Puzzling is Good for You

 Solution by Joey Hwang Since a marble can't jump over two or more marbles and the marbles can move forward only, if we have the pattern OOB(OBB) with the hole located left(right) to the pattern (O-orange and B-blue), the blue(orange) marble in the pattern will not have any chance to move. So our goal is to prevent generating the above situations. Keeping this in mind, you will find that there is only one way to move the marbles once you finished your first move. So the only two solutions for the problem are: 4 - 5, 6 - 4, 7 - 6, 5 - 7, 3 - 5, 2 - 3, 4 - 2, 6 - 4, 8 - 6, 9 - 8, 7 - 9, 5 - 7,  3 - 5, 1 - 3, 2 - 1, 4 - 2, 6 - 4, 8 - 6,  7 - 8, 5 - 7, 3 - 5, 4 - 3, 6 - 4, 5 - 6 and 6 - 5, 4 - 6, 3 - 4, 5 - 3, 7 - 5, 8 - 7,  6 - 8, 4 - 6, 2 - 4, 1 - 2, 3 - 1, 5 - 3,  7 - 5, 9 - 7, 8 - 9, 6 - 8, 4 - 6, 2 - 4,  3 - 2, 5 - 3, 7 - 5, 6 - 7, 4 - 6, 5 - 4 Both are done in 24 moves.
 Solution by Jensen Lai if the holes are numbered 1-9 starting from the left, the sequence is as follows: 4-5, 6-4, 7-6, 5-7, 3-5, 2-3,  4-2, 6-4, 8-6, 9-8, 7-9, 5-7,  3-5, 1-3, 2-1, 4-2, 6-4, 8-6,  7-8, 5-7, 3-5, 4-3, 6-4, 5-6. Total 24 moves. In words the solution is as follows: If there arises a situation where the marbles are arranged OOBB, where two oranges are next to two blues, there is a "block" where none of the marbles can move around the others. To finish the puzzle follow these steps: 1. move all possible oranges without creating a block. 2. move all possible blues without creating a block. 3. move all possible oranges without creating a block. 4. move all possible blues without creating a block. etc. Step 1 makes move 1.  Step 2 makes moves 2 and 3.  Step 3 makes moves 4-6.  Step 4 makes moves 7-10.  Step 5 makes moves 11-14.  Step 6 makes moves 15-18.  Step 7 makes moves 19-21.  Step 8 makes moves 22 and 23.  Step 9 makes move 24. Following this algorithm gives a solution to a puzzle with any number of marbles on each side of the gap. The least number of moves needed to complete the puzzle can be given by the formula N=(X+1)squared -1 where X is the number of marbles on one side of the gap.
 Solution by Michele Ely The only solution I have come up with takes 24 moves. 1-4 is the orange marbles, _ is the blank spot and 6-9 are the blue marbles. 1 2 3 4 _ 6 7 8 9  1 2 3 _ 4 6 7 8 9     01. (move 4 to the blank spot) 1 2 3 6 4 _ 7 8 9     02. (jump over 4 using 6)  1 2 3 6 4 7 _ 8 9     03. (move 7 to the blank spot) 1 2 3 6 _ 7 4 8 9     04. (jump over 7 using 4) 1 2 _ 6 3 7 4 8 9     05. (jump over 6 using 3) 1 _ 2 6 3 7 4 8 9     06. (move 2 to the blank spot) 1 6 2 _ 3 7 4 8 9     07. (jump over 2 using 6) 1 6 2 7 3 _ 4 8 9     08. (jump over 3 using 7) 1 6 2 7 3 8 4 _ 9     09. (jump over 4 using 8) 1 6 2 7 3 8 4 9 _     10. (move 9 to the blank spot) 1 6 2 7 3 8 _ 9 4     11. (jump over 9 using 4) 1 6 2 7 _ 8 3 9 4     12. (jump over 8 using 3) 1 6 _ 7 2 8 3 9 4     13. (jump over 7 using 2) _ 6 1 7 2 8 3 9 4     14. (jump over 6 using 1) 6 _ 1 7 2 8 3 9 4     15. (move 6 to the blank spot) 6 7 1 _ 2 8 3 9 4     16. (jump over 1 using 7) 6 7 1 8 2 _ 3 9 4     17. (jump over 2 using 8) 6 7 1 8 2 9 3 _ 4     18. (jump over 3 using 9) 6 7 1 8 2 9 _ 3 4     19. (move 3 to the blank spot) 6 7 1 8 _ 9 2 3 4     20. (jump over 9 using 2) 6 7 _ 8 1 9 2 3 4     21. (jump over 8 using 1) 6 7 8 _ 1 9 2 3 4     22. (move 8 to the blank spot) 6 7 8 9 1 _ 2 3 4     23. (jump over 1 using 9) 6 7 8 9 _ 1 2 3 4     24. (move 1 to the blank spot)
