15 Puzzle
A puzzle introduced by Sam Loyd in 1878. It consists of 15 squares numbered from 1 to 15 which are placed in a 
box leaving one position out of the 16 empty. The goal is to rearrange the squares from a given arbitrary startingarrangement by sliding them one at a time into the configuration shown above. For some initial arrangements, thisrearrangement is possible, but for others, it is not.
To address the solubility of a given initial arrangement, proceed as follows. If the Square containing the number
appears ``before'' (reading the squares in the box from left to right and top to bottom) 
numbers which are lessthan , then call it an inversion of order , and denote it 
. Then define
where the sum need run only from 2 to 15 rather than 1 to 15 since there are no numbers less than 1 (so
must equal0). If 
is Even, the position is possible, otherwise it is not. This can be formallyproved using Alternating Groups. For example, in the following arrangement
(2 precedes 1) and all other 
, so 
and the puzzle cannot be solved.
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 312-316, 1987. Bogomolny, A. ``Sam Loyd's Fifteen.'' http://www.cut-the-knot.com/pythagoras/fifteen.html. Bogomolny, A. ``Sam Loyd's Fifteen [History].'' http://www.cut-the-knot.com/pythagoras/history15.html. Johnson, W. W. ``Notes on the `15 Puzzle. I.''' Amer. J. Math. 2, 397-399, 1879. Kasner, E. and Newman, J. R. Mathematics and the Imagination. Redmond, WA: Tempus Books, pp. 177-180, 1989. Kraitchik, M. ``The 15 Puzzle.'' §12.2.1 in Mathematical Recreations. New York: W. W. Norton, pp. 302-308, 1942. Story, W. E. ``Notes on the `15 Puzzle. II.''' Amer. J. Math. 2, 399-404, 1879.
- Conformal Latitude
- Conformal Map
- Conformal Solution
- Conformal Tensor
- Conformal Transformation
- Congruence
- Congruence Axioms
- Congruence (Geometric)
- Congruence Transformation
- Congruent
- Congruent Incircles Point
- Congruent Isoscelizers Point
- Congruent Numbers
- Congruum
- Congruum Problem
- Conic
- Conical Coordinates
- Conical Frustum
- Conical Function
- Conical Spiral
- Conical Wedge
- Conic Constant
- Conic Double Point
- Conic Equidistant Projection
- Conic Projection