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The Knight's Tour

Volume Number: 14 (1998)
Issue Number: 11
Column Tag: Programming Puzzles

The Knight's Tour

by F.C. Kuechmann

A seemingly simple problem with thousands of solutions

There is a class of problems that, though seemingly simple in concept, involve numbers so large that the time or material required for solution renders them effectively impossible to solve manually within a human lifetime. The "grains of wheat on a chessboard" described by Gamow [1988] is one of the simpler and more easily explained examples of this sort of problem. We start with a single grain of wheat on the first square of the board, two grains on the second, four on the third, eight on the fourth, and so on. Each square receiving twice the number of grains as the previous square, until all 64 squares are occupied. Simple enough, right? Gamow suggests that it would take the entire world's wheat production for 2000 years to fill the board! In this article we're going to look at a similar challenge and show how to solve it with your Macintosh.

The Problem

The "knight's tour" is another chessboard problem that involves deceptively large numbers (this one is simpler though - we don't need any wheat). In the game of chess the knight can move only in L-shaped patterns consisting of two squares one direction and a single square perpendicular to the direction of the first two. There are eight possible moves from any given starting square, shown in Figure 1. From the 16 squares at the middle of a chessboard all eight moves can be executed without leaving the board; Away from the center fewer moves are executable because the knight would end up off the board entirely. In general, a knight in row 1 or 8, or column 1 or 8, can execute only 4 moves. A knight in row 2 or 7, or column 2 or 7, can execute 6 moves. A knight on a corner square has only two executable moves.

Figure 1. The eight possible moves of a knight.

The object of the knight's tour is, from a given starting square, to visit each square on the board exactly once.

From many of the 64 possible starting squares no complete tours are possible, whereas others offer thousands. My experiments have shown that, if there is at least one complete tour from a given starting square, there is a large number of complete tours from that square.

Starting at square one, we test eight possible moves. Each time a move is executed, we must test another eight moves, then another eight, and another, until we have either visited all 64 squares or exhausted the possible moves. At square 63, eight moves must be considered. At square 62, 8^2 moves must be tested. More generally, at any given square n, the number of moves to be tested is 8^(64-n), with a significantly smaller number of executable moves. If the knight's tour were as straightforward as the grains of wheat problem, determining all possible solutions from any given starting square would be described by a geometric progression of 8+(8^2)+(8^3)..+(8^62)+(8^63) - or 8^64 tests. That is not a small number! In fact, we get eight new tests only when we actually execute a move, so the number of required tests is somewhat smaller.

A human with a chessboard, a knight, and a pad of paper to record moves, together with a well-conceived systematic method and fast hands, would require so much time to derive even a single solution that it is practically if not theoretically impossible using hand methods.

1 38 59 36 43 48 57 52
60 35 2 49 58 51 44 47
39 32 37 42 3 46 53 56
34 61 40 27 50 55 4 45
31 10 33 62 41 26 23 54
18 63 28 11 24 21 14 5
9 30 19 16 7 12 25 22
64 17 8 29 20 15 6 13

Figure 2. One complete knight's tour.

Niklaus Wirth [1976,1986] describes a trial-and-error approach to the problem using recursion and backtracking, with soucecode in Pascal [1976] and Modula-2 [1986]. Unlike a human, a computer can find solutions quite easily, although it can still take a great deal of time. With Wirth's method and CodeWarrior Pascal, the solution shown in Figure 2 requires 66,005,601 possible moves to be considered, 8,250,732 moves to be executed, and occupies 35 seconds of time on a PPC Mac 6500/225. A total of 107 solutions for the same starting square were found in just under two hours with 16,114,749,106 total position tests and 2,014,343,776 moves; 12 hours got more than a thousand solutions without testing more than a fraction of the possible moves. At that rate finding all solutions for the entire board would take a very long time. Because of symmetries, however, we could simply divide the board into four 16-square quadrants, Figure 3, and find the solutions for any quadrant, then calculate the solutions for the remaining quadrants by mirroring.

