Tile Puzzles
The odd family out. Everywhere else on this site you work out a value; here you move pieces around until the picture is right, and the only question is how few moves it takes.
Half Of All Sliding Puzzles Cannot Be Solved
Scramble a 15 puzzle at random and there is an even chance it can never be finished, however long you play. The reason is parity: a legal move always changes the arrangement in a way that preserves one particular property, so exactly half of the 362,880 arrangements of a 3×3 are reachable and the other half are not.
Most sites avoid this by only ever scrambling with legal moves. This one lets you test any arrangement and shows the arithmetic, because knowing why a board is impossible is more useful than being protected from ever meeting one.
Why We Say The Shortest Solution And Not Just A Solution
Finishing a sliding puzzle is easy; finishing it in the fewest possible moves is not. Every 3×3 board here has its true minimum computed by exhaustive search, so finishing in that number means something. The furthest any 3×3 board can be from solved is 31 moves — a fact that falls out of the same search rather than being looked up.
Where the search cannot honestly finish, on the deeper 4×4 and 5×5 boards, the page says so rather than inventing a figure.
Tangram Is Placement, Not Deduction
A tangram silhouette is normally drawn by a person, and then somebody has to work out whether the seven pieces actually fit — most outlines of the right area cannot be filled at all. Here it runs the other way round: the pieces are placed first and the puzzle is their outline, so a solution exists before the puzzle does.
| Type | The goal | What the site can prove |
|---|---|---|
| Sliding puzzle | Get the tiles in order | Whether the board is solvable, and the exact fewest moves |
| Picture sliding puzzle | Reassemble the picture | The same, on the same engine |
| Tangram | Fill the outline with all seven pieces | That a solution exists, because the outline was traced round one |
Why These Do Not Print
Honesty first: this is the one family on the site that does not work on paper. A sliding puzzle is defined by moving pieces, and a tangram needs seven shapes you can pick up and turn. Printing either gives you a picture of a puzzle rather than a puzzle.
If you want something to take away, the shading and number grids are the families built for it. What these pages give you instead is something paper cannot: a solver that knows the true shortest route and will tell you whether your arrangement can be finished at all.
What The Exact Solver Is Actually For
It is not there to finish the puzzle for you. It is there so that "I solved it" means something — finishing a board in 24 moves is a different achievement depending on whether the minimum was 24 or 12, and no other sliding puzzle site tells you which.
It also settles the argument that comes up whenever somebody hands you a scrambled physical puzzle: quite often the thing genuinely cannot be solved, because half of all arrangements cannot, and the checker shows the arithmetic rather than asking you to take its word.
| Board | Solvability | Shortest solution |
|---|---|---|
| 3×3 (8 puzzle) | Always, with the arithmetic shown | Exact, by exhaustive search |
| 4×4 (15 puzzle) | Always | Exact on shallow boards; says so when it cannot |
| 5×5 (24 puzzle) | Always | Not attempted — and the page says so |
| Tangram | Guaranteed — the outline was traced round a solution | One valid arrangement, on request |
Tile Puzzles Frequently Asked Questions
What is a tile puzzle?
A puzzle where you move or place physical pieces rather than work out a number. The sliding puzzles ask you to shuffle numbered or pictured tiles into order through a single gap; tangram asks you to fit seven flat shapes into an outline. Both are about arrangement rather than deduction.
Why can some sliding puzzles never be solved?
Because a legal slide preserves a property mathematicians call parity, so the arrangements split into two halves and you can never cross from one to the other. Exactly half of the 362,880 arrangements of a 3×3 board are reachable. The page shows the arithmetic for whatever board you are looking at.
What does a perfect solve mean here?
Finishing in the exact minimum number of moves, which is computed for your specific board by exhaustive search rather than estimated. If you use the undo button the result is still recorded but Perfect is withheld, because a rewound run is not a run in the minimum.
Why are the pictures drawings rather than photographs?
Because a picture puzzle only works if the pieces can be told apart. Cut a photograph of a face into sixteen and several pieces are nothing but cheek; cut a landscape and half are sky. Illustrations can put a recognisable landmark in nearly every piece, which is what makes them placeable by reasoning instead of by trial and error.
Do the tangram shapes look like anything?
They are abstract rather than pictorial, and that is an honest trade. A hand-drawn set gives you a cat or a boat, but only as many shapes as somebody drew. These are generated, so the supply does not run out — and because the pieces are placed before the outline is traced, none of them is impossible.
Related Practice
Paper folding and rotation of figures train the same hold-it-in-your-head habit without the pieces. Loop puzzles are the closest thing here that is still drawn rather than deduced.