The 15 Puzzle, Fifteen Puzzle, 14-15 Puzzle, Boss Puzzle
Contents of this Page
What is the 15Puzzle?
Solution of the Puzzle
Some Mathematics
Sliding Block Puzzles
The Eight Puzzle
Sliding Block Puzzles with Different Tiles
Tower of Babylon 
The 15 Puzzle on the Internet
To the Main Page   "Mathematische Basteleien"
What is the 15 Puzzle? 
.. .. The fifteen puzzle has 15 pieces, which are numbered from 1 to 15 and which lay in a square frame. 
You cannot take away the pieces; you slide them by using a free square. At the beginning the 15 numbers are mixed (left). You must order the numbers by sliding (right). 
.. ..

Solution of the Puzzle top
You don't need special advice to find a solution. Everybody, who has some patience, can arrange the pieces in the right order.
To get to know the puzzle and to practice, it is best to build a model. You can move the numbers easily. The pieces don't get stuck while moving. 

You draw a 4x4 square freehand on a piece of cardboard, cut out 15 suitable cards from paper, label them and lay them on the squares (left). 

The usual method is moving the numbers from 1 to 15 one after the other, line after line.

You must not destroy a line. You must keep the order of the numbers at least. 

You start with moving 1 to the left-hand corner above. 

Therefore you must make a gap in front of the 1, so that 1 can go forward. You quickly notice, that it is good to take other numbers with you, too.

Then the next numbers follow... 
This is a main move, the circle: A number is going around in a 2x2 square. 
Here you can see, how 7 is going to its right place. 

If 7 is going around the other way, 8 is also on the right place.

Sometimes you don't have a 2x2 square for moving a piece (1).

You can see with 8 how to move the incomplete line to the left and down and then 8 up in a 2x2 square (2,3,4).

If you have solved the third line, sometimes the fourth line is also solved and you are finished. 

Otherwise the numbers of the last line aren't in the right order (1).

It is helpful to compress the third line into a 2x2 square (2). Then it is easy to order the last three numbers 13,14 15.

...... The solution is more systematical, if you know how to move the pieces inside a 2x3 rectangle. 

You always can succeed in moving any two numbers (here a and b) to the left, also changed (book 09).

Programs   top
...... If you have solved the puzzle, your next problem can be to order it with as few moves as possible.
I have written a small program in Visual Basic V3, which simulates and counts the moves. 

The pieces are already mixed from the start. 

You can download  the program. You need Vbrun300.dll.

There are many programs of the 15 puzzle on the net, especially one made by Karl Hörnell. You can play them online or offline.

You can also download programs to find out as few moves as possible. I used Ken'ichiro Takahashi 's program (thanks, URL below) to solve the pattern on this website. You need at least 59 moves: 9, 1, 6, 9, 1, 14, 12, 1, 2, 15, 11, 5, 1, 2, 15, 7, 9, 4, 10, 3, 13, 9, 3, 10, 8, 6, 14, 15, 4, 3, 7, 11, 5, 1, 2, 4, 3, 8, 6, 14, 15, 12, 4, 3, 8, 6, 14, 15, 12, 8, 6, 7, 11, 6, 7, 11, 10, 14, 15.

The problem of counting the moves of a solution with as few moves as possible is difficult (12). Today it is known that you can solve the 15 puzzle with at least 80 moves (12). Thus computers can manage the huge number of cases. 

Some Mathematics   top
Look at a 2x2 square for the sake of simplicity. 
Only three pieces are used. The free square is always at the bottom right-hand corner.
...... There are three possibilities under these circumstances to alter the position of a piece by sliding. You can write them as 123, 312 and 231. 
...... You can make 3! = 6 changings (permutations) with three numbers. There are still the possibilities 132, 213 and 321. They bring you to the positions of the pieces on the left, which don't appear in the puzzle. 
How do the permutations differ?
You form all the pairs within the permutations, where the bigger number is in front of the smaller one (inversions). 


all pairs:

12 13 23
31 32 12
23 21 31


no pair
2 pairs: 31 32
2 pairs: 21 31

These permutations are called even, because the number of pairs is even. 
(The even permutations make the alternate subgroup with the order 3 of the symmetric group. The basic set must have an order.) 

The other permutations are odd: 


all pairs:

13 12 32
21 23 13
32 31 21


1 pair: 32
1 pair: 21
3 pairs: 32 31 21


You can transfer these facts to the 4x4 square.
There are 15 pieces, so you have 15! = 1 307 674 368 000 permutations. Only half of them are even and will appear as positions. 
If you also count the empty square, you have 16! = 20 922 789 888 000 possibilities. 

Sliding Block Puzzles top
I found an insoluble 15puzzle in Italy, where the numbers 14 und 15 are changed.

This puzzle is insoluble. The permutation (1,2,...,13,15,14) is odd because of the pair (15,14). You can't change it by sliding to the even permutation (1,2,...,13,14,15). 

I thought that this was intented for making the puzzle insoluble. Now in August 2004 Martin Beckenkamp sent me an email that a 15puzzle is installed at his handy [England: mobile (phone), USA: cell (phone) ;-)] and shows "solved", if the numbers have this order

01 02 03 __
04 05 06 07
08 09 10 11
12 13 14 15
You can reach this order with the Italian puzzle above, where 14 and 15 are changed.

Most 15 puzzles, which you can buy nowadays, have picture pieces instead of numbers. You must find the completed picture.

