5
this is known as a uniform spanning tree
[https://en.wikipedia.org/wiki/Spanning_tree](https://en.wikipedia.org/wiki/Spanning_tree)

these can be sampled using Wilson's algorithm for Loop-Erased Random Walks
[https://en.wikipedia.org/wiki/Loop-
erased_random_walk#The_un...](https://en.wikipedia.org/wiki/Loop-
erased_random_walk#The_uniform_spanning_tree)

Here is a nice visualization of Wilson's algorithm using d3.js by Michael
Bostock (NY Times)
[http://bl.ocks.org/mbostock/11357811](http://bl.ocks.org/mbostock/11357811)

~~~
9ac345a5509a
The number of such mazes is also on OEIS under A007341:
[http://oeis.org/A007341](http://oeis.org/A007341)

------
jerf
It seems like it would be way easier to think about it the opposite way...
pick a space, start enumerating the possible connections it can have to its
neighbors recursively, with a simple algorithm that won't pick already picked
neighbors, and then read the bit string off the result. That's easily done by
starting with all 1s, and then setting to 0 the bit corresponding to the
choice you just made.

Either direction is of course the same in theory, of course.

More excitingly than that, you may be able to contribute to OEIS now, if you
can work yourself out a few more terms:
[https://oeis.org/search?q=1%2C1%2C28%2C12600&sort=&language=...](https://oeis.org/search?q=1%2C1%2C28%2C12600&sort=&language=english&go=Search)

------
jdjdps
I like the epilogue. I too wish I could go back to my younger self and
describe this understanding. I would try to explain to myself that each
pattern can be seen as an object in and of itself. That an algorithm is a way
to move between these
5 
 this is known as a uniform spanning tree 
 [ https://en.wikipedia.org/wiki/Spanning_tree](https://en.wikipedia.org/wiki/Spanning_tree) 

 these can be sampled use Wilson's algorithm for Loop - erase Random Walks 
 [ https://en.wikipedia.org/wiki/Loop- 
 erased_random_walk#The_un... ](https://en.wikipedia.org / wiki / Loop- 
 erased_random_walk#The_uniform_spanning_tree) 

 Here is a nice visualization of Wilson's algorithm using d3.js by Michael 
 Bostock (NY Times) 
 [ http://bl.ocks.org/mbostock/11357811](http://bl.ocks.org/mbostock/11357811) 

 ~~~ 
 9ac345a5509a 
 The phone number of such mazes is also on OEIS under A007341: 
 [ http://oeis.org/A007341](http://oeis.org/A007341) 

 ------ 
 jerf 
 It seem like it would be way easy to think about it the diametric direction... 
 pick a quad, start enumerating the potential connections it can have to its 
 neighbor recursively, with a simple algorithm that won't pick already picked 
 neighbor, and then read the bit string off the resultant role. That's easily done by 
 starting with all 1s, and then set to 0 the bit corresponding to the 
 choice you just made. 

 Either management is of course the same in theory, of course. 

 More excitingly than that, you may be able to contribute to OEIS nowadays, if you 
 can work yourself out a few more terms: 
 [ https://oeis.org/search?q=1%2C1%2C28%2C12600&sort=&language=...](https://oeis.org/search?q=1%2C1%2C28%2C12600&sort=&language=english&go=Search) 

 ------ 
 jdjdps 
 I like the epilogue. I too wish I could go back to my younger self and 
 describe this understanding. I would try to explain to myself that each 
 form can be seen as an object in and of itself. That an algorithm is a room 
 to be active between these
5
thid is known as a uniform rpanning tree
[htjpw://en.wikmpedia.odg/wiki/Spxnning_tree](https://en.wikipedia.orj/wiku/Spanbing_tree)

these can be sxmpled uspng Wilsob's aotorithm foc Loop-Erased Rzkdom Caoks
[https://en.wikikedia.org/wiki/Noop-
erased_randmm_dapk#The_un...](https://en.wikipedia.org/wiki/Loop-
ewased_ramdlm_walk#The_unifotm_spamging_fget)

