 14 only have 5, 3, Maximum, 5 to in the middle

Number of outer rings left: 7,8,9,10, number of inner laps left: 1,2,4.3 on the same line as 11, only 10+1=11

Number of outer rings left: 7,8,9, number of inner laps left: 2,4.1 on the same line as 13, only 9+4=11

Number of outer rings left: 7, 8, number of inner laps left: 2

So here's the thing.

The answer is: 6531031914842725

Below with Python expose cracked

Import a package that produces an arrangement

 from Import permutations

You should know what to do next.

Traverse all permutations, select the number that satisfies the above criteria

All Programs:

 fromItertoolsImportpermutationsImport time as timedefrun (): Digits=['1','2','3','4','5','6','7','8','9','Ten'] Perm=permutations (digits) Res=0 forPinchPerm:arr=Map (int,p) candidate= P[0] + p[1] + p[2] +p[3] + p[2] + p[4]+p[5] + p[4] + p[6]+p[7] + p[6] + p[8]+p[9] + p[8] + p[1]; #Print candidate        ifArr[0] > Arr[3]orArr[0]> Arr[5]orArr[0] > Arr[7]orArr[0]> Arr[9]:Continue        ifarr[1]==10orarr[2]==10orarr[4]==10orARR[8] ==10:Continue        if(arr[0]+arr[1]+arr[2])! = (Arr[3] + arr[2] + arr[4]):Continue        if(arr[0]+arr[1]+arr[2])! = (Arr[5] + arr[4] + arr[6]):Continue        if(arr[0]+arr[1]+arr[2])! = (Arr[7] + arr[6] + arr[8]):Continue        if(arr[0]+arr[1]+arr[2])! = (Arr[9] + arr[8] + arr[1]):Continue        ifInt (candidate) >res andLen (candidate) ==16: Res=Int (candidate)#Print res, arr                PrintResif __name__=='__main__': Start=Time.time () run ()Print "Running time:", (Time.time ()-start),'s'

Results

653103191484272530.3180000782 s

It's been a long time.

 Packageproject61; Public classp68{int[] p = {1,2,3,4,5,6,7,8,9,10}; voidrun () { for(inti=1;i<20;i++) {getnextperm (); System.out.println ("" +p[0]+p[1]+p[2]+p[3]+p[4]+p[5]+p[6]+p[7]+p[8]+p[9]); The String result="";  while(true){            if(! Getnextperm ()) Break; if(Checkresult ()) {String candidate= "" + p[0] + p[1] + p[2]                                + p[3] + p[2] + p[4]                                + p[5] + p[4] + p[6]                                + p[7] + p[6] + p[8]                                + p[9] + p[8] + p[1];            SYSTEM.OUT.PRINTLN (candidate); }        }            }     //similar to Quick sort    Booleangetnextperm () {intN =p.length; intI= N-1;  while(p[i-1]>=P[i]) {i--; if(i<1)return false; }        intj =N;  while(p[j-1]<=p[i-1]) {J= J-1; } Swap (I-1,j-1); I++; J=N;  while(i<j) {Swap (I-1,j-1); I++; J--; }                return true; }    BooleanCheckresult () {if(p[1]==10| |p[2]==10| |p[4]==10| |p[8]==10)return false; if(P[0] > p[3]| |p[0] > P[5]| |p[0] > P[7]| |p[0] > P[9])return false; if(P[0] + p[1]+ p[2]! = P[3] + p[2] +p[4])return false; if(P[0] + p[1]+ p[2]! = P[5] + p[4] +p[6])return false; if(P[0] + p[1]+ p[2]! = P[7] + p[6] +p[8])return false; if(P[0] + p[1]+ p[2]! = p[9] + p[8] +p[1])return false; return true; }    voidSwapintIintj) {        intk=P[i]; P[i]=P[j]; P[J]=K; }     Public Static voidMain (string[] args) {LongT0 =System.currenttimemillis (); NewP68 (). run (); LongT1 =System.currenttimemillis (); System.out.println ("Running Time:" + (T1-T0) + "MS"); }}

The arrangement of this own parameter, and then find the number that satisfies the condition

The program runs fast.

6531031914842725running Time:77ms

Reference links

Euler Project question 68th: Magic 5-gon Ring