1

2

3

4

Figure 3. The board as quadrants.

Solutions for quadrant 2 mirror those for quadrant 1 on the vertical axis. Quadrants 3 and 4 mirror quadrants 1 and 2 on the horizontal axis.

The Program

The chess board occupies the leftmost 2/3 of the window, with a control panel on the right. Top left of the control panel are three times...

  1. total time
  2. time on current starting square
  3. time since the most recent solution was found.

Tour 1 has only number 1; Tour 2 has numbers 1 and 3.

Top right is the current rotation pattern for testing possible moves; the patterns are toggled between 1 and 2 by clicking the Pat button, and the position is selected by clicking the Rot button. Below the time are statistics on move tests, moves, backtracks, and the maximum backtrack level.

Buttons for starting, pausing, pattern and rotation selection, update board status, tour selection [eight options], and speed 1-9 follow at bottom right.

The EvUp/SolUp button toggles update status. In the default EvUp mode the board is updated whenever the events are tested, and those intervals are determined by the speed setting; in SolUp mode the board is updated only if and when a complete tour is achieved. Since testing for events and updating the board require large amounts of time relative to calculating the knight's moves, higher execution speeds offer less frequent event testing and board updating.

The Tours

The eight tour options are:

  • Tour 1 - finds one solution from the starting square and terminates; starting square selected by clicking on it.
  • Tour 2 - finds all solutions from starting square; starting square selected by clicking on it.
  • Tour 3 - starts at row 1, column 1 and moves through the entire board; if a solution is found, the next square becomes the starting square.
  • Tour 4 - starts at row 1, column 1 and moves through the entire board; finds all solutions for each square.
  • Quad 1 - finds all solutions for upper left quadrant.
  • Quad 2 - finds all solutions for upper right quadrant.
  • Quad 3 - finds all solutions for lower left quadrant.
  • Quad 4 - finds all solutions for lower right quadrant.

Menus

Several program options are selected by menu.

  • The File menu offers Run and Quit options.
  • The Delay menu allows selection of the pause after each complete tour. The minimum is no pause, the maximum 30 seconds, the default 1 second.
  • The Time menu determines the maximum time spent touring from a given starting square. In the cases of tours 1 and 2, pursuit of solutions ceases at that point; with the other six tours execution continues at the next square. The minimum selection is 2 minutes; the maximum specific interval 4 weeks. The default in no time limit.
  • The Save menu offers options of No save, Save as text, and Save as records. The default is No save.

Drawing the Chess Board

The empty chess board is drawn and optionally initialized by calling the procedure in Listing 1. It calls the code in Listing 2 64 times, passing row, column and flag values. The flag determines whether the global array ggChessBd, which stores the current moves, is affected. If the flag is TRUE, the array location ggChessBd[row,column] is set to zero. The flag is TRUE during initialization, FALSE during all other updates.

Listing 1.

ClearTheBoard
   procedure ClearTheBoard(flag:boolean);
         {clear the board by drawing blank squares at}
         {all 64 positions}
   var
      row,column:integer;
   begin
      for column:=1 to ggN do
         for row:=1 to ggN do
            SnuffKnight(row,column,flag);
   end;

The SnuffKnight procedure in Listing 2 erases the square at the location given by row and column by drawing a light or dark empty square.

Listing 2.

SnuffKnight
   procedure SnuffKnight (row,column:integer;flag:boolean);
      {erases the move number at the specified row and column}
      {position by drawing an empty square there}
   var
      vOffset,hOffset,height,width:integer;
      pictureRect:Rect;
      thePicture:PicHandle;
   begin
      SetPort(ggKnightWindow);
      ggTourRect:=ggKnightWindow^.portRect;