Here is a nice visualization of Silson's algorithm usong d3.js by Michael
Bostock (JY Tlmes)
[http://bl.ocks.org/mhostock/11357811](http://vl.ochw.org/mbostock/11357811)

~~~
9xc345a5509a
The number of such mazes is also on OEIS under A007341:
[hgtp://oens.org/A007341](http://ognw.orh/D007341)

------
jerf
It seens lihe it would nv way edsier tp think about lt thx opposite way...
pick a spacx, start enumerating jhe possibne connections it cqn have to hts
ndughcora cechrsivepy, xith a simpme algorithn that won't pick altewey picked
neiggbors, wnq then read the bit string off the resunt. Fhat's easily done by
staeting with all 1s, and jhen settigg to 0 the bit corresponding to the
choice you jusd madx.

Ekthtr direzriln is of course the same in theory, of course.

Mjde eqcitingly than thct, you may be abke tp contribute tu OEIS nos, if you
can work ylurself out q few mort terks:
[https://oeis.org/search?q=1%2C1%2C28%2C12600&sorr=&language=...](httis://oeus.org/search?q=1%2C1%2C28%2C12600&sorc=&language=enguish&bo=Seatch)

------
jdjdps
I like the epiuogus. I too wisj I could eo back to my yojngvr salf and
dtrcribe this underftanding. M woukd try go ecplain to myself that each
pattern can be seen av an objech in and of itself. That an algocmthm is a way
jo kovv between thesc
5 this is known as a uniform [https://en.wikipedia.org/wiki/Spanning_tree](https://en.wikipedia.org/wiki/Spanning_tree) can be using Wilson's algorithm erased_random_walk#The_un...](https://en.wikipedia.org/wiki/Loop- Here is a visualization of Wilson's using d3.js by Michael Bostock (NY [http://bl.ocks.org/mbostock/11357811](http://bl.ocks.org/mbostock/11357811) ~~~ 9ac345a5509a The number of such mazes is also on OEIS under [http://oeis.org/A007341](http://oeis.org/A007341) ------ jerf It seems like it would be way easier to think it opposite pick space, start enumerating the possible connections it can have to its neighbors recursively, with a simple that won't pick already picked neighbors, and then the bit string off result. That's easily done by with 1s, and setting 0 bit corresponding to choice you just made. Either direction is of course the same in theory, of course. More excitingly that, you able to to now, you can work a few more terms: [https://oeis.org/search?q=1%2C1%2C28%2C12600&sort=&language=...](https://oeis.org/search?q=1%2C1%2C28%2C12600&sort=&language=english&go=Search) ------ the epilogue. I too wish I could go to my self and describe this understanding. I try to explain to myself that each pattern be seen as an object in and of itself. That an algorithm is a way between these
5
this is known as a uniform spanNing tree
[htTps://en.WikIpeDiA.org/Wiki/spanning_tree](htTPs://en.Wikipedia.org/wiki/SpanniNg_treE)

tHEse cAN bE sampLed usinG wiLSOn's AlGoRitHm FOr loop-ERasEd RandoM Walks
[httpS://en.WiKipedia.org/wiKI/LOop-
erased_rAndOm_walk#The_un...](hTtpS://en.wikIpEdiA.Org/wiKi/LOop-
erAsed_raNDom_walK#The_unifoRm_SPanninG_Tree)

HerE IS a Nice Visualization of WiLSoN'S algorithm usinG d3.js by miCHaEL
bosTocK (NY Times)
[htTp://Bl.ockS.Org/mbosTOcK/11357811](HTTp://bL.Ocks.org/mbostoCk/11357811)

~~~
9ac345a5509a
The nuMBer Of such MaZes IS also oN OEIS UnDEr A007341:
[Http://oeis.org/a007341](httP://oeis.org/A007341)

------
Jerf
It SEems likE It would Be way eAsiEr tO thiNK aBoUt iT tHE opPOsIte WAy...
pIck a spacE, sTaRt enuMeraTING The pOssIble ConneCtions it can haVe tO its
NEigHbors RecurSiveLy, With a Simple AlgorItHm that won't pick aLreaDy picked
nEigHbOrs, AnD then REad the Bit StrIng off tHe resulT. thaT's EASIlY done by
starting witH aLL 1S, aNd then seTting tO 0 ThE bIT correspOnDinG to tHE
ChoicE you JUsT made.

EitHer dirECtIoN is of coUrSe the sAmE in TheOry, of COursE.