 Solution by John Birch Nifty puzzle ! There are some interesting mathematics in it. Thanks for the clean & handy Flash implementation. ------------------------------------------------------------ ... I solved it intuitively, in 24 moves, and found that backward it was move #    123456789 hole# 24 bbbbxoooo 23 bbbboxooo 22 bbbxobooo 21 bbxbobooo 20 bbobxbooo 19 bbobobxoo 18 bboboboxo 17 bboboxobo 16 bboxobobo 15 bxobobobo 14 xbobobobo 13 obxbobobo 12 obobxbobo 11 obobobxbo 10 obobobobx  9 oboboboxb  8 oboboxobb  7 oboxobobb  6 oxobobobb  5 ooxbobobb  4 ooobxbobb  3 ooobobxbb  2 oooboxbbb  1 oooxobbbb  0 ooooxbbbb    123456789 hole# Most simply put, I can specify the piece to be moved, in the move order : 467532468975312468753465 Since there is always only 1 hole to receive that piece, no other punctuation nor digits are needed by anyone, to perform the moves. This solution is topologically identical to three other solutions, since the holes & the move numbers (order) can each be reversed without changing the nature of the solution. As far as I can see, this is the only solution, which implies that rule 6 is unneeded. Here are the rules to solve the puzzle: { Only rules 2, 3, & 4 are necessary. } ------------------------------------------------------------ Move the Blue marbles to the left and the Orange - to the right, again leaving an empty hole in the middle.  1. The marbles move (or jump) one at a time, into the empty hole. {unneeded rule?} 2. The marbles move toward the opposing side only. (Blue - to the left, Orange - to the right), and never backward. {Colors are unnecessary, if you follow the rules, but make completion easier to verify.} 3. Any marble can move one step into the empty hole next to it.  4. A marble can jump over only the marble next to it, landing on the empty hole {immediately} beyond it. 5. They can't jump over two or more marbles. {except on separate moves. see rule #4} 6. The puzzle should be solved with a minimum number of moves.  {This might be a goal, but is not a usable rule. Is it possible to solve with other than 24 moves ?} ------------------------------------------------------------ In the format you requested the solution was  4-5, 6-4, 7-6, 5-7, 3-5, 2-3,  4-2, 6-4, 8-6, 9-8, 7-9, 5-7,  3-5, 1-3, 2-1, 4-2, 6-4, 8-6,  7-8, 5-7, 3-5, 4-3, 6-4, 5-6 24 moves
 Solution by Ian Pedder The puzzle can be solved in 24 moves as follows: 4 - 5 6 - 4 7 - 6 5 - 7 3 - 5 2 - 3 4 - 2 6 - 4 8 - 6 9 - 8 7 - 9 5 - 7 3 - 5 1 - 3 2 - 1 4 - 2 6 - 4 8 - 6 7 - 8 5 - 7 3 - 5 4 - 3 6 - 4 5 - 6 It seems that for puzzles of this sort, with N pieces of each color then the minimum number of moves required is N * (N + 2). Thus with N = 4 in this case requires 4 * 6 = 24 moves.
 Solution by Emrah Baskaya Thanks for your great puzzles page. It seems there can only be two solutions, the one I am sending and its mirror. I am extremely curious if there can be other ways to solve it (other than the mirror.) I'll go as far as saying nobody can solve it with more than 24 moves. The puzzle seems pretty much fixed and the only way is using "avoid moves leading to two marbles of same color next to each other in front of opposite color" pattern... ...here goes the solution: 6-5,4-6,3-4,5-3,7-5,8-7, 6-8,4-6,2-4,1-2,3-1,5-3, 7-5,9-7,8-9,6-8,4-6,2-4, 3-2,5-3,7-5,6-7,4-6,5-4 Done 24 moves. And the mirror of the solution starts of course with: 4-5, and goes on.
