## Sliders

### Problem 244

Published on Saturday, 9th May 2009, 02:00 pm; Solved by 831You probably know the game *Fifteen Puzzle*. Here, instead of numbered tiles, we have seven red tiles and eight blue tiles.

A move is denoted by the uppercase initial of the direction (Left, Right, Up, Down) in which the tile is slid, e.g. starting from configuration (**S**), by the sequence **LULUR** we reach the configuration (**E**):

(S) | , (E) |

For each path, its checksum is calculated by (pseudocode):

checksum = 0

checksum = (checksum × 243 +

checksum = (checksum × 243 +

…

checksum = (checksum × 243 +

where checksum = (checksum × 243 +

`m`_{1}) mod 100 000 007checksum = (checksum × 243 +

`m`_{2}) mod 100 000 007…

checksum = (checksum × 243 +

`m`_{n}) mod 100 000 007`m`

_{k}is the ASCII value of the

`k`

^{th}letter in the move sequence and the ASCII values for the moves are:

L | 76 |

R | 82 |

U | 85 |

D | 68 |

For the sequence **LULUR** given above, the checksum would be 19761398.

Now, starting from configuration (**S**),
find all shortest ways to reach configuration (**T**).

(S) | , (T) |

What is the sum of all checksums for the paths having the minimal length?