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问题:

I just came across with this interesting question from my colleague. I'm trying now, but meanwhile I thought I could share it here.

With the password grid shown in the Android home screen, how many valid passwords are possible? min password length: 4 max: 9 (correct me if I'm wrong)

回答1:

Summary

The full combinations of 4 to 9 distinctive numbers, minus the combinations which include invalid "jump"s.

The Long Version

The rule for Android 3x3 password grid:

one point for once
cannot "jump" over a point

The author of the original post used Mathematica to generate all 985824 combinations.

Because there is no "jump", several pairs of consecutive points are invalid.

Delete all invalid combinations to reach the result.

The combinations for 4-to-9-point paths are respectively 1624, 7152, 26016, 72912, 140704, 140704.

The Original Post In Chinese

The reference is from guokr, a site alike Stack Exchange Skeptics in the form of blogs.

回答2:

I know this question is old, but I answered it in another question (before finding this question) with a brute force approach in python, so adding it here for posterity:

pegs = {
    1: {3:2, 7:4, 9:5},
    2: {8:5},
    3: {1:2, 7:5, 9:6},
    4: {6:5},
    5: {},
    6: {4:5},
    7: {1:4, 3:5, 9:8},
    8: {2:5},
    9: {1:5, 3:6, 7:8}
}

def next_steps(path):
    return (n for n in range(1,10) if (not path or n not in path and 
                                       (n not in pegs[path[-1]] 
                                        or pegs[path[-1]][n] in path)))

def patterns(path, steps, verbose=False):
    if steps == 0:
        if verbose: print(path)
        return 1
    return sum(patterns(path+[n], steps-1) for n in next_steps(path))

So you can list all the # of patterns for any number of steps:

>>> [(steps, patterns([], steps)) for steps in range(1,10)]
[(1, 9),
 (2, 56),
 (3, 320),
 (4, 1624),
 (5, 7152),
 (6, 26016),
 (7, 72912),
 (8, 140704),
 (9, 140704)]
>>> sum(patterns([], steps) for steps in range(4,10))
389112

This is not the most efficient way of solving it because you could use reflections and only calculate a 4*corner + 4*mid-edge + 1*middle, e.g.:

>>> patterns([], 6) == 4*patterns([1], 5) + 4*patterns([2], 5) + patterns([5], 5)
True

回答3:

I brute forced the answer with a recursive search and i found a bigger answer, 487272. The algorithm is simple: trying it all. I quoted it down here. I didn't found any error in my code (but I'm not very skilled with c++). Sorry for the grammatical error I'm not English.

#include <iostream>
#include <stdlib.h>
using namespace std;

int combo;  //counter

void research(int Ipoints /*number of points already took*/, bool Icheck[9]/*points matrix*/,int Ilast/*last took point*/,
                   int Icomboval/*combination representation, only for printing purpose*/, int deep/*number of iteration, only for printing purpose*/)
{

    //  int numcall = 0;  //DEBUG


     for( int i=0; i<9; i++) //Controlling every free point in search of a valid way to contimue
          if( Icheck[i] == false )
          {  
              //Just for security, coping every variable in a new variable. I don't know how c++ works but I will make it works
              int points = Ipoints;
              int last = Ilast;
              int comboval = Icomboval;
              bool check[9];
                   for( int j=0; j<9; j++)
                        check[j] = Icheck[j];  

              int e1,e2;
              int middle = -1;
              e1=i; e2=last;  //Ccontrolling duble jumps
              if( e1 == 0 && e2 == 2 ) middle = 1;
              if( e1 == 3 && e2 == 5 ) middle = 4;
              if( e1 == 6 && e2 == 8 ) middle = 7;
              if( e1 == 0 && e2 == 6 ) middle = 3;
              if( e1 == 1 && e2 == 7 ) middle = 4;
              if( e1 == 2 && e2 == 8 ) middle = 5;
              if( e1 == 0 && e2 == 8 ) middle = 4;
              if( e1 == 6 && e2 == 2 ) middle = 4;

              e2=i; e1=last;  // in both way
              if( e1 == 0 && e2 == 2 ) middle = 1;
              if( e1 == 3 && e2 == 5 ) middle = 4;
              if( e1 == 6 && e2 == 8 ) middle = 7;
              if( e1 == 0 && e2 == 6 ) middle = 3;
              if( e1 == 1 && e2 == 7 ) middle = 4;
              if( e1 == 2 && e2 == 8 ) middle = 5;
              if( e1 == 0 && e2 == 8 ) middle = 4;
              if( e1 == 6 && e2 == 2 ) middle = 4;

              if((middle != -1) && !(check[middle])) {      
                        check[middle] = true;
                        points++;                      //adding middle points
                        comboval *= 10;
                        comboval += middle;
              }       

              check[i] = true;
              points++;           // get the point

              comboval*=10;
              comboval += i+1;

              if(points > 3)
              {
                  combo++; // every iteration over tree points is a valid combo

                // If you want to see they all, beware because printing they all is truly slow:
                    // cout << "Combination n. " << combo << " found: " << comboval  << " , points " << points << " with " << deep << " iterations\n";
              }

              if(points > 9)   //Just for sure, emergency shutdown,
              { exit(1); }


              research(points,check,i,comboval,deep+1); /*Recursive, here is the true program!*/

              // numcall++; //DEBUG
          }

       //   cout << "Ended " << deep << " , with " << numcall << " subs called\n";   // Only for debug purposes,remove with all the //DEBUG thing

}



int main ()
{
    combo = 0; //no initial knows combo
    bool checkerboard[9];
    for( int i=0; i<9; i++) checkerboard[i]=false; //blank initial pattern

    research(0/*no point taken*/,checkerboard,-1/*just a useless value*/,0/*blank combo*/,1/*it's the firs iteration*/); //let's search!

    cout << "\n"  ;            
    cout << "And the answer is ... " << combo << "\n"; //out

    char ans='\0';
    while(ans=='\0')
    {                   //just waiting
    cin >> ans;
    }

    return 0;
}