Last time I discussed a pattern involving an interaction between a box and a row or column that passes through that box. If the only places left in the box for a particular value all lie in the same row or column, it's possible to eliminate the value from the part of the row or column that lies outside the box. By localizing a value in one type of group, we manage to localize it in an intersecting group.
Today I'm going to write about a pattern that involves the intersections between rows and columns, specifically where we know something about four cells that share two rows and two columns (i.e., they lie at the corners of a rectangle).
Consider this grid (you may want to try to solve it before reading further):
Without using any new rules, we can get to this situation:
In column 2, the number 3 can only appear in the cell (2, 2) or (2, 5). Similarly, in column 5, 3 can only appear in (5, 2) or (5, 5). I‘ve circled the possible cells in red, and highlighted the columns in red as well.
Suppose cell (2, 2) has the value 3. Then (5, 2) can't take a 3 because it is in the same row as (2, 2); thus (5, 5) is the only place left in column 5 that can take a 3. Conversely, if (2, 5) has the 3 for column 2 then (5, 2) must have the 3 for column 5. So either (2, 2) and (5, 5) contain 3's, or else (2, 5) and (5, 2) contain 3's. In short, there must be 3's at two opposite corners of the "X" formed by (2, 2), (2, 5), (5, 2), and (5, 5). This is the "X-Wings" (or "X-Wing") pattern, named for the way the values appear at the corners of an "X" shape. By coincidence, I'm writing this entry from the Electronic Theater at Siggraph 2005 , where a replica of a Star Wars T-65 X-Wing Starfighter is sitting in the lobby a couple of hundred feet away.
Now that we know that there are 3's at two opposite corners, we have accounted for the 3's in row 2 and row 3. We can remove all the other 3's in row 2 and row 5. I've highlighted the pencil marks that we can now erase in blue.
Note that we started out knowing that a number can be located at two places within each of two columns, and we ended up determining that it can occur in only two places in each of two rows. Again, localizing within one group or set of groups (the columns) enables us to localize in a related group or group (the rows). The key is that the cells involved form the corners of a rectangle.
Another example occurs later in the solution of the same grid (I won't draw circles around the corners anymore in order to simplify the diagrams a bit):
The corners of the red rectangle show the only places for a 5 in columns 2 and 3. The blue squares show where we can remove 5's in row 2. There's yet another X-Wing hidden in the grid:
Columns 5 and 7 have only two places for the number 9, which are in rows 2 and 3 for each column. We can remove 9's from the rest of those rows.
We can also start by locating a value in two cells of a each of two rows, and use this information to eliminate values up and down the columns. I call this type "horizontal X-Wings," as opposed to the "vertical X-Wings" discussed above.
This board contains a couple of examples of horizontal X-Wings:
Take a look at this stage in the solution:
The only places for the number 8 in row 1 and row 2 are in columns 2 and 4, so we remove the other remaining 8 in column 4.
The same grid also contains this pattern:
Note that we remove 6's from all of column 3, apart from
rows 2 and 4 that make up the X-Wing pattern. So the 6 at (3, 3) is
removed, along with the 6 at (3, 9). Don‘t be fooled by the "box" shape
— only two rows and two columns are involved in the X-Wing pattern
itself.
Next time I'll show how to extend the same reasoning to a more difficult (but rarer) pattern known as the Swordfish.
riezftveplvcmph, Sizegenetics scam, OjdCNNB, [url=http://sizegeneticblog.com/]Sizegenetics discount[/url], kbxtlEG, http://sizegeneticblog.com/ Sizegenetics enlargement guide, qWCpQiL.
Posted by: Size Genetics | January 26, 2012 at 12:45 AM
riezftveplvcmph, Sizegenetics scam, OjdCNNB, [url=http://sizegeneticblog.com/]Sizegenetics discount[/url], kbxtlEG, http://sizegeneticblog.com/ Sizegenetics enlargement guide, qWCpQiL.
