Rules

1. Play on a 4×4 frame with fifteen numbered square tiles (1–15) and one blank cell.
2. A legal move slides one tile that shares an edge with the blank into the blank. Orthogonal slides only.
3. Do not lift tiles out of the frame, swap two tiles by hand, jump, or move diagonally.
4. Some physical and digital versions allow sliding a whole row/column segment into the blank in one gesture; that still counts as successive legal slides of adjacent tiles.
5. The standard goal is tiles in reading order 1–15 left-to-right, top-to-bottom, with the blank at the bottom-right corner (some puzzles set a different goal; follow the puzzle).

Allowed moves

How to play

1. Start from a scrambled (but reachable) board, or from the solved board and scramble by sliding.
2. Move tiles into the blank until numbers read 1–15 in order.
3. Common human strategy: solve row by row (or spiral), then finish the last 2×2 or 3×3 subproblem.
4. Confirm the blank sits in the prescribed corner — done.

Board

Picture-tile “sliding photo” variants use the same moves; the goal is a restored image instead of numerals.

Play modes

Origins (c. 1874–1879)

The sliding 15-puzzle (also Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square) took shape in the United States in the mid-to-late 1870s. The best-supported inventor is Noyes Palmer Chapman, postmaster of Canastota, New York. By about 1874 he had shown friends a precursor with sixteen numbered blocks arranged so each row summed to 34 (a magic-square style challenge). The improved sliding form—fifteen tiles plus one blank in a 4×4 frame—circulated through family and trade connections to Syracuse, Watch Hill, and Hartford. Students at the American School for the Deaf in Hartford were making and selling copies by December 1879; Boston woodworker Matthias J. Rice began manufacturing a commercial “Gem Puzzle” the same month.

Chapman applied for a U.S. patent on a “Block Solitaire Puzzle” in February–March 1880. The application was rejected, reportedly as insufficiently distinct from Ernest U. Kinsey’s earlier Puzzle-Blocks patent (U.S. 207124, August 1878). The craze did not wait for a Chapman patent: cash prizes for solutions (e.g. Charles Pevey in Worcester in late January 1880) and rapid retail spread turned the toy into a national fad. Early trade names included Gem Puzzle, Boss Puzzle, Game of Fifteen and Mystic Square; all shared equal square tiles and a single blank rather than unequal Klotski-style blocks.

The 1880 craze

From January 1880 in the United States and from April 1880 in Europe, the 15-puzzle became a short, intense popular craze—political cartoons, newspaper jokes, and shop counters all featured it—then cooled by about July 1880. That first wave established the sliding numbered square as a mass-market mechanical puzzle long before later advertising myths attached a single celebrity inventor’s name to it.

Sam Loyd and the “14-15” advertising myth

Sam Loyd did not invent the 15-puzzle and took no part in its 1879–1880 launch. His first article on the puzzle appeared in 1886; from 1891 until his death in 1911 he publicly claimed invention. Later retellings often credit him with offering a $1,000 prize for restoring a board that differs from the solved state only by swapping tiles 14 and 15 (the so-called 14-15 puzzle). That swapped start is unreachable by legal slides: it is an odd permutation relative to the goal when the blank is home, so the prize—if framed that way—could never be claimed. Jerry Slocum and Dic Sonneveld’s archival study (2006) documents Loyd’s long self-attribution campaign and restores Chapman, Hartford makers, and Rice to the early commercial story. Japanese and other popular accounts that still center Loyd as inventor or as the sole driver of the 1880 boom are repeating the advertising myth, not the primary-source timeline. Separating “who sold the craze in 1880” from “who later claimed credit” is the main editorial correction this encyclopedia page maintains.

Parity mathematics and popular science

In 1879, Wm. Woolsey Johnson and William E. Story published “Notes on the ‘15’ Puzzle” in the American Journal of Mathematics, proving that legal slides preserve a parity invariant: only half of all labeled arrangements are reachable from a given start. With the blank fixed at bottom-right, the permutation of the fifteen tiles must be even. Equivalently, one tracks the parity of the full 16-position permutation together with the taxicab distance of the blank from its home cell. The 14-15 swap is the classroom example of an impossible instance. Later writers (Archer 1999 and others) gave cleaner proofs; group-theoretic treatments identify the reachable configurations with the alternating group (A_{15}) action on the tiles. Optimal-length solving for general (n \times n) sliding puzzles is NP-hard; for the classic 15-puzzle, God’s number is 80 single-tile moves (seventeen positions require 80).

Into Japan and the digital era

In Japan the puzzle appears by Meiji 40 (1907) in Sekai Yūgi-hō Taizen under the name 十五置き換え遊び; today it is usually called 15パズル or generically a スライドパズル. Twentieth-century plastic pocket versions, picture-tile variants, and computer implementations made random scrambling common—so implementers must generate only even (reachable) permutations or reject odd ones. The same rules underlie 8-puzzles, photo-slide apps, and many AI search demos (Manhattan distance, IDA*, A*). Bobby Fischer’s televised solve in seventeen seconds (1972) is a well-known pop-culture footnote.

References

1. English Wikipedia, 15 puzzle
2. Wolfram MathWorld, 15 Puzzle
3. Johnson & Story, Notes on the “15” Puzzle (1879), JSTOR / DOI
4. Slocum & Sonneveld, The 15 Puzzle (2006) — MAA review
5. Japanese Wikipedia, 15パズル
6. OI Wiki, 15-puzzle (parity / AI notes)
7. Chinese Wikipedia, 数字推盘游戏