Problem: Professor Random, known for his research on randomized algorithms, is now conducting an experiment on biased dice. His experiment consists of dropping a number of dice onto a plane, one after another from a fixed position above the plane. The dice fall onto the plane or dice already there, without rotating, and may roll and fall according to their property. Then he observes and records the status of the stack formed on the plane, specifically, how many times each number appears on the faces visible from above. All the dice have the same size and their face numbering is identical, which we show in Figure C-1.

<image>

Figure C-1: Numbering of a die

The dice have very special properties, as in the following.

(1) Ordinary dice can roll in four directions, but the dice used in this experiment never roll in the directions of faces 1, 2 and 3; they can only roll in the directions of faces 4, 5 and 6. In the situation shown in Figure C-2, our die can only roll to one of two directions.

<image>

Figure C-2: An ordinary die and a biased die

(2) The die can only roll when it will fall down after rolling, as shown in Figure C-3. When multiple possibilities exist, the die rolls towards the face with the largest number among those directions it can roll to.

<image>

Figure C-3: A die can roll only when it can fall

(3) When a die rolls, it rolls exactly 90 degrees, and then falls straight down until its bottom face touches another die or the plane, as in the case [B] or [C] of Figure C-4.

(4) After rolling and falling, the die repeatedly does so according to the rules (1) to (3) above.

<image>

Figure C-4: Example stacking of biased dice

For example, when we drop four dice all in the same orientation, 6 at the top and 4 at the front, then a stack will be formed as shown in Figure C-4.

<image>

Figure C-5: Example records

After forming the stack, we count the numbers of faces with 1 through 6 visible from above and record them. For example, in the left case of Figure C-5, the record will be "0 2 1 0 0 0", and in the right case, "0 1 1 0 0 1".

Input

The input consists of several datasets each in the following format.

> n
>  t1  f1
>  t2  f2
>  ...
>  tn  fn
>

Here, n (1 ≤ n ≤ 100) is an integer and is the number of the dice to be dropped. ti and fi (1 ≤ ti, fi ≤ 6) are two integers separated by a space and represent the numbers on the top and the front faces of the i-th die, when it is released, respectively.

The end of the input is indicated by a line containing a single zero.

Output

For each dataset, output six integers separated by a space. The integers represent the numbers of faces with 1 through 6 correspondingly, visible from above. There may not be any other characters in the output.

Sample Input


4
6 4
6 4
6 4
6 4
2
5 3
5 3
8
4 2
4 2
4 2
4 2
4 2
4 2
4 2
4 2
6
4 6
1 5
2 3
5 3
2 4
4 2
0


Output for the Sample Input


0 1 1 0 0 1
1 0 0 0 1 0
1 2 2 1 0 0
2 3 0 0 0 0






Example

Input

4
6 4
6 4
6 4
6 4
2
5 3
5 3
8
4 2
4 2
4 2
4 2
4 2
4 2
4 2
4 2
6
4 6
1 5
2 3
5 3
2 4
4 2
0


Output

0 1 1 0 0 1
1 0 0 0 1 0
1 2 2 1 0 0
2 3 0 0 0 0

Solution (in JAVA):

Solution:
import java.io.IOException;
import java.io.InputStream;
import java.util.*;
import java.util.function.BiFunction;
import java.util.function.Function;
import java.util.function.Supplier;
import java.util.stream.Collectors;
import java.util.stream.IntStream;

public class Main {
  final static int INF = 1 << 28;
  final static long MOD = 1_000_000_007;
  final static double GOLDEN_RATIO = (1.0 + Math.sqrt(5)) / 2.0;
  Scanner sc = new Scanner(System.in);

  public static void main(String[] args) {
    new Main().run();
  }


  static class Dice implements Comparable<Dice> {
    int val;
    int[] tonari;
    int offset;

    Dice() {
    }

    static Dice valueOf(int top, int front) {
      if (!LUT.containsKey(top)) {
        return null;
      }
      Dice me = new Dice();
      Dice atom = LUT.get(top);
      me.val = atom.val;
      me.tonari = atom.tonari;
      for (int i = 0; i < 4; ++i) {
        if (me.tonari[i] == front) {
          me.offset = i;
        }
      }
      return me;
    }