      if column mod 2=0 then
         begin
               {even # columns}
            if row mod 2=0 then
               thePicture:=GetPicture(ggcDK_ERASE_ID)
            else
               thePicture:=GetPicture(ggcLT_ERASE_ID);
         end
      else
         begin
               {odd # columns}
            if row mod 2>0 then
               thePicture:=GetPicture(ggcDK_ERASE_ID)
            else
               thePicture:=GetPicture(ggcLT_ERASE_ID);
         end;

      pictureRect:=thePicture^^.picFrame;
      hOffset:=(column-1) * ggcSQUARE_SIZE;
      vOffset:=(row-1) * ggcSQUARE_SIZE;
      height:=pictureRect.bottom-pictureRect.top;
      width:=pictureRect.right-pictureRect.left;
      PlacePict(ggTourRect,vOffset,hOffset,height,width);
      DrawPicture(thePicture,ggTourRect);
      if flag then
         ggChessBd[row,column]:=0;
   end;

Listing 3 shows how the board is refreshed after an update event. If the value in location ggChessBd[row,column] is non-zero (i.e. it holds a move number for the knight), that number is drawn by calling the DrawKnight code in Listing 4; Otherwise Listing 2 is called with a flag value of FALSE.

Listing 3.

UpDateBoard
   procedure UpDateBoard;
   var
      row,column,index:integer;
   begin
      for row:=1 to ggN do
         for column:=1 to ggN do
            begin
               index:=ggChessBd[row,column];
               if index>0 then
                  DrawKnight(row,column,index)
               else
                  SnuffKnight(row,column,FALSE);
            end;
   end;

Listing 4.

DrawKnight
   procedure DrawKnight(row,column,index:integer);
      {draws a square with a knight move # at the specified row}
      {and column}
   var
      vOffset,hOffset,height,width:integer;
      S:Str255;
   begin
      SetPort(ggKnightWindow);
      ggTourRect:=ggKnightWindow^.portRect;
      hOffset:=(column-1) * ggcSQUARE_SIZE;
      vOffset:=(row) * ggcSQUARE_SIZE;
      NumToString(index,S);
      if index<10 then
         begin
               {selectively erase squares with 1-9 when backtracking}
            if (ggIndex<10) and (ggMaxBak<ggNsqr) then
               SnuffKnight(row,column,FALSE);
            MoveTo(hOffset+20,vOffset);
         end
      else
         MoveTo(hOffset+14,vOffset);
      TextSize(20);
      ForeColor(blackColor);
      
      if column mod 2=0 then
         begin
            if row mod 2=0 then
               BackColor(redColor)
            else
               BackColor(whiteColor);
         end
      else
         begin   
            if row mod 2>0 then
               BackColor(redColor)
            else
               BackColor(whiteColor);
         end;      
      
      TextMode(srcCopy);
      DrawString(S);
      BackColor(whiteColor);
   end;

The Core Procedure

Most of the work in Knight's Tour is accomplished in Listing 5a. Testing for events, tracking the time, updating the board and statistics is accomplished by calling Listing 5b. The variable index holds the move number, x and y the column and row numbers. Variable k counts the possible moves 1-8. The global arrays gDeltaX and gDeltaY hold the number of squares to be moved on the X and Y axes to get to the position to be tested. Those values are added to the current row and column values to get the position to be tested. If the new position is on the board (i.e. both row and column in the 1..8 range, Listing 5c) and that board location unoccupied (Listing 5d), the move is made (Listing 5e); then if we don't have a complete tour (Listing 5f) the code in Listing 5a calls itself to make the next move. If the tested move can't be executed, the next possible move is tested. When no more possibilities exist, we drop out of loop and backtrack.

Listing 5a.

