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GridWorks Extensions: Symmetries

One important concept in geometry is the notion of symmetry. There are several important forms of symmetry, including translational symmetry, reflectional symmetry, and rotational symmetry. To understand these forms of symmetry, you must first understand the operations of translation (sliding an image), reflection (flipping an image, like in a mirror), and rotation (turning an image).

Let's take a look at some brand new geometrical extensions to the GridWorks system. These new clue types are challenging, and a whole lot of fun. While working on these puzzles, you will develop a deeper understanding of the transformations that play such a critical role in geometry.

Rotation

Rotation is the process of turning an image. To explore the idea of rotation within GridWorks, we now introduce a new rotation clue type.
When you see a pattern that is marked with a special rotation icon, you must consider the pattern in in each of its four rotations (0, 90, 180, 270 degrees).
Now take a look at these rotation clues.
Examples of Positive Clues
OR    
         
OR    
    This clue tells us that at least one of these four patterns can be found in the grid. In other words, there is a blue piece in at least one of the four corners.
OR OR OR
    This clue tells us that at least one of these four patterns can be found in the grid. In other words, the blue triangle is adjacent to the yellow X, but the orientation, as well as the exact position, is not given by the clue.
Examples of Negative Clues
    With a rotation clue on the negative side of the clue card, we now have to make sure that this pattern doesn't exist anywhere in the grid, in any of its four possible orientations.
OR


POSSIBLE LOCATIONS

OR
   

CLUE PLACEMENT

   
    This clue tells us that these four patterns are prohibited. Therefore, the green triangle is not in any corner of the grid.
OR


POSSIBLE LOCATIONS

OR
   

CLUE PLACEMENT

   
    This clue tells us that these four patterns are prohibited. Therefore, the only spot left for the blue circle is the grid's central cell.

Reflection

Reflection is the process of flipping an image. You can flip an image along any axis, but in the context of GridWorks, we'll focus on two main types of reflection.
Reflection along a vertical axis results in left/right flipping. Reflection along a horizontal axis results in top/bottom flipping.
When you see a pattern that is marked with one of these reflection icons, you must consider both the pattern and the flipped pattern. Now take a look at these reflection clues.
Examples of Positive Clues
OR    
    This clue tells us that at least one of these two patterns can be found in the grid. If you consider all the possible places these patterns can fit within the grid, you can conclude that the yellow X is in either the top row or the bottom row (but not the middle row).
OR    
    This clue tells us that at least one of these two patterns can be found in the grid. So this clue means that the green circle is to the left or the right of another green piece (and possibly both).
Examples of Negative Clues
OR
   

CLUE PLACEMENT

  POSSIBLE LOCATIONS
    This clue tells us that these two patterns are forbidden in the grid. If you consider all the possible places these patterns can fit within the grid, you can conclude that since the yellow X is not permitted in the top or bottom rows, the yellow X must be in the middle row.

Combinations

In some puzzles, you'll even see clues that are marked with both rotation and reflection icons! Now take a look at these combination clues.
Examples of Positive Clues
OR OR
    With this clue, we know that the solution is one of these three possibilities.
   
       
   

THIS CLUE PLACEMENT IS NOT VALID

       
    This grid is arrived at by flipping the grid top/bottom, and then left/right. When you see a grid that is marked with multiple geometrical transformations, you don't need to concern yourselves with the possibilities involving one transformation applied to the result of another transformation. Only consider patterns that immediately follow from transforming the clue directly with one of the marked transformations.
OR OR
         
OR    
    With this clue, we know that the solution is one of these five possibilities.
   
       
   

THIS CLUE PLACEMENT IS NOT VALID

       
    Again, remember that you don't need to consider flipped versions of the rotated grids, because this would involve consecutive applications of rotation and reflection transformations.
OR OR
         
OR OR 
    This clue yields the following six possibilities.
Examples of Negative Clues
    Much like the positive combination clues, you must consider a pattern and all the transformations indicated by the icons. But this time, the resulting possibilities are prohibited!
OR OR
   

CLUE PLACEMENT

OR OR
         
OR    
OR OR
         
OR OR 
Last Updated: January 13, 2009 top
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