    Pair<Integer, Dice> roll(Point now, Set<Point> done) {
      int idx = -1;
      int max = 0;
      for (int i = 0; i < 4; ++i) {
        int n = tonari[(i + offset) % 4];
        if (n < 4) {
          continue;
        }
        int[] d = ofs[i];
        Point next = new Point(now);
        next.x += d[0];
        next.y += d[1];
        next.z--;
        if (done.contains(next) || next.z < 0) {
          continue;
        }
        if (max < n) {
          max = n;
          idx = i;
        }
      }
      int front = 0;
      switch (idx) {
        case 1:
        case 3:
          front = tonari[offset];
          break;
        case 0:
          front = val;
          break;
        case 2:
          front = 7 - val;
          break;
      }
      return new Pair<>(idx, Dice.valueOf(7 - max, front));
    }

    final static Map<Integer, Dice> LUT;

    final static int[][] ofs = {
        {0, 1},
        {1, 0},
        {0, -1},
        {-1, 0}
    };

    static {
      HashMap<Integer, Dice> map = new HashMap<>();
      Dice one = new Dice();
      one.val = 1;
      one.tonari = new int[]{
          3, 5, 4, 2
      };
      map.put(1, one);
      Dice two = new Dice();
      two.val = 2;
      two.tonari = new int[]{
          3, 1, 4, 6
      };
      map.put(2, two);
      Dice three = new Dice();
      three.val = 3;
      three.tonari = new int[]{
          6, 5, 1, 2
      };
      map.put(3, three);
      Dice four = new Dice();
      four.val = 4;
      four.tonari = new int[]{
          1, 5, 6, 2
      };
      map.put(4, four);
      Dice five = new Dice();
      five.val = 5;
      five.tonari = new int[]{
          3, 6, 4, 1
      };
      map.put(5, five);
      Dice six = new Dice();
      six.val = 6;
      six.tonari = new int[]{
          4, 5, 3, 2
      };
      map.put(6, six);
      LUT = Collections.unmodifiableMap(map);
    }

    @Override
    public int compareTo(Dice dice) {
      return this.val - dice.val;
    }

    @Override
    public String toString() {
      return "{top = " + val + " front = " + tonari[offset] + "}";
    }
  }

  static class Point implements Comparable<Point> {
    int x, y, z;

    Point(int x, int y, int z) {
      this.x = x;
      this.y = y;
      this.z = z;
    }

    Point(Point p) {
      x = p.x;
      y = p.y;
      z = p.z;
    }

    @Override
    public int compareTo(Point p) {
      if (x != p.x) {
        return x - p.x;
      }
      if (y != p.y) {
        return y - p.y;
      }
      return z - p.z;
    }

    @Override
    public String toString() {
      return "(" + x + ", " + y + ", " + z + ")";
    }
  }

  void run() {
    for (; ; ) {
      int n = ni();
      if (n == 0) {
        break;
      }
      TreeMap<Point, Dice> map = new TreeMap<>();
      Point atom = new Point(0, 0, INF);
      for (int i = 0; i < n; ++i) {
        int t = ni();
        int f = ni();
        Dice dice = Dice.valueOf(t, f);
        Point top = map.floorKey(atom);
        Point now = new Point(0, 0, 0);
        if (top != null) {
          now.z = top.z + 1;
        }
//        debug(i, now, map);
        for (; ; ) {
          Pair<Integer, Dice> ret = dice.roll(now, map.keySet());
          if (ret.s == null) {
            break;
          }
          int idx = ret.f;
          int[] d = Dice.ofs[idx];
          now.x += d[0];
          now.y += d[1];
          dice = ret.s;
          for (; ; ) {
            Point next = new Point(now);
            next.z--;
            if (next.z < 0) {
              break;
            }
            if (map.containsKey(next)) {
              break;
            }
            now = next;
//            debug("\t\t", now);
          }
//          debug("\t", idx, d, now, dice);
        }
//        debug(i, now);
        map.put(now, dice);
      }
      int[] cnt = new int[6];
      for (Point p : map.keySet()) {
        Point ue = new Point(p);
        ue.z++;
        if (map.containsKey(ue)) {
          continue;
        }
        Dice dice = map.get(p);
        cnt[dice.val - 1]++;
      }
      String ans = IntStream.of(cnt).mapToObj(Integer::toString).collect(Collectors.joining(" "));
      System.out.println(ans);
    }
  }

  int ni() {
    return Integer.parseInt(sc.next());
  }

  void debug(Object... os) {
    System.err.println(Arrays.deepToString(os));
  }

  /**
   * ??????????????????????????????
   *
   * @return a ??¨ b ????????§??¬?´???°
   */
  long gcd(long a, long b) {
    if (b == 0) {
      return a;
    }
    return gcd(b, a % b);
  }

  /**
   * ????????????????????????????????????
   *
   * @return mx + ny = gcd(m, n)??¨???????????????(x, y)?????????
   */
  Pair<Long, Long> gcd_ex(long m, long n) {
    long[][] mat = _gcd_ex(m, n);
    return new Pair<>(mat[0][0], mat[0][1]);
  }

  long[][] _gcd_ex(long m, long n) {
    if (n == 0) {
      return new long[][]{{1, 0}, {0, 1}};
    }
    long k = m / n;
    long[][] K = new long[][]{{0, 1}, {1, -k}};
    long[][] r = _gcd_ex(n, m % n);
    long[][] dst = new long[2][2];
    for (int y = 0; y < 2; ++y)
      for (int x = 0; x < 2; ++x)
        for (int i = 0; i < 2; ++i)
          dst[y][x] += r[y][i] * K[i][x];
    return dst;
  }