Try
   procedure Try(index,x,y:integer;var q:boolean);
   var
      k,column,row,dX,dY:integer;
      q1:boolean;
   begin
       k:=0;
       ggIndex:=index;
       q1:=FALSE;
       repeat
        Inc(gLoopCount);
        if gLoopCount>=ggUpdateInterval then
               DoUpdates(index:integer);
        Inc(gTests);
        if gTests>=ggcTenTo7th then
              begin
                 gTests:=0;
                 Inc(gTestOvr);
                 EraseTestCount;
                 UpdateTests(gTests,gTestOvr);
                 end;
        Inc(k);
        dX:=gDeltaX[k];
        dY:=gDeltaY[k];
        column:=x+dX;
        row:=y+dY;
        if SquareIsOnBoard(row,column) and
                           SquareNotOccupied(row,column) then
           begin
              MakeTheMove(row,column,index);
              Inc(gMoves);
              if gMoves>=ggcTenTo7th then
                 begin
                    gMoves:=0;
                    Inc(gMoveOvr);
                     EraseMoveCount;
                     UpdateMoves(gMoves,gMoveOvr);
                 end;

              if not CompleteTour(index) then
                 begin
                    Try(index+1,column,row,q1);

                     if (not q1) and (not ggQuitFlag) then
                        begin
                          ggChessBd[row,column]:=0;
                          Inc(gBakTrax);
                          if gBakTrax>=ggcTenTo7th then
                             begin
                                gBakTrax:=0;
                                  EraseBakTrax; 
                                Inc(gBakOvr);
                                UpDateBakTrax(gBakTrax,gBakOvr);
                             end;

                          if index<gLowestSoFar then
                             begin
                                gLowestSoFar:=index;
                                ggMaxBak:=index;
                                UpdateLowestRecurse(gLowestSoFar);
                             end;
                       end;
                 end
              else if ggTourNum in [1,3] then
                 begin
                    q1:=TRUE;
                    DoSolution(index,q1);
                    GetTime(gStime);
                     Inc(ggSolNum);
                     Inc(gSolNum);
                    UpdateSolNum(gSolNum);
                    DoTime;
                 end
              else
                 begin
                    DoTime;
                    DoSolution(index,TRUE);
                    if ggTourNum>3 then
                       DrawElapsed(0,3);
                    ggChessBd[row,column]:=0;
                    GetTime(gStime);
                     Inc(ggSolNum);
                     Inc(gSolNum);
                    UpdateSolNum(gSolNum);
                    DoTime;           
                 end;
           end;
      until (k>=ggcNumKnightMoves) or ggQuitFlag 
                             or ggErrFlag or gTimeFlag or q1;
      q:=q1;
   end;

Listing 5b.

DoUpdates
   procedure DoUpdates(index:integer);
   begin
      gLoopCount:=0;
      repeat
       if ggUpdateFlag or ggRedrawFlag then
          begin
              UpDateBoard;
              UpdateStats(index);
              Stall(ggStallVal);
              ggRedrawFlag:=FALSE;
           end;
         HandleEvent;
         DoTime;
      until (not ggPauseFlag) or gTimeFlag;
   end;

Listing 5c.

SquareIsOnBoard
   function SquareIsOnBoard(row,column:integer):boolean;
   begin
      If (column in [1..ggN]) and (row in [1..ggN]) then
         SquareIsOnBoard:=TRUE
      else
         SquareIsOnBoard:=FALSE;
   end; 

Listing 5d.

SquareNotOccupied
   function SquareNotOccupied(row,column:integer):boolean;
   begin     
      if ggChessBd[row,column]=0 then
          SquareNotOccupied:=TRUE
      else
          SquareNotOccupied:=FALSE;
   end;

Listing 5e.

MakeTheMove
   procedure MakeTheMove(row,column,index:integer);
   begin
      ggChessBd[row,column]:=index;
   end;

Listing 5f.

CompleteTour
   function CompleteTour(index:integer):boolean;
   begin
      if index<ggNsqr then
         CompleteTour:=FALSE
      else
         CompleteTour:=TRUE;
   end;

Initializing the Offsets

The values in arrays gDeltaX and gDeltaY are initialized to -2,-1,1 or 2 by passing them to the procedure in Listing 6. Two conditions must be accomodated in the initialization: the move pattern 1-2 held in the variable ggPattern, and the rotation 1-8 of that pattern held in ggRot. The move values are held in two, two-by-eight integer constant arrays, cDeltaX and cDeltaY. Using ggPattern and ggRot as indices, the values are transfered from the constant arrays to the proper locations in deltaX and deltaY.

