4 deleted 32 characters in body
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Since we must always turn 90 degrees, every odd-numbered step we take is always vertical and every even-numbered step is horizontal. Since the last step has to be horizontal, it means our total path has to be an even number of steps. In other words, once we've reached the red squares, we have made the same amount of vertical and horizontal steps.

After each odd-numbered vertical step, we are an odd number of rows away from the starting point. After each even-numbered step, we are an even number of rows away. The same applies for horizontal steps.

In the end, we have taken the same number of steps both horizontally and vertically (i.e. either both are even or both are odd). We are 4 columns away from the starting point, so we must have taken an even number of horizontal steps. However, we are 5 rows away from the starting point, so we must have taken an odd number of vertical steps. This is impossible, so we can conclude that there is no such path through the maze.

Note though that this only works because this particular maze has no straight moves available (i.e. intersections where you can't turn but only go forward). For example, using the same rules, if you remove the vertical line directly below the 2x1 block...

...there actually is a solution:

solution.

Since we must always turn 90 degrees, every odd-numbered step we take is always vertical and every even-numbered step is horizontal. Since the last step has to be horizontal, it means our total path has to be an even number of steps. In other words, once we've reached the red squares, we have made the same amount of vertical and horizontal steps.

After each odd-numbered vertical step, we are an odd number of rows away from the starting point. After each even-numbered step, we are an even number of rows away. The same applies for horizontal steps.

In the end, we have taken the same number of steps both horizontally and vertically (i.e. either both are even or both are odd). We are 4 columns away from the starting point, so we must have taken an even number of horizontal steps. However, we are 5 rows away from the starting point, so we must have taken an odd number of vertical steps. This is impossible, so we can conclude that there is no such path through the maze.

Note though that this only works because this particular maze has no straight moves available (i.e. intersections where you can't turn but only go forward). For example, using the same rules, if you remove the vertical line directly below the 2x1 block...

...there actually is a solution:

solution

Since we must always turn 90 degrees, every odd-numbered step we take is always vertical and every even-numbered step is horizontal. Since the last step has to be horizontal, it means our total path has to be an even number of steps. In other words, once we've reached the red squares, we have made the same amount of vertical and horizontal steps.

After each odd-numbered vertical step, we are an odd number of rows away from the starting point. After each even-numbered step, we are an even number of rows away. The same applies for horizontal steps.

In the end, we have taken the same number of steps both horizontally and vertically (i.e. either both are even or both are odd). We are 4 columns away from the starting point, so we must have taken an even number of horizontal steps. However, we are 5 rows away from the starting point, so we must have taken an odd number of vertical steps. This is impossible, so we can conclude that there is no such path through the maze.

Note though that this only works because this particular maze has no straight moves available (i.e. intersections where you can't turn but only go forward). For example, using the same rules, if you remove the vertical line directly below the 2x1 block...

...there actually is a solution.

3 added 39 characters in body
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Since we must always turn 90 degrees, every odd-numbered step we take is always vertical and every even-numbered step is horizontal. Since the last step has to be horizontal, it means our total path has to be an even number of steps. In other words, once we've reached the red squares, we have made the same amount of vertical and horizontal steps.

After each odd-numbered vertical step, we are an odd number of rows away from the starting point. After each even-numbered step, we are an even number of rows away. The same applies for horizontal steps.

In the end, we have taken the same number of steps both horizontally and vertically (i.e. either both are even or both are odd). We are 4 columns away from the starting point, so we must have taken an even number of horizontal steps. However, we are 5 rows away from the starting point, so we must have taken an odd number of vertical steps. This is impossible, so we can conclude that there is no such path through the maze.

Note though that this only works because this particular maze has no straight moves available (i.e. intersections where you can't turn but only go forward). For example, using the same rules, if you remove the vertical line directly below the 2x1 block...

...there actually is a solution (N, E, and then N-W until the end).:

solution

Since we must always turn 90 degrees, every odd-numbered step we take is always vertical and every even-numbered step is horizontal. Since the last step has to be horizontal, it means our total path has to be an even number of steps. In other words, once we've reached the red squares, we have made the same amount of vertical and horizontal steps.

After each odd-numbered vertical step, we are an odd number of rows away from the starting point. After each even-numbered step, we are an even number of rows away. The same applies for horizontal steps.

