Solution to the impossible puzzles

A few weeks ago, I proposed two Alcazar puzzles that have no solution. Some readers came up with very clever ways to explain why they are impossible. Here is my summary.

 

The key idea is to notice that all the possible lines fall into two categories: those that fill an even number of squares, and those that fill an odd number of squares. When the puzzle is colored like a checkerboard (black, white, black, white...), you can extract important information about these lines (see image).


Solution to the first puzzle

With 18 white squares and 17 black ones, we need an odd path to fill this room, and it needs to start and end on a white square. Unfortunately, there is only one door on a white square. This is why the puzzle has no solution.


Solution to the second puzzle

It was meant to be much harder than the first one, but "jh" (one of the blog's reader) came up with a very elegant solution that I summarize here. Congratulations, jh!

 

Like the first puzzle, this one has an extra white square, which means that the solution must start and end on a white square. We can close all the black exits of the puzzle.

 

Then, let's have a look at the 6x6 area in the bottom right. It has as many white squares as black ones, which means that if we want to fill this area with a single line, this line will need to start on a white square and end on a black one (or vice versa). But now that the puzzle's black exits are closed, this area has only white exits. In conclusion: we cannot fill it with a single line.


One last question remains: could we fill this area with more than one than one line? In other words, could we enter this area, exit, and then enter it again? Let's see...


There are only 3 squares that link this area to the rest of the puzzle and they are white. It means that we must enter and exit on white squares with our first path, and then, enter again on a white square before exiting the whole puzzle (again, on a white square). Each of our lines will cross one extra white square, but the area doesn't have extra white squares.


And that's why the puzzle has no solution.


What about normal puzzles?

Click to play in your browser
Click to play in your browser

When playing some of the advanced puzzles in the real game, using the square colors can be very useful. Look at the first level of the book "The Six Rooms", for example. The upper right room has an even number of squares and 4 exits: 3 are white and one is black. Conclusion: you must use the black one.



I hope that this article was helpful. Don't forget to get the Alcazar app (iOS and Android), and if you like it, leave a comment and a rating!

> Click here for the iOS version

> Click here for the Android version


In other news, you can now get the new edition of my game Manifold from Brainwrights' website.


Have fun!


Jérôme


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