Listing 6.

InitDelta
            {x value increases left-to-right}
            {y value increases top-to-bottom}
   procedure InitDelta(var deltaX,deltaY:ggDeltaType);
   type
      knightMoves=array[1..2,1..8] of integer;
   const
      cDeltaX:knightMoves=((-2,-1,1,2,2,1,-1,-2),
                                     (2,1,-1,-2,-2,-1,1,2));
      cDeltaY:knightMoves=((-1,-2,-2,-1,1,2,2,1),
                                     (-1,-2,-2,-1,1,2,2,1));
   var
      n,m,p:integer;
   begin
      n:=ggRot;
      m:=ggPattern;
      for p:=1 to 8 do
         begin
            deltaX[p]:=cDeltaX[m,n];
            deltaY[p]:=cDeltaY[m,n];
            Inc(n);
            if n>8 then
               n:=1;
         end;
   end;

Displaying the Test Pattern

Displaying the testing order for possible moves in the upper right corner of the window and changing that display as the Pat and Rot buttons are clicked is handled by the code in Listing 7. Pattern 1 tests begin at 10 o'clock and rotate clockwise. Pattern 2 tests begin at 2 o'clock and rotate counter-clockwise. Listing 7a copies the test position numbers from integer constant array cPat1 or cPat2 into the display position array gRotPos. Display positions are numbered 1-8 starting at ten o'clock and moving clockwise. The value of rotation variable ggRot is used to index into the appropriate constant array, determined by the value of ggPattern. Listing 7b is then called.

Listing 7a.

DrawPattern
   procedure DrawPattern;
   type
      patArray=array[1..15] of integer;
   const
      cPat1:patArray=(2,3,4,5,6,7,8,1,2,3,4,5,6,7,8);
      cPat2:patArray=(4,3,2,1,8,7,6,5,4,3,2,1,8,7,6);
   var
      n,p:integer;
   begin
      p:=ggRot-1;
      case ggPattern of
         1:
            begin   
               for n:=1 to 8 do
                  gRotPos[n]:=cPat1[n+(7-p)];
            end;
         2:
            begin   
               for n:=1 to 8 do
                  gRotPos[n]:=cPat2[n+p];
            end;
      end; {case}
      DrawPat;
   end;

Listing 7b sets the text size and colors, then calls Listing 7c to clear the display rectangle. Next it calls Listing 7d to draw the grid of white lines. Finally, using the values stored in array gRotPos, it draws the test order numbers on the grid.

Listing 7b.

DrawPat

   procedure DrawPat;
   var
      leftEdge,topEdge,squareSize,x,y,z,n:integer;
      S:Str255;
   const
      cXoff:integer=6;
      cYoff:integer=15;
   begin
      TextSize(gcPatSize);
      ForeColor(whiteColor);
      BackColor(blackColor);
      ClearRotRect;
      DrawMatrix;
      BackColor(whiteColor);
      leftEdge:=ggcClockLeft+gcPatLeft;
      topEdge:=10;
      TextMode(srcOr);
      squareSize:=20;
      S:='K';
      x:=(2*squareSize)+cXoff;
      y:=(2*squareSize)+cYoff;
      MoveTo(leftEdge+x,topEdge+y);
      DrawString(S);
      
