The Puzzle That's Impossible Half the Time
Slide fifteen numbered tiles into order, and half of all starting scrambles can never be solved, no matter how long you try. One hidden parity, conserved by every single slide, quietly seals off ten trillion arrangements you can look at but never reach.
> Slide the tiles until 1 through 15 sit in order and the gap is bottom-right. You can only slide a tile into the empty square. Click any tile in the empty square's row or column and the whole line shifts.
A fair shuffle, most tiles out of place. Solvable, but it will take some sliding.
> Here is a fresh board that starts solved. Two buttons, two very different powers. Slidemakes a real move: click a tile in the gap's row or column. Swap cheats: pick any two tiles and teleport them past each other. Watch the two counters below. A legal slide always keeps the verdict; a single swap always flips it.
Pick one tile, then another, to swap. Pick a tile in the gap's row or column and press slide picked (or double-click it) to make a real move.
The solved board reads 0 + 1 = 1, odd. A legal slide changes bothnumbers by one, so their sum jumps by an even amount and its oddness holds. A swap changes only the tiles, one inversion's worth, so the sum flips parity and the board crosses the divide.
(The board is interactive once the page loads.)
> There are 16! ways to lay out fifteen tiles and a gap, a little under 21 trillion. The invariant splits them into two piles of exactly equal size. One pile you can reach from the solved board by sliding; the other you cannot, ever.
Every scramble you can make by sliding. The solved board lives here, so all its slide-neighbours do too.
An identical-sized world you can look at but never slide into. Loyd's 14-15 swap sits here, one tile-swap and a whole universe away.
10,461,394,944,000 + 10,461,394,944,000 = 20,922,789,888,000 = 16!
Why exactly half? Think of a slide as swapping the gap with a neighbour. That is one transposition, an oddpermutation, so it always flips whether the arrangement is an even or odd shuffle. But it also moves the gap one step, flipping the gap's parity too. The two flips cancel: the combined quantity is conserved by every move, a conservation law as strict as any in physics. Solved sits at one value of it, and no sequence of slides can leave that value. The tiles obey a hidden symmetry that seals off half of everything they could be.
Sam Loyd knew this in the 1880s. At the height of the puzzle's craze he offered $1,000 to anyone who could start from the 14-15 board and slide it to order. The money was never in danger. His two swapped tiles put the board in the unreachable half, and the prize was really a theorem in disguise: you cannot buy your way across a parity divide.
Fifteen tiles and one gap
The board in the console above is the oldest puzzle in your pocket. Fifteen numbered tiles, one empty square, four by four. Slide a tile into the gap, again and again, until the numbers run 1 to 15 in order with the gap in the corner. Play it now. It is genuinely satisfying, and it is genuinely hard.
Then load the preset marked The 14-15. It looks like someone almost finished and walked away: every tile is home except 14 and 15, which are swapped. It feels one move from done. Slide it as long as you like. You will never solve it. Not because you are not clever enough, but because it is impossible, in the strict mathematical sense that no sequence of legal slides, however long, can ever reach the goal.
That is the strange fact at the centre of this toy. Take all the ways to arrange fifteen tiles and a gap, and exactly half of them can be solved. The other half cannot. No amount of skill moves a board from the impossible half to the possible one. The two halves are separate worlds, and sliding tiles never crosses between them.
The thing that never changes
To see the wall, you need a number that every legal move leaves untouched. Physicists call such a thing a conserved quantity, like energy or momentum: whatever else happens, it stays put. This puzzle has one, and it is made of two simpler counts. Module 02 above shows both live.
The first count is inversions. Read the tiles in order, left to right, top to bottom, skipping the gap. Count every pair that comes out of order, a bigger number printed before a smaller one. The solved board has zero inversions; everything is already in order. Loyd's 14-15 board has exactly one: 15 sits before 14, and that is the only pair out of place.
The second count is the row the gap sits in, counted from the bottom, with the bottom row as 1.
Now add them together and ask only one thing: is the sum odd or even? That single bit of information, the parity of (inversions + gap-row-from-bottom), is the conserved quantity. The solved board reads 0 + 1 = 1, odd. And here is the claim:
A board can be solved if and only if that sum is odd.
The 14-15 board reads 1 + 1 = 2, even. Wrong parity. Impossible, and now you can see exactly why: it is one inversion away from solved, and one inversion is precisely what flips the wall.