 Solution by Brian M. Dailey I had a bunch of fun with your oxbow puzzle. Here is the solution I came up with: number of moves 24, 4-5,6-4,7-6,5-7,3-5,2-3, 4-2,6-4,8-6,9-8,7-9,5-7, 3-5,1-3,2-1,4-2,6-4,8-6, 7-8,5-7,3-5,4-3,6-4,5-6: it works backwards too: 6-5,4-6... Thanks for the fun PS. The solutions can be written shorter, because there is only one open space on a board at any time. As long as you know the first move the rest the second numbers should match the first numbers. eg. solution 24 moves: 4-5,6,7,5,3,2, 4,6,8,9,7,5, 3,1,2,4,6,8, 7,5,3,4,6,5
 Solution by Nigel Wilson 4-5,6-4,7-6,5-7,3-5,2-3, 4-2,6-4,8-6,9-8,7-9,5-7, 3-5,1-3,2-1,4-2,6-4,8-6, 7-8,5-7,3-5,4-3,6-4,5-6 24 moves The secret is to avoid 2 consecutive of the same colour.
 Solution by Carol It took me 24 moves to win the game. I played it 5 times and won, this is a supper game!!!!!! Thank you
 Solution by Sue Hinman Thanks for a fun puzzle and a great website. My students had a great time figuring this out! Though difficult at first, once they figured out the secret, they could solve it quickly. It was a good confidence builder. Other than a mirror image solution, we can't imagine there's any other way to solve this, so we'll be interested to see if there are other solutions or not. Thank you again!  4-5  6-4 7-6 5-7 3-5 2-3 4-2 6-4 8-6 9-8 7-9 5-7 3-5 1-3 2-1 4-2 6-4 8-6 7-8 5-7 3-5 4-3 6-4 5-6 24 moves
 Solution by Russell Baum Thanks for the great site which I have just come across... The solution to mini contest 20 is as follows: This is actually very straight forward since the balls cannot move backwards. Consequently at every stage there are very few legal moves available and incorrect moves are easy to spot before they become problems. Obviously the puzzle can be solved moving orange first or blue first with the same pattern of moves. Starting the puzzle with a jump is no good as it then creates a buffer that stops the second colour from moving. The first move therefore is the first ball into the middle. Moving the same colour again, either by a single move or a jump creates another buffer so the second move must be a jump by the first ball in the second colour. Using the same logic from then on keeping colours separate provides the solution below. 4-5, 6-4, 7-6, 5-7, 3-5, 2-3,  4-2, 6-4, 8-6, 9-8, 7-9, 5-7,  3-5, 1-3, 2-1, 4-2, 6-4, 8-6,  7-8, 5-7, 3-5, 4-3, 6-4, 5-6 24 moves.
 Solution by David Atkinson I had a go at your Oxbow Puzzle. I used a pen and paper to write down my moves, and marked where there was more than one move that wouldn't result in getting stuck, so I could easily go back and try the other move. I hope that isn't considered cheating :) Anyway here is the result I got: I solved it in 24 moves. 6-5, 4-6, 3-4, 5-3, 7-5, 8-7,  6-8, 4-6, 2-4, 1-2, 3-1, 5-3,  7-5, 9-7, 8-9, 6-8, 4-6, 2-4,  3-2, 5-3, 7-5, 6-7, 4-6, 5-4. After solving it with pen & paper, I then wrote a C program to see if there were any other ways of solving it. There is 1 other way, which appears to be by starting with the opposite move ie. 4-5, 6-4, 7-6 (which incidentally is what you get if you read my solution backwards). Thanks for the puzzle!

Oxbow Puzzle
(solution)

There are just two symmetrical solutions to this puzzle; both count 24 moves. Some of your correct solutions with interesting comments are shown below.

Also there were some great comments and suggestions regarding the set of rules to the Oxbow Puzzle and its solution's notation. Thanks a lot for them! We'd only like to point out that our main goal was to save the true puzzle "smell" of this nice puzzle gem. So we've just used the rules accompanying the real old sample of this puzzle, and tried to presented them as clear as possible.