Go: // BFS

package main

import (
	"bufio"
	"os"
		"fmt"
	"strconv"
	"strings"
)

var scanner = bufio.NewScanner(os.Stdin)

var p = [15]int{}
var g = [13][4]int {
	// -1は存在しないマス目
	{2,-1,-1,-1},
	{2,5,-1,-1},{0,1,3,6},{2,7,-1,-1},
	{5,-1,-1,-1},{1,4,6,9},{2,5,7,10},{3,6,8,11},{7,-1,-1,-1},
	{5,10,-1,-1},{6,9,11,12},{7,10,-1,-1},
	{10,-1,-1,-1},
}

/*
　  ０ １ ２ ３ ４
 0 |／|／|  |／|／|
 1 |／|　|  |　|／|
 2 |　|　|  |　|　|
 3 |／|　|  |　|／|
 4 |／|／|  |／|／|
*/

var row = [13]int{0,1,1,1,2,2,2,2,2,3,3,3,4}
var col = [13]int{2,1,2,3,0,1,2,3,4,1,2,3,2}

type wolf struct {
	m [13]int
}

type pair struct {
	first wolf
	second int
}

func abs(a int) int {
	if a < 0 {
		return -a
	}
	return a
}

func print_board(board [13]int) {
	fmt.Println("--------------------------")
	fmt.Printf("        [%2d]\n", board[0])
	fmt.Printf("    [%2d][%2d][%2d]\n", board[1], board[2], board[3])
	fmt.Printf("[%2d][%2d][%2d][%2d][%2d]\n", board[4], board[5], board[6], board[7], board[8])
	fmt.Printf("    [%2d][%2d][%2d]\n", board[9], board[10], board[11])
	fmt.Printf("        [%2d]\n", board[12])
	fmt.Println("--------------------------")
}

func main() {
	for scanner.Scan() {
		p1, _ := strconv.Atoi(scanner.Text())
		if p1 == -1 {
			break
		}
		p[0]= p1
		scanner.Scan()
		buf1 := strings.Split(scanner.Text(), " ")
		p2, _ := strconv.Atoi(buf1[0])
		p3, _ := strconv.Atoi(buf1[1])
		p4, _ := strconv.Atoi(buf1[2])
		p[1], p[2], p[3] = p2, p3, p4
		scanner.Scan()
		buf2 := strings.Split(scanner.Text(), " ")
		p5, _ := strconv.Atoi(buf2[0])
		p6, _ := strconv.Atoi(buf2[1])
		p7, _ := strconv.Atoi(buf2[2])
		p8, _ := strconv.Atoi(buf2[3])
		p9, _ := strconv.Atoi(buf2[4])
		p[4], p[5], p[6], p[7], p[8] = p5, p6, p7, p8, p9
		scanner.Scan()
		buf3 := strings.Split(scanner.Text(), " ")
		p10, _ := strconv.Atoi(buf3[0])
		p11, _ := strconv.Atoi(buf3[1])
		p12, _ := strconv.Atoi(buf3[2])
		p[9], p[10], p[11] = p10, p11, p12
		scanner.Scan()
		p13, _ := strconv.Atoi(scanner.Text())
		p[12] = p13
		S := make(map[wolf]struct{})
		st := wolf{}
		for i := 0; i < 13; i++ {
			st.m[i] = p[i]
		}
		Q := make([]pair, 0)
		S[st] = struct{}{}
		Q = append(Q, pair{st, 0})
		yet := true
		for len(Q) > 0 {
			at := Q[0].first
			print_board(at.m)
			cost := Q[0].second
			Q = Q[1:]
			ok := true
			for i :=1; i < 12; i++ {
				if at.m[i] != i {
					ok = false
				}
			}
			if ok {
				print_board(at.m)
				yet = false
				fmt.Println(cost)
				break
			}
			now := cost
			for i := 0; i < 13; i++ {
				if at.m[i] != 0 {
					now += abs(row[i]-row[at.m[i]]) + abs(col[i]-col[at.m[i]])
				}
			}
			if now > 20 {
				continue
			}
			for i := 0; i < 13; i++ {
				if at.m[i] == 0 {
					for j := 0; j < 4; j++ {
						if ^g[i][j] != 0 {
							at.m[i], at.m[g[i][j]] = at.m[g[i][j]], at.m[i]
							if _, ok := S[at]; !ok {
								S[at] = struct{}{}
								Q = append(Q, pair{at, cost+1})
							}
							at.m[i], at.m[g[i][j]] = at.m[g[i][j]], at.m[i]
						}
					}
				}
			}
		}
		if yet {
			fmt.Println("NA")
		}
	}
}

