Sliding Puzzle

How does the Rising sliding puzzle work?

Rising uses the usual sliding moves with a looser goal. You win when every row counts up from left to right and every column counts up from top to bottom. The gap is skipped when you read a line, and it can finish in any cell.

• Moves: a tile next to the gap slides into it. A click further along the gap's row or column slides the whole line and counts one move per tile.
• Goal: 210 different 3×3 layouts win, and 168,168 on 4×4. The classic order is just one of them.
• Progress: a tick beside a row or under a column shows that line already counts up. The text under the board gives the same count in words.
• Fair starts: each board is made by sliding tiles away from a random winning layout, so it can always be won. On 3×3 the nearest win is at least 8 moves away.

What does a winning Rising board look like?

A winning Rising board reads upwards along every line. Take the start below. The rows 1 2 5 and 3 4 (gap) are fine, but the bottom row reads 6 8 7, so it fails. Slide the 7 up into the gap and the board reads 1 2 5 / 3 4 7 / 6 8 (gap).

Now every row and every column rises, so the game stops the clock after one move. Classic mode would not accept that finish, because 5 sits in the top right corner where Classic wants 3. The third board shows the opposite case: its top row reads 4 2 6 and its left column 4 1 5, so it is not solved, and the engine finds its nearest win 10 moves away.

Where should tile 1 and the highest tile go?

Tile 1 always finishes in the top left corner, and the highest tile always finishes in the bottom right corner. The one exception is a gap that ends in that corner. With the gap top left, tile 1 sits just right of the gap or just below it instead, and the same mirrored rule holds for the highest tile and a gap in the bottom right.

So place those two tiles first, then grow the rising rows from the top left. The rule follows from the goal: a tile in the top left has nothing above it or to its left, so it must be the smallest. The engine tests check it on all 210 winning 3×3 boards and on thousands of random 4×4 wins.

How many Rising boards win, and how hard is a 3×3 board?

The number of winning Rising boards depends on where the gap ends. We counted every layout with the game's own code. A gap in the top left or bottom right corner allows the most winning layouts.

A breadth-first search over all 362,880 arrangements of the 3×3 board measured how many single-tile moves each one needs. Rising boards need fewer moves than Classic ones, and the hardest still needs 26.

Moves are single-tile moves to the nearest win. The Classic maximum of 31 matches the published figure in OEIS A087725.

What is Classic mode?

Classic mode is the 15 puzzle as it was first made in the 1870s: put the tiles in order from 1 to 15, row by row, with the gap in the bottom right corner. On 3×3 it becomes the 8 puzzle, with tiles 1 to 8. Exactly one layout wins.

How do you solve the classic 15 puzzle?

1. Solve the top row. Bring tiles 1, 2 and 3 into place, then park 4 below its spot and slide 3 and 4 in together.
2. Solve the second row the same way: 5, 6, then 7 and 8 as a pair.
3. The last two rows are now a small 2-row puzzle. Solve them column by column from the left: place 9 above 13, then slide both in, then 10 and 14, and so on.
4. Finish the last 2×2 corner by cycling the three remaining tiles around the gap until they line up.

On 3×3 the same idea is shorter: solve the top row, then the left column of the bottom two rows, then cycle the last corner.

Why can half of all classic boards never be solved?

Half of all classic boards can never be solved because every move keeps a hidden property, the parity. Write the tiles out row by row and skip the gap. An inversion is any pair where a larger number comes before a smaller one.

On a 3×3 board a classic position can be solved only if the number of inversions is even. The board above has 1 inversion (6 before 5), so no sequence of moves will ever fix it. On 4×4 the row of the gap counts too. Classic mode checks the rule before you play, so you never get an impossible shuffle.

Exactly half of all arrangements can be reached: 9! ÷ 2 on 3×3 and 16! ÷ 2 on 4×4. Rising has no such split, because both halves contain winning layouts.

Who invented the 15 puzzle?

The 15 puzzle was designed by Noyes Chapman, a postmaster from Canastota, New York, according to the Lilly Library at Indiana University. It was created in upstate New York and first made in Hartford and then Boston in the 1870s.

• The craze: in early 1880 the puzzle swept across America and other countries.
• The proof: W. W. Johnson and W. E. Story explained its mathematics in the American Journal of Mathematics, in the December 1879 issue published in April 1880.
• The false claim: Sam Loyd claimed he created it. Starting in the 1890s he offered a $1000 prize for a solution to a board that has none, and Keith Conrad of the University of Connecticut calls the belief that Loyd invented it false.
• The records: the hardest 15 puzzle positions need 80 moves, proved by computer search in 1999, and no 8 puzzle position needs more than 31.

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