      for n:=1 to 8 do
         begin
            z:=gRotPos[n];
            NumToString(z,S);
            case n of
               1:
                  begin
                        {position 1}
                     x:=cXoff;
                     y:=(squareSize)+cYoff;
                     MoveTo(leftEdge+x,topEdge+y);
                     DrawString(S);
                  end;
               2:
                  begin
                        {pos 2}
                     x:=(squareSize)+cXoff;
                     y:=cYoff;
                     MoveTo(leftEdge+x,topEdge+y);
                     DrawString(S);
                  end;
               3:
                  begin
                        {pos 3}
                     x:=(3*squareSize)+cXoff;
                     y:=cYoff;
                     MoveTo(leftEdge+x,topEdge+y);
                     DrawString(S);
                  end;
               4:
                  begin
                        {pos 4}
                     x:=(4*squareSize)+cXoff;
                     y:=(squareSize)+cYoff;
                     MoveTo(leftEdge+x,topEdge+y);
                     DrawString(S);
                  end;
               5:
                  begin
                        {pos 5}
                     x:=(4*squareSize)+cXoff;
                     y:=(3*squareSize)+cYoff;
                     MoveTo(leftEdge+x,topEdge+y);
                     DrawString(S);
                  end;
               6:
                  begin
                        {pos 6}
                     x:=(3*squareSize)+cXoff;
                     y:=(4*squareSize)+cYoff;
                     MoveTo(leftEdge+x,topEdge+y);
                     DrawString(S);
                  end;
               7:
                  begin
                        {pos 7}
                     x:=(squareSize)+cXoff;
                     y:=(4*squareSize)+cYoff;
                     MoveTo(leftEdge+x,topEdge+y);
                     DrawString(S);
                  end;
               8:
                  begin
                        {pos 8}
                     x:=cXoff;
                     y:=(3*squareSize)+cYoff;
                     MoveTo(leftEdge+x,topEdge+y);
                     DrawString(S);
                  end;
            end;
         end;
      ForeColor(blackColor);
      TextMode(srcCopy);   
   end;

Listing 7c calls Listing 7e to set the boundaries of the display rectangle and clears it to black, then adds the pattern and rotation numbers at the bottom.

Listing 7c.

ClearRotRect
   procedure ClearRotRect;
   var
      width,height,leftEdge,topEdge:integer;
      myRect:Rect;
      S:Str255;
   begin
      SetPort(ggKnightWindow);
      myRect:=ggKnightWindow^.portRect;
      leftEdge:=ggcClockLeft+gcPatLeft;
      topEdge:=10;
      width:=100;
      height:=100;
      SetRect(myRect,leftEdge,topEdge,width,height);   
      EraseRect(myRect);
      NumToString(ggPattern,S);
      S:=concat('Pattern #',S);
      MoveTo(leftEdge+10,topEdge+height+gcPatSize+5);
      DrawString(S);
      NumToString(ggRot,S);
      S:=concat('Rotation ',S);
      MoveTo(leftEdge+10,topEdge+height+(gcPatSize*2)+10);
      DrawString(S);      
   end;

Listing 7d.

DrawMatrix
   procedure DrawMatrix;
   var
      leftEdge,topEdge,n:integer;
   begin
      leftEdge:=ggcClockLeft+gcPatLeft;
      topEdge:=10;
      for n:=1 to 4 do
         begin
            MoveTo(leftEdge+(20*n),topEdge);
            Line(0,100);
         end;
      for n:=1 to 4 do
         begin
            MoveTo(leftEdge,(20*n)+topEdge);
            Line(100,0);   
         end;
   end;

Listing 7e.

SetRect
   procedure SetRect(var myRect:Rect;
                      leftEdge,topEdge,width,height:integer);
   begin
      myRect.top:=topEdge+myRect.top;
      myRect.bottom:=myRect.top+height+2;
      myRect.left:=leftEdge+myRect.left;
      myRect.right:=myRect.left+width;
   end;

Running the Program

There are four options that cannot be changed once a tour is underway, although three can be used at default values. The three that can be used at defaults are whether or not to save solutions (Save menu), move test pattern (Pat button), and the rotation of that pattern (Rot button). The fourth, mandatory option, is the tour number. The amount of additional housekeeping required depends on your choice of tour. Tours 1 and 2 require selection of the starting square by clicking on it. The other six tours have fixed starting squares as described previously. Once a sufficient number of selections have been made, the Run button becomes active. Clicking on it starts the search for solutions.