Two examples follow: Ads of Mc Donald's and Kaiser Bier.


Hint: You can take off the tiles with some power. You must slightly lift the middle pieces at the beginning.

The Eight Puzzle   top
....... Problem: 
Order the numbers going backwards to the normal order 1 to 8 with a free place on the bottom right.
Henry Ernest Dudeney invented this problem. He needed 36 moves. Martin Gardner asked readers of the American magazine  "Scientific American" for solving the puzzle in less steps. He got many solutions.
...... Result: You only need 30 moves. - There are 10 solutions. Two solutions (on the left and in the centre) form a pair and are reverse. You recognize this, if the moves run backwards (on the right).

The corresponding 15 puzzle is insoluble. 
But if 1 and 2 change their places, it is soluble. You need at least 70 moves (Ken'ichiro Takahashi): 2, 5, 9, 13, 14, 10, 11, 7, 3, 1, 6, 11, 10, 15, 7 ,3, 1, 6, 5, 9, 13, 14, 15, 7, 3, 1, 6, 5, 9, 13, 14, 15, 12, 8, 4, 14, 15, 12, 8, 4, 12, 8, 7, 3, 1, 6, 5, 9, 13, 15, 14, 2, 15, 14, 2, 12, 8, 2, 11, 10, 2, 7, 3, 2, 6, 5, 9, 13, 14, 15. 

Sliding Block Puzzles with Different Tiles      top
...... Many puzzles followed the 15 puzzle in the19th century. You also had to move pieces, but pieces with different sizes. 

You must transport the left top corner square to the left bottom corner. 

You need at least 59 moves (solution in book 07).

This sliding block puzzle is called Dad's Puzzle. 

Tower of Babylon  (German: Zauberturm)   top

The "Tower of Babylon" is one of the many puzzles that followed Rubik's cube. In principle it is a sliding puzzle.

You must order 36 balls, so that there are 6 balls with the same colour in a column and ordered by shades. You can easily move them horizontally. They are fastened in a ring. Moving vertically is only possible with a trick: You can push a ball at the bottom to the middle (left arrow), so it disappears. Then the balls move down and form a gap at the top (right arrow). If you hold the tower horizontally, you can move the gap to another place. This is enough to solve the puzzle. This is a stupid work because of the many balls.

My tower has a screw at the top, so that I can easily put the tower into pieces and arrange the balls properly ;-). It's a pity I have lost the ball-bearing between the slices :-(.

The 15 Puzzle on the Internet       top





Alexander Bogomolny (cut-the-knot)
Sam Loyd's FifteenHistory, Lucky7, Happy8, Slider

Don Taylor
The 14-15 Puzzle

Ed Pegg Jr. (MAA online)
sliding-block Puzzles
Sliding Block Puzzle online

Harry Broeders

Herbert Kociemba
Fifteen Puzzle Optimal Solver

Jaap Scherphuis
14-15 puzzle / Boss puzzle

Jerry Slocum & Dic Sonneveld
The 15 Puzzle

Jim Loy 
The 15 Puzzle (The possible puzzle; The impossible puzzle) 

Karl Hörnell's Applet Center
The 15 Puzzle

Nick Baxter
Sliding Block Home Page

Ken'ichiro Takahashi (takaken) 
15Puzzle Optimal Solver

Fifteen puzzle, Sliding puzzle

References top
(00) ermann Schubert: Mathematische Mußestunden, Walter de Gruyter Verlag Berlin 1941 (1.Auflage 1897)
(01) W.Ahrens: Mathematische Unterhaltungen und Spiele, Leipzig 1918
(02) G.Kowalewski: Mathematica delectans, Band 1 , Leipzig 1921
(03) G.Kowalewski: Alte und neue Spiele, Leipzig 1930 (Nachdruck: Martin Sändig, Walluf 1978 (ISBN 3-500-19830-9)
(04) Martin Gardner: Mathematical Puzzles & Diversions, New York 1959 
(05) Walter Lietzmann: Lustiges und Merkwürdiges von Zahlen und Formen, Göttingen 1961
(06) Bruno Kerst: Mathematische Spiele, Berlin 1933 (Nachdruck: Martin Sändig, Wiesbaden 1968)
(07) Martin Gardner: Mathematisches Labyrinth, Braunschweig 1979 (ISBN 3-528-08402-2)
(08) Michael Mrowka: Zauberturm, Teufelstonne und Magische Pyramide, Niedernhausen/Ts.1981 (ISBN 3-8068-0606-3)
(09) Monika Dewess und Günter Dewess: Summa Summarum, Thun; Frankfurt am Main, 1986 (ISBN 3-87144-898-2)
(10) Johannes Lehmann (Hrsg.): Rechnen und Raten , Köln 1986 (ISBN 3-7614-0930-3). 
(11) L. Edward Hordern: Sliding Piece Puzzles, 249pp, hb, Oxford, England, 1993, Oxford University Press
(12) F. R. W. Karlemo and P. R. J. Östergård : On sliding block puzzles, Journal of Combinatorial Mathematics and Combinatorial Computing 34 (2000), 97-107
(13) Jerry Slocum & Dic Sonneveld: The 15 Puzzle, Slocum Puzzle Foundation, 257 South Palm Drive, Beverly Hills, CA 90212, 2006

Thank you Gail from Oregon Coast for supporting me in my translation.

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©  2000 Jürgen Köller