Posted by: Size Genetics | January 26, 2012 at 12:45 AM
jupnjtveplvcmph, Viagra side effects, CVZPosP, [url=http://www.brooksdesignassociates.com/]Viagra no prescription canada[/url], OSleIgw, http://www.brooksdesignassociates.com/ Viagra pills, PVCrNmJ, Organa Slim, whqZsEk, [url=http://healthydiettrends.com/organa-slim-review]Organa Slim[/url], TgKVODh, http://healthydiettrends.com/organa-slim-review Organa Slim, TIkVpLU, Carisoprodol and breastfeeding, VGJgMZX, [url=http://www.la-cadenatricolor.com/carisoprodol.html]Buying carisoprodol[/url], AMeLMuS, http://www.la-cadenatricolor.com/carisoprodol.html Carisoprodol online, rTwrezx, Clonazepam discount, EeccJIl, [url=http://lightbodies.com/products.html]Clonazepam[/url], vCCnCse, http://lightbodies.com/products.html Snorting adderall and clonazepam together, HMLaLRn, Cialis, SSNbMNE, [url=http://www.jeffcadwell.com/cialis.html]Forms of cialis[/url], DBReuGG, http://www.jeffcadwell.com/cialis.html Cialis lifestyle pharmaceuticals, FFdTTzz, Cellucor d4 thermal shock, RiErhuZ, [url=http://healthymensproducts.com/cellucor-no3-chome-review]Cellucor[/url], HnakGHe, http://healthymensproducts.com/cellucor-no3-chome-review cellucor, SmJHffq.
Posted by: Cellucor d4 t7 ws1 | January 26, 2012 at 12:47 AM
jupnjtveplvcmph, Viagra side effects, CVZPosP, [url=http://www.brooksdesignassociates.com/]Viagra no prescription canada[/url], OSleIgw, http://www.brooksdesignassociates.com/ Viagra pills, PVCrNmJ, Organa Slim, whqZsEk, [url=http://healthydiettrends.com/organa-slim-review]Organa Slim[/url], TgKVODh, http://healthydiettrends.com/organa-slim-review Organa Slim, TIkVpLU, Carisoprodol and breastfeeding, VGJgMZX, [url=http://www.la-cadenatricolor.com/carisoprodol.html]Buying carisoprodol[/url], AMeLMuS, http://www.la-cadenatricolor.com/carisoprodol.html Carisoprodol online, rTwrezx, Clonazepam discount, EeccJIl, [url=http://lightbodies.com/products.html]Clonazepam[/url], vCCnCse, http://lightbodies.com/products.html Snorting adderall and clonazepam together, HMLaLRn, Cialis, SSNbMNE, [url=http://www.jeffcadwell.com/cialis.html]Forms of cialis[/url], DBReuGG, http://www.jeffcadwell.com/cialis.html Cialis lifestyle pharmaceuticals, FFdTTzz, Cellucor d4 thermal shock, RiErhuZ, [url=http://healthymensproducts.com/cellucor-no3-chome-review]Cellucor[/url], HnakGHe, http://healthymensproducts.com/cellucor-no3-chome-review cellucor, SmJHffq.
Posted by: Cellucor d4 t7 ws1 | January 26, 2012 at 12:47 AM
wvqxatveplvcmph, . with a maximum of up to 65 or 70 points reached the final or . in addition each can still further 5 to 10 points will be achieved, FgmsZPX, [url=http://healthymensproducts.com/maximum-10-review]maximum 10 reviews[/url], FKsWndx, http://healthymensproducts.com/maximum-10-review maximum 10 reviews, MjlONqJ.
