Parimony
Understanding parity
On a checkered board, the concept of parity can be seen simply as the relative quantities of tiles in each colour that exist in a region.
While the checkered pattern is kept, permuting colours will not change this relative amount, neither will moving nor rotating a region. This works on both square and hex grids, respectively with 2 and 3 colours. Indeed, parity is an invariant of the system (thus very useful for proving certain mathematical theorems).
Sets of regions by parity
Equal Parity 0
The following regions have an equal amount of cells in each colour:
Unequal Parity 1
The following regions have exactly one more cell of any colour:
In the hex grid, a maximum difference of 1 falls into two different cases: (0,1) and (1,1).
Unequal Parity 2
And here are couple regions with a difference of 2 in the square grid:
In the hex grid this maximum difference of 2 can spread over three cases (0,2), (1,2), (2,2):
Parimony puzzles
Anurag Sahay
To celebrate the implementation of Parimony, Anurag Sahay kindly created this three-puzzle set:
PinkHoodie
Variants
hex
In the hex grid, there are 3 tile colours to account for:
territory-unique
This variant requires each territory to appear exactly once:
Note: modnads is a doppel with a Different Rectangles puzzle.
territory-no-unique
This variant requires each territory to appear more than once.
territory-repeat-2
This variant requires each territory to appear exactly twice.
territory-adjacent-area-consecutive
full
lohkous
Acknowledgements and Changes
hex variant and to Skyeislime for noticing a missing sentence in the rules, just added. territory-unique by Anurag Sahay and a hex by Pedro were added. Also, Random 8 and hutthutthutt found non-uniquenesses and proposed corrections, now shown territory-adjacent-area-consecutive variant puzzle by Anurag Sahay was added - thank you! hex, territory-unique, territory-no-unique, territory-repeat and classic puzzles (8), and found an incorrectly assigned region, corrected. Thank you! full variant. lohkous variant puzzle!