  /**
   * ??°?????????2???????????¨??????????????????????£?
   *
   * @return a^r (mod 1,000,000,007)
   */
  long pow(long a, long r) {
    long sum = 1;
    while (r > 0) {
      if ((r & 1) == 1) {
        sum *= a;
        sum %= MOD;
      }
      a *= a;
      a %= MOD;
      r >>= 1;
    }
    return sum;
  }

  /**
   * ???????????????
   * O(n)
   *
   * @return {}_nC_r
   */
  long C(int n, int r) {
    long sum = 1;
    for (int i = n; 0 < i; --i) {
      sum *= i;
      sum %= MOD;
    }
    long s = 1;
    for (int i = r; 0 < i; --i) {
      s *= i;
      s %= MOD;
    }
    sum *= pow(s, MOD - 2);
    sum %= MOD;

    long t = 1;
    for (int i = n - r; 0 < i; --i) {
      t *= i;
      t %= MOD;
    }
    sum *= pow(t, MOD - 2);
    sum %= MOD;

    return sum;
  }

  /**
   * ??????????????¢?´¢
   *
   * @param left  ??????
   * @param right ??????
   * @param f     ??¢?´¢????????¢??°
   * @param comp  ??????????????¢??°?????¢?´¢????????¨?????????Comparator.comparingDouble(Double::doubleValue)
   *              ??????????????¢??°?????¢?´¢????????¨?????????Comparator.comparingDouble(Double::doubleValue).reversed()
   * @return ?\?????????§?¨?x
   */
  double goldenSectionSearch(double left, double right, Function<Double, Double> f, Comparator<Double> comp) {
    double c1 = divideInternally(left, right, 1, GOLDEN_RATIO);
    double c2 = divideInternally(left, right, GOLDEN_RATIO, 1);
    double d1 = f.apply(c1);
    double d2 = f.apply(c2);
    while (right - left > 1e-9) {
      if (comp.compare(d1, d2) > 0) {
        right = c2;
        c2 = c1;
        d2 = d1;
        c1 = divideInternally(left, right, 1, GOLDEN_RATIO);
        d1 = f.apply(c1);
      } else {
        left = c1;
        c1 = c2;
        d1 = d2;
        c2 = divideInternally(left, right, GOLDEN_RATIO, 1);
        d2 = f.apply(c2);
      }
    }
    return right;
  }

  /**
   * [a,b]???m:n???????????????????????????
   */
  double divideInternally(double a, double b, double m, double n) {
    return (n * a + m * b) / (m + n);
  }

  /**
   * http://alexbowe.com/popcount-permutations/
   * bit????????£???????????°????°?????????????????????????????????¨???????????????
   * ex)
   * <pre>
   * for (int i = 0; i < 25; ++i) {
   *   int bits = (1 << i) - 1;
   *   long m = C(25, num);
   *   for (j = 0; j < m; ++j) {
   *     ...(25???????????????i???bit????????£?????????)
   *     if (bits != 0)
   *       bits = next_perm(bits);
   *   }
   * }
   * </pre>
   *
   * @param v ?????¨???bit???
   * @return ?¬????bit???
   */
  int next_perm(int v) {
    int t = (v | (v - 1)) + 1;
    return t | ((((t & -t) / (v & -v)) >> 1) - 1);
  }

  /**
   * http://qiita.com/p_shiki37/items/65c18f88f4d24b2c528b
   */
  static class FastScanner {
    private final InputStream in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

    public FastScanner(InputStream in) {
      this.in = in;
    }

    private static boolean isPrintableChar(int c) {
      return 33 <= c && c <= 126;
    }

    private boolean hasNextByte() {
      if (ptr < buflen) {
        return true;
      } else {
        ptr = 0;
        try {
          buflen = in.read(buffer);
        } catch (IOException e) {
          e.printStackTrace();
        }
        if (buflen <= 0) {
          return false;
        }
      }
      return true;
    }

    private int readByte() {
      if (hasNextByte()) return buffer[ptr++];
      else return -1;
    }

    private void skipUnprintable() {
      while (hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;
    }

    public boolean hasNext() {
      skipUnprintable();
      return hasNextByte();
    }

    public String next() {
      if (!hasNext()) throw new NoSuchElementException();
      StringBuilder sb = new StringBuilder();
      int b = readByte();
      while (isPrintableChar(b)) {
        sb.appendCodePoint(b);
        b = readByte();
      }
      return sb.toString();
    }