Posted by: Mastercraft maximum 10 miter saw | January 26, 2012 at 12:48 AM
suttitveplvcmph, Ambien no prescription mastercard, mUXNZcM, [url=http://www.la-cadenatricolor.com/ambien.html]Ambien sleep[/url], YfbVwNC, http://www.la-cadenatricolor.com/ambien.html Ambien cr, ILwBuWK, Clonazepam vs alprazolam, RIMtJiZ, [url=http://cokercreekgallery.com/new/pages/contact.htm]Can you take clonazepam and alprazolam together safely?[/url], vetYYXC, http://cokercreekgallery.com/new/pages/contact.htm Clonazepam pregnancy, aTgWeHp, Effet levitra, obhnWNY, [url=http://www.oleani.com/products.cfm]Generic Levitra[/url], vnxOfFz, http://www.oleani.com/products.cfm Buy Levitra, lBTZseX, Melatrol sleep aid saftey, ihgCgrU, [url=http://melatrol2k.com/]Melatrol sleep aid review[/url], bHadDdi, http://melatrol2k.com/ Melatrol side effects, MBmUIhV, Free little girls naked preteen porn pussys, nQHNgVQ, [url=http://pornfaq.net/]Free porn videos[/url], VLOLnMt, http://pornfaq.net/ Porn, zXImbur, Buy Tramadol, SaiaDCz, [url=http://whiteker.com/]Tramadol[/url], KaElwHf, http://whiteker.com/ Tramadol, EnivQiI.
Posted by: Melatrol | January 26, 2012 at 12:51 AM
oqkootveplvcmph, Eazol ingredients, aonYXlz, [url=http://eazol4u.net/]Eazol[/url], OZqVCRg, http://eazol4u.net/ Eazol, gDyhJOX, Webcam girls, ZhFlDyt, [url=http://123webcam.org/]Xxx msn webcam[/url], pVDbnkU, http://123webcam.org/ Free adult webcam, ZeWWBvC, Virility EX, IuJLenL, [url=http://buyvirilityexonline.com/]Virility ex review[/url], QJwZasM, http://buyvirilityexonline.com/ Where can i buy virility ex in australia, OpKoYSn, Free viagra, lKsnXuB, [url=http://viagra-guide.com/]How does viagra work[/url], zimcjWG, http://viagra-guide.com/ Pillsdeals.com]viagra, BzHQYSh, CigReds, yQJINNl, [url=http://smokelesscigarettestoday.com/cigreds-review]CigReds[/url], PWWrIuO, http://smokelesscigarettestoday.com/cigreds-review CigReds, oJpsiXH, Cheapest generic cialis, ZqoKZZe, [url=http://www.readyhealthgo.com/information-about-treating-erectile-dysfunction-using-cialis_41]Low cost cialis[/url], vSkEyJp, http://www.readyhealthgo.com/information-about-treating-erectile-dysfunction-using-cialis_41 Cialis lawyer columbus, UbSIdCI.
Posted by: Cialis injury lawyer columbus | January 26, 2012 at 12:53 AM
miizrtveplvcmph, Dietrol, DpdMaab, [url=http://dietrol4u.com/]Dietrol[/url], BioElZf, http://dietrol4u.com/ Dietrol, GQhfyks, Revitol scar cream, tQEwinv, [url=http://revitolscarcream-direct.com/]Revitol Scar Cream[/url], kBrBYAe, http://revitolscarcream-direct.com/ Revitol scar cream, LYAjoaE, Diet weight loss patch, egVbjkp, [url=http://populardiettrends.com/reviews/diet-patch-today-review]Apple patch diet program[/url], quBsahz, http://populardiettrends.com/reviews/diet-patch-today-review Apple patch diet scams, nJlfLoa, Movies, FatAery, [url=http://moviesabc.info/]Future movies[/url], ucyOIBx, http://moviesabc.info/ Watch movies net, gRjgRuS, Irs tax refund status, zgHECGz, [url=http://irsfaq.net/]Irs 2010 tax forms[/url], AqVcYlG, http://irsfaq.net/ Irs fraud reporting, PyWqhFF, web easy professional 8, oobNZgE, [url=http://topwebdesignsoftware.com/]web easy professional review[/url], OXnLtMm, http://topwebdesignsoftware.com/ web easy professional 8 download, uouzNLT.
Posted by: Charlie sheen movies | January 26, 2012 at 01:07 AM
nklcmtveplvcmph, Proactol, myvEOsU, [url=http://proactol-advice.net/]Is proactol safe and effec tine[/url], ATgfPzl, http://proactol-advice.net/ Is proactol safe and effective, aglHmvz.
