# Backlot (MIT Mystery Hunt 2020)

Backlot | |
---|---|

MIT Mystery Hunt 2020 | |

Creative Pictures Studios | |

Author(s) | Chris Yao |

Answer | Click to reveal🏴☠️ |

Statistics | |

No. solves | 16 |

No. total guesses | 33 |

Links | |

Puzzle | Link |

Solution | Link |

**Backlot** is a path-drawing Logic Puzzle from the Creative Pictures Studios round of the 2020 MIT Mystery Hunt. The logic puzzle contained within has a recursive structure.

## Solve Path[edit | edit source]

The structure of the logic puzzle is given on the surface:

- Draw a path between any two exits.
- The path must cross every cell with a dot.
- The path must cross exactly the clued number of squares in every row/column with such a number.
- The blue squares are a smaller copy of this grid, following the same rules.
- In the outermost case, the path enters from the north, and must enter at least one blue square.

Deductions for now are few and far between. The most important deduction to make without having decided on which exits to look at is that the 7 in the second row forces at least one square in the upper-left 2x4 region to be visited, and as the region only has two entrances both must be used.

Afterwards, though, the best thing to do is to pick a pair of exits and see if it leads to a solution. Since the rules specify an exit to start with, the choice of path should be heavily constrained by a choice of exits. For example, if the west exit isn't chosen, then entering R4C1 would use up all 6 cells in the column and leave the dotted cell in R8C1 without an exit. If the south exit isn't chosen, then there are only two possible sets of cells that could satisfy the bottom row.

An example solve path for each set of exits is presented below:

**TO DO** Images

The west exit and R8C1 require one additional cell each in order to escape. As such, R1C1 is inaccessible.

Since the south exit isn't used, exactly one of R9C1-3 and R9C4-6 is used. Therefore, R9C9 isn't used. R8C7 must exit left to avoid looping.

Bifurcate on which of R7C1 or R9C1 is used:

Taking R7C1 immediately forces the bottom two rows:

The rightmost column has only one unused cell left. Xing anything in that column adjacent to a corner also forces that corner to be unused, meaning those cells should be used. This completes the top two rows and the right two rows:

Row 7 now has one cell left on the path. It cannot be the subgrid, as that would strand the path. Taking standard deductions from there completes almost the entire grid:

Using R6C2 eventually disconnects the bottom-right segment, so the final path uses R4C2:

Taking R9C1 immediately forces the bottom two rows:

Note that R5C3 must escape using at least one other cell in its column; this forces R3C3 to be unused. However, the 8 clue also forces R3C2 to be used, and it has only one neighbor to exit to. Therefore, this cannot be a solution.

Having not selected the west exit means that R4C1 and R5C1 are not used. This means at least one of R6-7C1 is used, meaning both are used and R8C1 must escape north.

The last cell in the left column will directly affect the bottom row. Bifurcate on which of the corner cells is used:

Taking R1C1 immediately forces both the bottom row and the second column:

In the top row, four cells are already used by the loop. This prevents the top-right corner from being used. In the row below it, one unused cell is already present; if the other unused cell is in R2C8-9, the other cell in that group cannot escape, so both must be used.

In the eighth row, R8C7 must escape to the left to avoid forming a loop. This closes off the other two undetermined cells in that row.

From there, one can fill out the right two columns and the seventh row. Following the chain of deductions further, however, results in a contradiction:

The seventh column must use R4C7, but doing so would immediately close the loop.

Taking R9C1 immediately forces the bottom row. Column 2 is also forced—blocking R4C2 would result in a T-intersection in R6C2, so R4C2 must be used. This in turn forces almost every other cell on the board:

From here, experimenting with the path a little will result in three valid solutions to the grid:

Having not selected the west exit means that R4C1 and R5C1 are not used. This means at least one of R6-7C1 is used, meaning both are used and R8C1 must escape north. This forces several other cells in the left two columns:

Column 7 has one unused cell left. Placing it in R5 forces a loop; Placing it in R6 causes a dead end.

Placing it in R4 forces this arrangement that breaks C9:

If placed in R7, it leaves two entrances into the bottom right, which must be used. This eventually breaks as well; one way is shown.

Placing it in R2 can also be shown to break:

Taken together, this indicates that the north-south entrance combination has no solution.

The west exit and R8C1 require one additional cell each in order to escape. As such, R1C1 is inaccessible.

The exit in the south blocks usage of R9C1-3, forcing R8C1 north. This forces the left column.

R2C8 is unused (if it were used it would break the 7 in Row 2). If R2C9 is used under this scenario it breaks the puzzle:

Therefore the top right corner is left unused, forcing the following:

If R9C5 goes west, it blocks off the bottom right corner and forces a loop. As such, R8-9C4 are unused.

If R7C3 is unused, it eventually breaks the puzzle:

Setting R7C3 to used forces R7C6 to be unused, since if used the path cannot exit without creating another used cell in Row 7. This forces the following:

Column 7 thus cannot use R6C7 (or more problems with the path's route would arise). Standard deductions solve the rest of the puzzle.

The west exit and R8C1 require one additional cell each in order to escape. As such, R1C1 is inaccessible.

R2C8 is unused (if it were used it would break the 7 in Row 2). If R2C9 is used under this scenario it breaks the puzzle:

Therefore the top right corner is left unused, forcing the following:

Since the south exit isn't used, exactly one of R9C1-3 and R9C4-6 is used. Therefore, R9C9 isn't used. However, following this with standard deductions will fill up R8 and reveal that this arrangement cannot fulfill the bottom row:

As such, there are no solutions using both the east and west exits.

Having not selected the west exit means that R4C1 and R5C1 are not used. This means at least one of R6-7C1 is used, meaning both are used and R8C1 must escape north. The exit in the south blocks usage of R9C1-3. This forces the left two columns.

In the top row, three cells are already used by the loop. This prevents the top-right corner from being used. In the row below it, one unused cell is already present; if the other unused cell is in R2C8-9, the other cell in that group cannot escape, so both must be used.

At least one of the remaining cells in R1 will be in C5-7, and thus must escape to another cell in R1C5-7. This blocks R1C4, forcing the top three rows.

Row 7 is now satisfied, which forces the bottom two rows, leading to completion of most of the grid.

The left-side segment now has only two possible exits. This forces the rest of the path.

There are thus six solutions to the grid without considering the recursion. Of these, two of the N+E solutions pass straight through a blue square; however, no solutions can satisfy the embedded grid, so neither of these can be used. The remaining N+E solution is the only one that uses the north gate and passes through a blue square; it thus must be the outermost layer of the recursion, and contains both the S+E and a S+W solution. The S+W solution contains another S+E solution, and the S+E solution contains the N+W solution. The N+W solution does not enter a blue square, meaning the recursion is finite.

The puzzle also provides an Instructional Extraction below the grid: each solution used in the final path highlights some squares based on where it exits. Combining the highlighted squares draws a picture of the final answer.

## Puzzle Elements[edit | edit source]

Logic Puzzle - The puzzle is a non-standard path-drawing Logic Puzzle. Elements of the puzzle resemble the logic puzzle genre Snake, the prior Mystery Hunt puzzle Random Walk, and standard mazes.

True Recursion - The grid's main feature is the blue squares, which contain an identical copy of the grid.

Instructional Extraction - Below the puzzle are instructions that state to shade certain cells in white based on where a section of the path exits a grid or subgrid.

Bitmaps - Taking unidentified cells to be black, the final image depicts the answer.