The default speed, button 2, is deliberately rather slow, with frequent board updates and every move displayed. As speeds are increased by clicking higher-numbered buttons, updates and event tests come less frequently.

If you want to see some solutions as rapidly as possible, select Tour 2, pattern 1, rotation 5, speed 8, starting square row 1, column 8.

For additional information, see the operating manual with the aps.

Source Code

Source code for a Macified implementation of Wirth's algorithm for the knight's tour is supplied for CodeWarrior Professional Pascal. Those readers familiar with Dave Mark's books may notice some resemblances between the sourcecode and some of that found in Dave's books - things like some of the names of constants and general structure of the event loop. I used the Timer project from the Macintosh Pascal Programming Primer, Vol. 1, by Dave Mark and Cartwright Reed, as a "skeleton". Most of the overlying code is mine, but underneath there's a bit of Mark and Reed code doing some of the housekeeping.

Bibliography and References

  • Gamow, George, One Two Three...Infinity, (New York: Dover Books, 1988).
  • Wirth, Niklaus, Algorithms + Data Structures = Programs, (Englewood Cliffs NJ: Prentice-Hall, 1976).
  • Wirth, Niklaus, Algorithms and Data Structures, (Englewood Cliffs NJ: Prentice-Hall, 1986).

F.C. Kuechmann is a hardware designer, programmer and consultant with degrees from the University of Illinois at Chicago and Clark College who is currently trying to find the time to do the soldering required to make the programmers' clock that he has designed so that he can read the time in hexadecimal. You can reach him at fk@aone.com.

 

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MooVee - Your Movies Guru 1.0 Device: iOS iPhone Category: Entertainment Price: $1.99, Version: 1.0 (iTunes) Description: MooVee helps you effortlessly manage your movies, on your iPhone. | Read more »
Geometry Wars 3: Dimensions (Games)
Geometry Wars 3: Dimensions 1.0.0 Device: iOS Universal Category: Games Price: $4.99, Version: 1.0.0 (iTunes) Description: Enjoy the next chapter in the award-winning Geometry Wars franchise and enjoy stunning, console-quality... | Read more »
CHAOS RINGS Ⅲ (Games)
CHAOS RINGS Ⅲ 1.0.0 Device: iOS Universal Category: Games Price: $19.99, Version: 1.0.0 (iTunes) Description: The newest addition to the popular smartphone RPG series is finally here! ・CHAOS RINGS Overview | Read more »
The Popular Insight Series of Travel Gui...
Getting around in a country when you can't understand the primary language can be tough. Fortunately there are several options available to help wold travellers with the important stuff like giving directions to a cab driver or asking where the... | Read more »
Desktop Dungeons is Now on the iPad Desp...
Desktop Dungeons has been a well-loved roguelike on PC for quite some time, and now it's finally available for the iPad! Just the iPad, though. Sorry iPhone users. [Read more] | Read more »
Moleskine Timepage – Calendar for iCloud...
Moleskine Timepage – Calendar for iCloud, Google & Exchange 1.0 Device: iOS iPhone Category: Productivity Price: $4.99, Version: 1.0 (iTunes) Description: The most elegant calendar for your pocket and wrist, Timepage is a... | Read more »
QuizUp Gets Social in its New Update
Plain Vanilla Corp has released a new and improved version of their popular trivia game, QuizUp. The app now emphasizes social play so you can challenge friends from all over the world. [Read more] | Read more »
The Deep (Games)
The Deep 1.0 Device: iOS Universal Category: Games Price: $1.99, Version: 1.0 (iTunes) Description: Swipe Controls Delve into the deep in this retro rogue-like! Swipe to move your diver around and keep away from the enemies as you... | Read more »
Sproggiwood (Games)
Sproggiwood 1.2.8 Device: iOS Universal Category: Games Price: $9.99, Version: 1.2.8 (iTunes) Description: Sproggiwood was developed for devices with at least 1GB of RAM. We recommend you only download Sproggiwood if your device... | Read more »
Battle of Gods: Ascension (Games)
Battle of Gods: Ascension 1.0 Device: iOS Universal Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: TURN-BASED TACTICAL COMBATFight tactical battles against the forces of Hades! In Battle of Gods: Ascension you play... | Read more »