Posted by: Buy proactol | January 26, 2012 at 01:22 AM
nklcmtveplvcmph, Proactol, myvEOsU, [url=http://proactol-advice.net/]Is proactol safe and effec tine[/url], ATgfPzl, http://proactol-advice.net/ Is proactol safe and effective, aglHmvz.
Posted by: Buy proactol | January 26, 2012 at 01:22 AM
expnbtveplvcmph, What is ultram 50 mg, XsXlwxq, [url=http://www.ultrampharm.com/]Ultram er generic[/url], AoNmbOG, http://www.ultrampharm.com/ Snorting ultram, SDSJmfg.
Posted by: Ultram dose | January 26, 2012 at 01:23 AM
vadhmtveplvcmph, California car insurance quotes online, fXcmdzq, [url=http://phylliszullodesigns.com/]Car Insurance Quotes[/url], MPgfBne, http://phylliszullodesigns.com/ Car Insurance Quotes, UxenWGY.
Posted by: Car Insurance Quotes | January 27, 2012 at 12:26 AM
evlhetveplvcmph, SizeGenetics, lQpcoQR, [url=http://sizegeneticstore.com/]Sizegenetics discount[/url], mICVdkE, http://sizegeneticstore.com/ Sizegenetics does it work, IMzbYEl, Idol white, fNFEwWe, [url=http://idolwhitestory.com/]Idol White[/url], FbeOggk, http://idolwhitestory.com/ Idol White, mLCRHEf, Propecia side effect, ZRElzjc, [url=http://www.hairenigma.com/propecia-for-hair-loss-treatment_23/]Propecia merck[/url], MkZlxLI, http://www.hairenigma.com/propecia-for-hair-loss-treatment_23/ Among buy propecia comments e-mail name comment [b][/b] - [i][/i] - [u][/u]- [quote][/, uKcbCzF, Car Insurance, FCnbblc, [url=http://phylliszullodesigns.com/]Car Insurance Quotes[/url], KVRkyGS, http://phylliszullodesigns.com/ Nj car insurance quotes, mzWDjlI, acai berry diet, fUfGzwP, [url=http://healthydiettrends.com/revitaslim-free-trial-review]Revitaslim[/url], XPDmgeS, http://healthydiettrends.com/revitaslim-free-trial-review Revitaslim, yzKJDuO, Volume pills maine, HpXAlzt, [url=http://increasedsemenvolume.com/]Volume pills[/url], gbxCaUm, http://increasedsemenvolume.com/ Retail volume pills, AnbXtwo.
Posted by: Idol White | January 28, 2012 at 01:04 AM
evlhetveplvcmph, SizeGenetics, lQpcoQR, [url=http://sizegeneticstore.com/]Sizegenetics discount[/url], mICVdkE, http://sizegeneticstore.com/ Sizegenetics does it work, IMzbYEl, Idol white, fNFEwWe, [url=http://idolwhitestory.com/]Idol White[/url], FbeOggk, http://idolwhitestory.com/ Idol White, mLCRHEf, Propecia side effect, ZRElzjc, [url=http://www.hairenigma.com/propecia-for-hair-loss-treatment_23/]Propecia merck[/url], MkZlxLI, http://www.hairenigma.com/propecia-for-hair-loss-treatment_23/ Among buy propecia comments e-mail name comment [b][/b] - [i][/i] - [u][/u]- [quote][/, uKcbCzF, Car Insurance, FCnbblc, [url=http://phylliszullodesigns.com/]Car Insurance Quotes[/url], KVRkyGS, http://phylliszullodesigns.com/ Nj car insurance quotes, mzWDjlI, acai berry diet, fUfGzwP, [url=http://healthydiettrends.com/revitaslim-free-trial-review]Revitaslim[/url], XPDmgeS, http://healthydiettrends.com/revitaslim-free-trial-review Revitaslim, yzKJDuO, Volume pills maine, HpXAlzt, [url=http://increasedsemenvolume.com/]Volume pills[/url], gbxCaUm, http://increasedsemenvolume.com/ Retail volume pills, AnbXtwo.