In the end, we have taken the same number of steps both horizontally and vertically (i.e. either both are even or both are odd). We are 4 columns away from the starting point, so we must have taken an even number of horizontal steps. However, we are 5 rows away from the starting point, so we must have taken an odd number of vertical steps. This is impossible, so we can conclude that there is no such path through the maze.

Note though that this only works because this particular maze has no straight moves available (i.e. intersections where you can't turn but only go forward). For example, using the same rules, if you remove the vertical line directly below the 2x1 block...

...there actually is a solution (N, E, and then N-W until the end).

Since we must always turn 90 degrees, every odd-numbered step we take is always vertical and every even-numbered step is horizontal. Since the last step has to be horizontal, it means our total path has to be an even number of steps. In other words, once we've reached the red squares, we have made the same amount of vertical and horizontal steps.

After each odd-numbered vertical step, we are an odd number of rows away from the starting point. After each even-numbered step, we are an even number of rows away. The same applies for horizontal steps.

In the end, we have taken the same number of steps both horizontally and vertically (i.e. either both are even or both are odd). We are 4 columns away from the starting point, so we must have taken an even number of horizontal steps. However, we are 5 rows away from the starting point, so we must have taken an odd number of vertical steps. This is impossible, so we can conclude that there is no such path through the maze.

Note though that this only works because this particular maze has no straight moves available (i.e. intersections where you can't turn but only go forward). For example, using the same rules, if you remove the vertical line directly below the 2x1 block...

...there actually is a solution:

solution

2 added 330 characters in body
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Since we must always turn 90 degrees, every odd-numbered step we take is always vertical and every even-numbered step is horizontal. Since the last step has to be horizontal, it means our total path has to be an even number of steps. In other words, once we've reached the red squares, we have made the same amount of vertical and horizontal steps.

After each odd-numbered vertical step, we are an odd number of rows away from the starting point. After each even-numbered step, we are an even number of rows away. The same applies for horizontal steps.

In the end, we have taken the same number of steps both horizontally and vertically (i.e. either both are even or both are odd). We are 4 columns away from the starting point, so we must have taken an even number of horizontal steps. However, we are 5 rows away from the starting point, so we must have taken an odd number of vertical steps. This is impossible, so we can conclude that there is no such path through the maze.

Note though that this only works because this particular maze has no straight moves available (i.e. intersections where you can't turn but only go forward). For example, using the same rules, if you remove the vertical line directly below the 2x1 block...

...there actually is a solution (N, E, and then N-W until the end).

Since we must always turn 90 degrees, every odd-numbered step we take is always vertical and every even-numbered step is horizontal. Since the last step has to be horizontal, it means our total path has to be an even number of steps. In other words, once we've reached the red squares, we have made the same amount of vertical and horizontal steps.

After each odd-numbered vertical step, we are an odd number of rows away from the starting point. After each even-numbered step, we are an even number of rows away. The same applies for horizontal steps.

In the end, we have taken the same number of steps both horizontally and vertically (i.e. either both are even or both are odd). We are 4 columns away from the starting point, so we must have taken an even number of horizontal steps. However, we are 5 rows away from the starting point, so we must have taken an odd number of vertical steps. This is impossible, so we can conclude that there is no such path through the maze.

Since we must always turn 90 degrees, every odd-numbered step we take is always vertical and every even-numbered step is horizontal. Since the last step has to be horizontal, it means our total path has to be an even number of steps. In other words, once we've reached the red squares, we have made the same amount of vertical and horizontal steps.

After each odd-numbered vertical step, we are an odd number of rows away from the starting point. After each even-numbered step, we are an even number of rows away. The same applies for horizontal steps.

In the end, we have taken the same number of steps both horizontally and vertically (i.e. either both are even or both are odd). We are 4 columns away from the starting point, so we must have taken an even number of horizontal steps. However, we are 5 rows away from the starting point, so we must have taken an odd number of vertical steps. This is impossible, so we can conclude that there is no such path through the maze.

Note though that this only works because this particular maze has no straight moves available (i.e. intersections where you can't turn but only go forward). For example, using the same rules, if you remove the vertical line directly below the 2x1 block...

...there actually is a solution (N, E, and then N-W until the end).

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