${DynamicText("About this genre (loading...)")}
## [parimony] puzzles
### @AnuragSahay
To celebrate the implementation of [parimony], @AnuragSahay kindly created this three-puzzle set:
W=4x4&L=w1w5b4w1b1w1b2&G=parimony&A=anurag_sahay&D=2024-03-11&F=1&T=Rotated_Roadster
W=6x3&L=w3w1w1w1w2w1w4&G=parimony&A=Anurag_sahay&D=2024-03-12&F=2&T=Disparimony
W=6x6&L=w0w4w3w1b1b1w2b2b2w4w4b1w2b3w2w2&G=parimony&A=anurag_sahay&D=2024-03-12&F=4&N=3&T=P_for_Parimony&K=random8
W=4x4&L=w1b4w5b1w1b3&G=parimony&A=Anurag_sahay&D=2024-03-15&F=3
W=5x5&L=w0w1b4x4w1w2b2b1x4w1w1&G=parimony&A=Anurag_sahay&D=2024-03-25&F=1&T=Parimonial_elixir
### @PinkHoodie
W=5x5&L=w5b1w1w2b1w1b1w12&SSE=4RRRU2URRRD9RRDLLLURDDD12R7LU&G=parimony&A=PinkHoodie&D=2024-03-13&F=3&T=Brekad
W=7x7&L=x1w9w1x1w1b1w1w3w2x1x2x2x2w1w5x3w4w1w2x4&SSE=5RDRU5URUR16LDRU4LU5RD1RRR12U2UR4LDLUR&G=parimony&A=PinkHoodie&D=2024-03-13&F=3&T=Creckolda
## Variant puzzles
### ´´hex´´
V=hex
In the hex grid, there are 3 tile colours to account for:
W=3x3x3&L=b0w3b2b3w5b2&SSE=3EREREREULULU1REREULU14ULU12UEU14R&V=hex&A=Pedro&D=2024-03-13&K=random8&G=parimony
W=3x3x2&L=b0w2b1b5w4&SSE=4REU3ERERER7RDRE24WLUE&V=hex&G=parimony&A=PinkHoodie&D=2024-03-14&F=2&O=Hex&T=Parithree
W=4x4x4&L=w3w3w4b8b2w3b1b3w5w2w2&SSE=3EUERDRDRERDREULULWLULULUERDRDRD10EULUE8REU7U6U15EUERDWDRDR11U14E11LUER1ULUE&V=hex&G=parimony&A=PinkHoodie&D=2024-03-14&F=3&O=Hex&T=ReParithree
### ´´territory-unique´´
V=territory-unique
This variant requires each territory to appear exactly once:
W=4x4&L=w1b2w2b1w1&V=territory-unique&G=parimony&A=Anurag_Sahay&D=2024-03-13&F=3&N=7&T=odd-even_conundrum&K=hutthutthutt
W=4x4&L=b0b1b4b3&SSE=1RUURU5RD5URR&V=territory-unique&G=parimony&A=PinkHoodie&D=2024-03-13&F=2&O=Territory_Unique&T=Modnads
Note: modnads is a doppel with a [different-rectangles] puzzle.