Price Scanner via MacPrices.net

Apple refurbished 2014 13-inch Retina MacBook...
The Apple Store has Apple Certified Refurbished 2014 13″ Retina MacBook Pros available for up to $400 off original MSRP, starting at $979. An Apple one-year warranty is included with each model, and... Read more
What Would the ideal Apple Productivity Platf...
For the past four years I’ve kept a foot in both the Mac and iPad camps respectively. my daily computing hours divided about 50/50 between the two devices with remarkable consistency. However, there’... Read more
PageMeUp 1.2.1 Ten Dollar Page Layout Applica...
Paris, France-based Softobe, an OS X software development company, has announced that their PageMeUp v. 1.2.1, is available on the Mac App Store for $9.99. The license can be installed on up to 5... Read more
Eight New Products For USB Type-C Application...
Fresco Logic, specialists in advanced connectivity technologies and ICs, has introduced two new product families targeting the Type-C connector recently introduced across a number of consumer... Read more
Scripps National Spelling Bee Launches Buzzwo...
Scripps National Spelling Bee fans can monitor the action at the 2015 Spelling Bee with the new Buzzworthy app for iOS, Android and Windows mobile devices. The free Buzzworthy app provides friendly... Read more
13-inch 2.5GHz MacBook Pro on sale for $120 o...
B&H Photo has the 13″ 2.5GHz MacBook Pro on sale for $979 including free shipping plus NY sales tax only. Their price is $120 off MSRP, and it’s the lowest price for this model (except for Apple’... Read more
27-inch 3.3GHz 5K iMac on sale for $1899, $10...
B&H Photo has the new 27″ 3.3GHz 5K iMac on sale for $1899.99 including free shipping plus NY tax only. Their price is $100 off MSRP. Read more
Save up to $50 on iPad Air 2, NY tax only, fr...
B&H Photo has iPad Air 2s on sale for up to $50 off MSRP including free shipping plus NY sales tax only: - 16GB iPad Air 2 WiFi: $469 $30 off - 64GB iPad Air 2 WiFi: $549.99 $50 off - 128GB iPad... Read more
Updated Mac Price Trackers
We’ve updated our Mac Price Trackers with the latest information on prices, bundles, and availability on systems from Apple’s authorized internet/catalog resellers: - 15″ MacBook Pros - 13″ MacBook... Read more
New 13-inch 2.9GHz Retina MacBook Pro on sale...
B&H Photo has the 13″ 2.9GHz/512GB Retina MacBook Pro on sale for $1699.99 including free shipping plus NY tax only. Their price is $100 off MSRP, and it’s the lowest price for this model from... Read more

Jobs Board

Program Manager, *Apple* Community Support...
**Job Summary** Apple Support Communities ( discussions. apple .com) helps customers get the most from their Apple products and services by providing access to Read more
Senior Data Scientist, *Apple* Retail - Onl...
**Job Summary** Apple Retail - Online sells Apple products to customers around the world. In addition to selling Apple products with unique services such as iPad Read more
*Apple* Solutions Consultant - Retail Sales...
**Job Summary** As an Apple Solutions Consultant (ASC) you are the link between our customers and our products. Your role is to drive the Apple business in a retail Read more
*Apple* Watch SW Application Project Manager...
**Job Summary** The Apple Watch software team is looking for an Application Engineering Project Manager to work on new projects for Apple . The successful candidate Read more
Engineering Manager for *Apple* Maps on the...
…the Maps App Team get to take part in just about any new feature in Apple Maps, often contributing a majority of the feature work. In our day-to-day engineering work, we Read more
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