[Game of Fifteen][1], a.k.a. the 15-Puzzle, was very popular 100+ years ago. The game is still found in stores; for example, you can search Amazon for "15 Puzzle" and get it for about 5 dollars. 

In the paper [Notes on the 15 puzzle][2] (American Journal of Mathematics,
1879, Vol. 2, No. 4, 397-404) W.W. Johnson and W.E. Story showed, via parity argument, that half of the positions in the puzzle [are not solvable][3]. The parity argument is accessible even to high school students. 

 Editors' note accompanying the paper by Johnson & Story (1879) in American Journal of Mathematics is quite interesting:

>  The "15" puzzle for the last few weeks has been prominently before
> the American public, and may safely  be said to have engaged the
> attention of nine out of ten persons of both sexes and of all ages and
> conditions of  the community. But this would not have weighed with the
> editors to induce them to insert articles upon such  a subject in the
> American Journal of Mathematics, but for the fact that the principle
> of the game has its root  in what all mathematicians of the present
> day are aware constitutes the most subtle and characteristic
> conception  of modern algebra, viz: the law of dichotomy applicable to
> the separation of the terms of every complete  system of permutations
> into two natural and indefeasible groups, a law of the inner world of
> thought, which  may be said to prefigure the polar relation of left
> and right-handed screws, or of objects in space and their  reflexions
> in a mirror. Accordingly the editors have thought that they would be
> doing no disservice to their  science, but rather promoting its
> interests by exhibiting this *a priori* polar law under a concrete
> form, through  the medium of a game which has taken so strong a hold
> upon the thought of the country that it may almost be  said to have
> risen to the importance of a national institution. Whoever has made
> himself master of it may fairly be said to have taken his first lesson
> in the theory of determinants.  It may be mentioned as a parallel case
> that Sir William Rowan Hamilton invented, and Jacques & Co., the
> purveyors of toys and conjuring tricks, in London (from whom it may
> possibly still be procured), sold a  game called the "Eikosion" game,
> for illustrating certain consequences of the method of quaternions.
> -EDS.



  [1]: http://en.wikipedia.org/wiki/15_puzzle
  [2]: https://www.jstor.org/stable/2369492
  [3]: https://en.wikipedia.org/wiki/15_puzzle#Solvability