C++: #include <iostream>
#include <map>
#include <queue>
#include <cstdlib>
#include <vector>
using namespace std;
#define rep(i,n) for(int i=0;i<n;i++)

int to[13][5] = {
	{2,-1},
	{2,5,-1},
	{0,1,3,6,-1},
	{2,7,-1},
	{5,-1},
	{1,4,6,9,-1},
	{2,5,7,10,-1},
	{3,6,8,11,-1},
	{7,-1},
	{5,10,-1},
	{6,9,11,12},
	{7,10,-1},
	{10,-1}
};
int X[]={2,1,2,3,0,1,2,3,4,1,2,3,2};
int Y[]={0,1,1,1,2,2,2,2,2,3,3,3,4};

int solve(vector<int> data){
	queue< vector<int> > Q;
	map<vector<int>,int> done;
	Q.push(data);
	done[Q.front()] = 0;
	
	while(Q.size()){
		vector<int> q = Q.front(); Q.pop();
		int dif = 0;
		rep(i,13)if(q[i])dif += abs(X[q[i]]-X[i])+abs(Y[q[i]]-Y[i]);
		if(dif==0)return done[q];;
		if(dif+done[q]>20)continue;
		int curcost = done[q];
		rep(x,13){
			if(q[x] == 0){
				rep(i,5){
					if(!~to[x][i])break;
					swap(q[x],q[to[x][i]]);
					if(done.find(q) == done.end() ){
						done[q] = curcost+1;
						Q.push(q);
					}
					swap(q[x],q[to[x][i]]);
				}
			}
		}
	}
	return 21;
}
int main(){
	vector<int> data(13);
	while(1){
		rep(i,13)if(!(cin >> data[i]))return 0;
		int ans = solve(data);
		if(ans > 20)cout << "NA" << endl;
		else cout << ans << endl;
	}
}

C++: #include<iostream>
#include<algorithm>
#include<vector>
#include<queue>
#include<unordered_map>
#include<unordered_set>
#define lol(i,n) for(int i=0;i<n;i++)
#define mod 1000000007
typedef long long ll;
using namespace std;
int dx[13][4]={
  {3,0,0,0},
  {3,6,0,0},
  {1,2,4,7},
  {3,8,0,0},
  {6,0,0,0},
  {2,5,7,10},
  {3,6,8,11},
  {4,7,9,12},
  {8,0,0,0},
  {6,11,0,0},
  {7,10,12,13},
  {8,11,0,0},
  {11,0,0,0}
};
unordered_map<ll,int> mp;
ll Compress(int p[]){
  ll ans=0;
  for(int i=0;i<13;i++){
    ans*=13;
    ans+=p[i];
  }
  return ans;
}

int unzip[13];
void Unzip(ll x){
  for(int i=0;i<13;i++){
    unzip[12-i]=x%13;
    x/=13;
  }
}

int beflen=0;
queue<pair<ll,int> >Q;
void BFS(int p[],int len){
  //if(len!=beflen){cout<<len<<endl;beflen=len;}
  ll tmp=Compress(p);
  mp[tmp]=len;
  if(len==14)return;
  lol(i,13){
    if(p[i]==0){
      lol(j,4){
	int t=dx[i][j];
	if(t==0)break;
	t--;
	if(p[t]!=0){
	  swap(p[i],p[t]);
	  tmp=Compress(p);
	  if(mp[tmp]!=0)continue;
	  mp[tmp]=-1;
	  Q.push(make_pair(tmp,len+1));
	  swap(p[i],p[t]);
	}
      }
    }
  }
}

unordered_set<ll> st;
int ans;
void EFS(int p[],int len){
  //if(len!=beflen){cout<<len<<endl;beflen=len;}
  ll tmp=Compress(p);
  if(mp[tmp]>0){
    ans=min(ans,len+mp[tmp]-1);
    return;
  }
  if(len==10)return;
  lol(i,13){
    if(p[i]==0){
      lol(j,4){
	int t=dx[i][j];
	if(t==0)break;
	t--;
	if(p[t]!=0){
	  swap(p[i],p[t]);
	  tmp=Compress(p);
	  Q.push(make_pair(tmp,len+1));
	  swap(p[i],p[t]);
	}
      }
    }
  }
}
int main(){
  int p[13];
  lol(i,13)p[i]=i;p[12]=0;
  Q.push(make_pair(Compress(p),1));
  while(!Q.empty()){
    ll data=Q.front().first;
    int len=Q.front().second;
    Q.pop();
    if(mp[data]>0)continue;
    Unzip(data);
    BFS(unzip,len);
  }
  //cout<<"#"<<mp.size()<<endl;
  while(cin>>p[0]&&p[0]!=-1){
    for(int i=1;i<13;i++)cin>>p[i];
    ans=mod;
    Q.push(make_pair(Compress(p),0));
    while(!Q.empty()){
      ll data=Q.front().first;
      int len=Q.front().second;
      Q.pop();
      if(st.find(data)!=st.end())continue;
      st.insert(data);
      Unzip(data);
      EFS(unzip,len);
    }
    st.clear();
    if(ans>20)cout<<"NA"<<endl;
    else cout<<ans<<endl;
  }
  return 0;
}