### ´´territory-no-unique´´
V=territory-no-unique
This variant requires each territory to appear more than once
W=4x4&L=b0w1x1x1w4w1b2b2w3&SSE=1RRD7RRDLLD5U2UUR&V=territory-no-unique&G=parimony&A=PinkHoodie&D=2024-03-14&F=1&O=Territory_Repeat&T=Renfasd
W=4x5&L=b0b2b2b1w2w7b2&SSE=2RRDRUUR2RRRDLLDDD12U3R&V=territory-no-unique&G=parimony&A=PinkHoodie&D=2024-03-14&F=2&O=Territory_Repeat&T=Rionfasc
### ´´territory-repeat-2´´
V=territory-repeat-2
This variant requires each territory to appear exactly twice
W=9x3&L=b0x2w1b3w4b1w4w5w1w2b3&SSE=4UU4UUU2R2UURDRUU12ULD6RU1DLDRRR&V=territory-repeat-2&G=parimony&A=PinkHoodie&D=2024-03-14&F=2&O=Territory_Repeat_2&T=Gemixies
### ´´territory-adjacent-area-consecutive´´
V=territory-adjacent-area-consecutive
W=4x4&L=w2b1w12&V=territory-adjacent-area-consecutive&G=parimony&A=Anurag_Sahay&D=2024-03-14&F=2&T=Min
## 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:
W=8x5&L=x0x2x3x2x1x1x1x2x3x2x3x1x1x1x1x3x1x4x5x2&T=Example_regions_with_parity_(0),_square_grid&G=parimony
W=7x3x3&L=x1x2x3x4x1x1x2x2x3x2x1x3x1x5x3x2x1x1&V=hex&T=Example_regions_with_parity_(0,0),_hex_grid&G=parimony
#### Unequal Parity 1
The following regions have exactly one more cell of any colour:
W=10x6&L=x3x4x2x2x4x1x1x1x2x1x3x1x1x1x6x2x4x1x1x4x1x2x1x1x1x1x3x1&T=Example_regions_with_parity_(1),_square_grid&G=parimony
In the hex grid, a maximum difference of 1 falls into two different cases: (0,1) and (1,1)
W=8x1x3&L=x2x2x1x2x2x3x1x4x2x2x1x1&V=hex&T=Example_regions_with_parity_(0,1),_hex_grid&G=parimony
W=8x1x3&L=x4x1x2x2x4x4x2x2x2&V=hex&T=Example_regions_with_parity_(1,1),_hex_grid&G=parimony
#### Unequal Parity 2
And here are couple regions with a difference of 2 in the square grid:
W=7x3&L=x0x1x2x3x1x2x1x1x1x1x2&T=Example_regions_with_parity_(2),_square_grid&G=parimony
In the hex grid this maximum difference of 2 can spread over three cases (0,2), (1,2), (2,2):
W=7x1x3&L=x0x5x1x2x1x1x7x2&V=hex&T=Example_region_with_parity_(0,2),_hex_grid&G=parimony
W=3x3x2&L=x3x1x1x1x4&V=hex&T=Example_region_with_parity_(1,2),_hex_grid&G=parimony
W=4x4x2&L=x0x1x3x1x1x5x1x3x1x1x2x3&V=hex&T=Example_regions_with_parity_(2,2),_hex_grid&G=parimony
### ´´full´´
W=5x5&L=b0b5b1b1w5b4b2&SSE=6URRUL1URURUU1UURU11RR3RDDLU&V=full&G=parimony&A=PinkHoodie&D=2024-03-24&F=2&O=Full&T=Inidisoa
### ´´lohkous´´
V=lohkous
W=4x4&L=w0(2)2b1(1)1(1)1b5(2)1w1(3)1b2&SSE=3RDDRURR5RU7UR2UU&G=parimony&V=lohkous&A=PinkHoodie&D=2024-04-13&F=2&O=Partial_Scattered_Lohkous&T=Splitting_ways
## Acknowledgements and Changes
**2024-03-12** Thanks to @AnuragSahay for feedback in the implementation and this page, finding a bug, and for brilliantly making the [rule panels|../panels#modern] neatly fit a 4x4 grid.
**2024-03-12** A three-puzzle set by @AnuragSahay was also added to celebrate the implementation of [Parimony]
**2024-03-13** Thanks to @JonyMeister for suggesting and exploring the ´´hex´´ variant and to @SkyeIsLime for noticing a missing sentence in the rules, just added.
**2024-03-13** Sketch lines moved from edge to centre, a suggestion by @EliDoris - thank you!
**2024-03-13** A ´´territory-unique´´ by @AnuragSahay and a ´´hex´´ by Pedro were added. Also, @Random8 and @HuttHuttHutt found non-uniquenesses and proposed corrections, now shown
**2024-03-14** A ´´territory-adjacent-area-consecutive´´ variant puzzle by @AnuragSahay was added - thank you!
**2024-03-14** A section [#Understanding Parity] was added, after feedback from @JanJakopu, @Yyao and @Xsheep.
**2024-03-15** :::puzzletters @PinkHoodie sent ´´hex´´, ´´territory-unique´´, ´´territory-no-unique´´, ´´territory-repeat´´ and classic puzzles (8), and found an incorrectly assigned region, corrected. Thank you!
**2024-03-15** :::puzzletters @AnuragSahay sent an additional [parimony] - thank you.
**2024-03-29** A classic [parimony] by @Anurag was added, plus a puzzle by @PinkHoodie using the newly corrected ´´full´´ variant.
**2024-04-19** Rotational parity antisymmetry was obtained by @PinkHoodie in the latest ´´lohkous´´ variant puzzle!