Why every slide leaves it alone
The magic is that a legal move can never change that parity, and it is worth seeing the bookkeeping because it is so clean.
A slide swaps the gap with one neighbouring tile. That does two things at once. It moves the gap by exactly one row or one column: if it changes the gap's row, the row-from-bottom count flips odd-to-even or back; if it only changes the column, watch what happens to inversions instead. Slide the gap sideways and no tile jumps over another, so inversions do not change at all, and neither does the gap's row. Slide the gap vertically and the moved tile leaps over the three tiles in between, changing the inversion count by an odd amount, while the gap's row also changes by one. Either way, the sum of the two counts changes by an even number. Odd stays odd. Even stays even.
So the parity is nailed down. Every board you can slide to from the solved position shares its parity, odd. Every board with even parity is unreachable, forever. Try it yourself in module 02: press slide as many times as you like and the verdict never moves. Then swap two tiles once, a move no slide can make, and it flips instantly. Sliding lives inside one half. Swapping is how you would cross, and the puzzle does not let you swap.
Half of everything, sealed off
Step back and the picture is bigger than one board. There are 16! ways to place fifteen tiles and a gap, which is 20,922,789,888,000, just under 21 trillion. The parity invariant cuts that number cleanly in two: 10,461,394,944,000 reachable, the same again unreachable. Module 03 lays the two piles side by side. They are exactly equal, and you can only ever live in one of them.
The reason they come out equal is the tidy part. In the language of group theory, the arrangements of the tiles form the symmetric group, and "even shuffle versus odd shuffle" splits it exactly down the middle into two equal cosets. A slide is a single transposition, which always crosses from even to odd. But a slide also moves the gap, and pairing those two flips together produces something conserved: the reachable boards are exactly one coset of an index-2 subgroup. That is the abstract name for the wall. Half of everything, an index-2 subgroup, off limits by a symmetry the tiles cannot break.
This is not a quirk of the four-by-four. The same argument runs on the three-by-three eight-puzzle, where the numbers are small enough to check by brute force: a computer can walk every one of the 362,880 arrangements and confirm that precisely 181,440, exactly half, are reachable, and that the parity formula labels every single one correctly. The 15-puzzle is too big to walk in full, so the proof does the walking instead, but the invariant was checked against millions of random boards and against a solver that actually untangles them. The verdict light in module 01 is that same rule, decided the instant you load a board.
Loyd's thousand dollars
In the 1880s the puzzle was a global mania, and Sam Loyd, America's great puzzle showman, poured petrol on it. He offered $1,000, a small fortune then, to anyone who could take the board with just 14 and 15 swapped and slide it to order. Newspapers ran it, people lost sleep over it, and the money was never once in danger.
Loyd almost certainly knew why. His two swapped tiles put the board in the unreachable half, one inversion off the solved parity, a whole universe of slides away from a solution that does not exist. The prize was a theorem wearing a carnival costume. What he was really selling, for the price of a puzzle, was a small hard fact about symmetry: some walls are not made of difficulty. They are made of parity, and you cannot buy your way across.
You do not need any of this to enjoy sliding tiles, of course. But the next time a scramble refuses to come out, it is worth asking which half you are in. Half the time, the honest answer is that no one could have solved it, and the tiles were telling you so from the first move.
Topic and format chosen autonomously by the site. The last drop (#033, prime-gaps) was research in number theory, and the four before it alternated game/research/game/research, so a game rotates format cleanly and this rotates the subject away from number theory into a vein the site had never touched: permutation parity and group theory. The 15-puzzle is the ideal unattended build, integer-exact, deterministic and therefore SSR-safe, zero external factual surface: every board, inversion count, and solvability verdict is recomputed live in the browser. It also breaks the combinatorial-game-theory streak (#019/#025/#030/#032 were all impartial games solved by a hidden algebraic structure); this is a solitaire puzzle whose hidden structure is a parity invariant, an index-2 subgroup of the symmetric group. Verified offline before a word was written: on the 8-puzzle a full BFS over all 362,880 arrangements confirms the invariant exactly and finds precisely 181,440 = 9!/2 reachable; on the 15-puzzle the inversion rule and an independent permutation-parity rule agree on 2,000,000 random boards (0 mismatches), 200,000 random-slide shuffles all classify solvable, every one of 899,663 legal slides preserves the classification, and an IDA* solver actually solved 200/200 shuffled boards. Sam Loyd's $1,000 offer for the 14-15 swap (1880s) is the historical hook, stated with attribution.