    public long nextLong() {
      if (!hasNext()) throw new NoSuchElementException();
      long n = 0;
      boolean minus = false;
      int b = readByte();
      if (b == '-') {
        minus = true;
        b = readByte();
      }
      if (b < '0' || '9' < b) {
        throw new NumberFormatException();
      }
      while (true) {
        if ('0' <= b && b <= '9') {
          n *= 10;
          n += b - '0';
        } else if (b == -1 || !isPrintableChar(b)) {
          return minus ? -n : n;
        } else {
          throw new NumberFormatException();
        }
        b = readByte();
      }
    }
  }

  class BIT<T> {
    int n;
    ArrayList<T> bit;
    BiFunction<T, T, T> bif;

    /**
     * 1-indexed ???Binary Indexed Tree????§??????????
     *
     * @param n   ??????
     * @param bif ?????¨???????????¢??°
     * @param sup ?????????
     */
    BIT(int n, BiFunction<T, T, T> bif, Supplier<T> sup) {
      this.n = n;
      bit = new ArrayList<>(n + 1);
      for (int i = 0; i < n + 1; ++i) {
        bit.add(sup.get());
      }
      this.bif = bif;
    }

    /**
     * i??????????????????v??§??´??°??????
     *
     * @param i index
     * @param v ??°?????????
     */
    void set(int i, T v) {
      for (int x = i; x <= n; x += x & -x) {
        bit.set(x, bif.apply(bit.get(x), v));
      }
    }

    /**
     * ?????¨??????
     *
     * @param defaultValue ?????????
     * @param i            index
     * @return [1, i]?????§f????????¨????????????
     */
    T reduce(T defaultValue, int i) {
      T ret = defaultValue;
      for (int x = i; x > 0; x -= x & -x) {
        ret = bif.apply(ret, bit.get(x));
      }
      return ret;
    }
  }

  class SegmentTree<T> {
    int n;
    ArrayList<T> dat;
    BiFunction<T, T, T> bif;
    Supplier<T> sup;

    /**
     * 0-indexed ???Segment Tree????§??????????
     *
     * @param n_  ????±???????
     * @param bif ?????¨???????????¢??°
     * @param sup ?????????
     */
    SegmentTree(int n_, BiFunction<T, T, T> bif, Supplier<T> sup) {
      n = 1;
      while (n < n_) n *= 2;

      dat = new ArrayList<>(2 * n - 1);
      for (int i = 0; i < 2 * n - 1; ++i) {
        dat.add(sup.get());
      }
      this.bif = bif;
      this.sup = sup;
    }

    /**
     * k??????????????????v??§??´??°??????
     *
     * @param k index
     * @param v ??°?????????
     */
    void set(int k, T v) {
      k += n - 1;
      dat.set(k, v);
      while (k > 0) {
        k = (k - 1) / 2;
        dat.set(k, bif.apply(dat.get(k * 2 + 1), dat.get(k * 2 + 2)));
      }
    }

    /**
     * ?????¨??????
     *
     * @param l ?????????
     * @param r ?????????
     * @return [l, r)??§?????????bif????????¨?????????????????????
     */
    T reduce(int l, int r) {
      return _reduce(l, r, 0, 0, n);
    }

    T _reduce(int a, int b, int k, int l, int r) {
      if (r <= a || b <= l) return sup.get();
      if (a <= l && r <= b) return dat.get(k);
      T vl = _reduce(a, b, k * 2 + 1, l, (l + r) / 2);
      T vr = _reduce(a, b, k * 2 + 2, (l + r) / 2, r);
      return bif.apply(vl, vr);
    }
  }

  static class Pair<F extends Comparable<F>, S extends Comparable<S>> implements Comparable<Pair<F, S>> {
    F f;
    S s;

    Pair() {
    }

    Pair(F f, S s) {
      this.f = f;
      this.s = s;
    }

    Pair(Pair<F, S> p) {
      f = p.f;
      s = p.s;
    }

    @Override
    public int compareTo(Pair<F, S> p) {
      if (f.compareTo(p.f) != 0) {
        return f.compareTo(p.f);
      }
      return s.compareTo(p.s);
    }

    @Override
    public int hashCode() {
      return f.hashCode() ^ s.hashCode();
    }

    @Override
    public boolean equals(Object o) {
      if (this == o) {
        return true;
      }
      if (o == null || this.f == null || this.s == null) {
        return false;
      }
      if (this.getClass() != o.getClass()) {
        return false;
      }
      Pair p = (Pair) o;
      return this.f.equals(p.f) && this.s.equals(p.s);
    }

    @Override
    public String toString() {
      return "{" + f.toString() + ", " + s.toString() + "}";
    }
  }
}