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!