Posted by: Idol White | January 28, 2012 at 01:04 AM
vubtbtveplvcmph, Propecia prescriptions, LhUufMD, [url=http://propeciagids.com/]Propecia costs[/url], lnXKNld, http://propeciagids.com/ Bower buy propecia comments e-mail name comment [b][/b] - [i][/i] - [u][/u]- [quote][/, rMiJFKq.
Posted by: Propecia | January 30, 2012 at 01:00 AM
jzxlmtveplvcmph, South Beach Smoke Electronic Cigarette, cbrOOGr, [url=http://ecigbrandsreview.com/reviews/south-beach-smoke-review]South Beach Smoke Review[/url], NVmIvIK, http://ecigbrandsreview.com/reviews/south-beach-smoke-review Southbeachsmoke, WzVVDIR.
Posted by: South Beach Smoke Electronic Cigarette | January 30, 2012 at 01:01 AM
qV louis vuitton handbag iR http://www.t33n.co.uk/blogs/4995/71821/louis-vuitton-handbags-really
Posted by: presceptisren | January 31, 2012 at 12:12 AM
xtxsetveplvcmph, South Beach Smoke Electric Cigarette, DFSKtHz, [url=http://ecigbrandsreview.com/reviews/south-beach-smoke-review]Southbeachsmoke[/url], uPAWIlV, http://ecigbrandsreview.com/reviews/south-beach-smoke-review Southbeachsmoke, VIsuIIW.
Posted by: South Beach Smoke Electric Cigarette | January 31, 2012 at 01:03 AM
deosmtveplvcmph, Viagra, TlHTRCG, [url=http://kapiolaniparkonline.com/]Viagra alternative[/url], pdCLiwg, http://kapiolaniparkonline.com/ Buy viagra now, mhDRoOT.
Posted by: Buy cheap viagra | January 31, 2012 at 01:07 AM
aipghtveplvcmph, Ativan and alcohol, PJOuKtB, [url=http://www.intermezzocoffeehouse.com/ativan.html]Generic ativan[/url], ZHowfwt, http://www.intermezzocoffeehouse.com/ativan.html Ativan prescription info, nTKjAZx.
Posted by: Ativan without a prescription | February 01, 2012 at 01:00 AM
bmccatveplvcmph, Gas Fireplace, vMNgTcL, [url=http://www.bestgasfireplaces.org/]Gas Fireplace[/url], JrVTypG, http://www.bestgasfireplaces.org/ Gas Fireplaces, NYdeUxq.
Posted by: Atlantic gas log inc fireplaces and service | February 01, 2012 at 01:00 AM
wncpqtveplvcmph, Casino games online, DVwtAOC, [url=http://onlinecasinoebook.com/]Casino internet online poker[/url], KzlzoDh, http://onlinecasinoebook.com/ Online casino poker, GRbVnAd.
Posted by: Online Casino | February 01, 2012 at 01:01 AM
efvpmtveplvcmph, Ultimate hgh, TcPkson, [url=http://personalhghblog.com/]Hgh dietary supplements[/url], uyLORMb, http://personalhghblog.com/ HGH, hPpiSFg.
Posted by: HGH | February 01, 2012 at 01:01 AM
wncpqtveplvcmph, Casino games online, DVwtAOC, [url=http://onlinecasinoebook.com/]Casino internet online poker[/url], KzlzoDh, http://onlinecasinoebook.com/ Online casino poker, GRbVnAd.
Posted by: Online Casino | February 01, 2012 at 01:01 AM
efvpmtveplvcmph, Ultimate hgh, TcPkson, [url=http://personalhghblog.com/]Hgh dietary supplements[/url], uyLORMb, http://personalhghblog.com/ HGH, hPpiSFg.
Posted by: HGH | February 01, 2012 at 01:01 AM