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 <bits/stdc++.h>
using namespace std;
const int pos_x[] = {0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4};
const int pos_y[] = {2, 1, 2, 3, 0, 1, 2, 3, 4, 1, 2, 3, 2};
const int table[][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, -1},
	{7, 10, -1},
	{10, -1},
};

struct Puzzle {
	vector<int> val;
	Puzzle(): val(13){}
	int score(){
		int res = 0;
		for (int i = 0; i < 13; i++){
			if (!val[i]) continue;
			int v = val[i];
			res += abs(pos_x[v] - pos_x[i]);
			res += abs(pos_y[v] - pos_y[i]);
		}
		return res;
	}
	void show(){
		for (int i = 0; i < 13; i++){
			printf("%d ", val[i]);
			if (i == 0 || i == 3 || i == 8 || i == 11 || i == 12){
				puts("");
			}
		}
	}
};

/*inline bool check(Puzzle &p)
{
	for (int i = 1; i <= 11; i++){
		if (p.val[i] != i) return false;
	}
	return true;
}*/

bool search(int d, Puzzle &p)
{
	if (p.score() == 0) return true;
	if (d == 0) return false;
	
	for (int i = 0; i < 13; i++){
		if (p.val[i]) continue;
		for (const int *to = &table[i][0]; ~(*to); to++){
			swap(p.val[i], p.val[*to]);
			//printf("%d %d\n", i, *to);
			//printf("%d\n", p.score());
			if (d - 1 - p.score() >= 0){
				if (search(d - 1, p)) return true;
			}
			swap(p.val[i], p.val[*to]);
		}
	}
	return false;
}

void solve(Puzzle &p)
{
	for (int d = 0; d <= 20; d++){
		if (search(d, p)){
			printf("%d\n", d);
			return;
		}
	}
	puts("NA");
}

int main()
{	
	Puzzle p;
	while (true){
		for (int i = 0; i < 13; i++){
			scanf("%d", &p.val[i]);
			if (p.val[i] == -1) return 0;
		}
		solve(p);
	}
}

C++: #include<bits/stdc++.h>
using namespace std;
   
typedef long long int64;
const int INF = 1 << 30;
   
vector< int > edge[13];
  
bool back(string str, int cnt){ //枝刈り大事だな
  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 diff = 0;
  for(int i = 0; i < 13; i++){
    if(str[i] != '0'){
      int pos = (str[i] >= 'A' ? str[i] - 'A' + 10 : str[i] - '0');
      diff += abs( X[pos] - X[i]) + abs( Y[pos] - Y[i]);
    }
  }
  return diff + cnt > 20;
}
  
  
int bfs(string mas){
  
  queue< string > que;
  map< string , int > memo;
  que.push( mas );
  memo[mas] = 0;
  while(!que.empty()){
  
    string p = que.front(); que.pop();
    if(back( p, memo[p])) continue;
    if(p == "0123456789AB0") return memo[p];
  
    for(int i = 0; i < 13; i++){
      if(p[i] == '0') {
        for(int j = 0; j < edge[i].size(); j++){
          string next = p;
          swap( next[i], next[edge[i][j]]);
          if(memo.find(next) != memo.end()) continue;
          que.push( next );
          memo[next] = memo[p] + 1;
        }
      }
    }
  }
  return -1;
}
  
int main(){
  vector< int > mas(13);
  
  edge[0].push_back(2);
  edge[1].push_back(2); edge[1].push_back(4);
  edge[2].push_back(0); edge[2].push_back(1); edge[2].push_back(3); edge[2].push_back(6);
  edge[3].push_back(2); edge[3].push_back(7);
  edge[4].push_back(5);
  edge[5].push_back(1); edge[5].push_back(4); edge[5].push_back(6); edge[5].push_back(9);
  edge[6].push_back(2); edge[6].push_back(5); edge[6].push_back(7); edge[6].push_back(10);
  edge[7].push_back(3); edge[7].push_back(6); edge[7].push_back(8); edge[7].push_back(11);
  edge[8].push_back(7);
  edge[9].push_back(5); edge[9].push_back(10);
  edge[10].push_back(6); edge[10].push_back(9); edge[10].push_back(11); edge[10].push_back(12);
  edge[11].push_back(7); edge[11].push_back(10);
  edge[12].push_back(10);
  
  while(cin >> mas[0], mas[0] != -1){
    for(int i = 1; i < 13; i++){
      cin >> mas[i];
    }
  
    string start = "";
    for(int i = 0; i < 13; i++){
      if(mas[i] >= 10) start += (char)(mas[i] - 10 + 'A');
      else start += mas[i] + '0';
    }
    int ret = bfs(start);
    if(~ret) cout << ret << endl;
    else cout << "NA" << endl;
  }
}