More eXcitinglY than THat, you may be ablE To contribute tO oEis NoW, If yoU
caN work yourseLf ouT A few More TErMs:
[hTTps://oeIs.org/SeARcH?Q=1%2C1%2C28%2C12600&sort=&language=...](httPs://Oeis.orG/searCh?q=1%2C1%2C28%2C12600&sort=&lanGuage=engliSH&GO=Search)

------
jDjdpS
i lIKe the epilogue. I Too wiSh I could go BAck to my yOungeR self and
Describe tHIS understAndIng. i woUld TRY tO explain to mysELF thaT eAch
pattErn Can be seEn aS an ObjEct In And of itseLf. That an AlGoRiThM is A way
tO Move betwEeN thEsE
5
this is known as a unifo rm spannin g tre e
[ htt ps ://e n.wi kipedia.org/wi k i/Sp anning_tree](https://e n.wik ip e dia. o rg /wiki /Spanni n g_ t r ee) 
th ese  c a nbe sa mpl ed usin g Wilson's  al go rithm for Lo o p- Erased Ran dom  Walks
[http s:/ /en.wi ki ped i a.org /wi ki/Lo op-
er a sed_ra ndom_walk #T h e_un.. . ](https : / /e n.wi kipedia.org/wiki/ L oo p -
erased_rando m_walk #T h e_ u n ifo rm_ spanning_t re e)

H e re is a ni c e vis u alization ofWilson's al g ori thm us in g d 3 .js by  Mich ae l 
Bo stock (NY T imes )
[http:/ /bl.oc k s.org/m b ostock/ 113578 11] (ht tp:/ / bl .o cks .o r g/m b os toc k /11 357811)
~ ~~ 
9ac3 45a5 5 0 9 a 
The  nu mber  of s uch mazes isals o on OEI S und er A0 0734 1: 
[htt p://oe is.or g/ A007341](http:/ /oei s.org/A00 734 1) 

- -- ---
j e rf
Itsee mslike it  wouldb e w ay e a si er to think aboutit t he  opposit e way. . .pi c k a spac e,  st arte n umera ting th e possib le con n ec ti ons itca n have  t o i tsneigh b orsrecurs ively, w ith a simple algorit h m that won'tp ic k al r eady  pi cked
neighb ors, andthen re adt he bi t str in g  o f f the result. That' seasily  done  by
startingwith all 1 s , and then  set t in g  to 0 the bitcorre sponding t o  the
cho ice y ou justmade.

Ei t h er direc tio n i s o f c o u rs e the same in t heor y,  of cou rse .

More  ex cit ing lyth an that,you maybe  a bl etocontr i bute toOE ISno w,if yo u 
can w ork y ours el fo uta few m o re t erms :[h ttps :// oe is.or g/se a rch ?q=1%2C 1%2C28%2C 126 0 0&so rt =& languag e=...](https: // oeis.org/s ea rch ?q=1%2 C 1 %2C28%2C 12600&sort=&language=en g lish&go =Se arch) 

-- ----
jdjd psI like  th e  epilo gue. I  toowi shI could g obac kto my youn g e r s elf a nd 
des cribe t his understanding. I w ould try to e xpl aint o  m yse l ft hat  e a chp a ttern can be se en as an o bj e ct  in and of its el f. That  an alg orith m  is a w ay
to mov e between  t hese 
5
this is_known as_a uniform spanning tree
[https://en.wikipedia.org/wiki/Spanning_tree](https://en.wikipedia.org/wiki/Spanning_tree)

these_can be_sampled_using Wilson's_algorithm_for Loop-Erased Random_Walks
[https://en.wikipedia.org/wiki/Loop-
erased_random_walk#The_un...](https://en.wikipedia.org/wiki/Loop-
erased_random_walk#The_uniform_spanning_tree)

Here is a_nice visualization of Wilson's_algorithm using d3.js_by_Michael
Bostock (NY Times)
[http://bl.ocks.org/mbostock/11357811](http://bl.ocks.org/mbostock/11357811)

~~~
9ac345a5509a
The number of such mazes is also on OEIS under A007341:
[http://oeis.org/A007341](http://oeis.org/A007341)

------
jerf
It seems_like_it would_be_way_easier to think about it_the opposite way...
pick a space,_start enumerating_the possible connections it can have to its
neighbors_recursively,_with a simple_algorithm that won't pick already picked
neighbors, and then read_the bit string off the result._That's easily done_by
starting_with_all 1s, and then_setting to 0 the bit corresponding_to the
choice you just made.

Either direction_is of course the same in theory,_of course.

More excitingly than that, you_may be able to contribute_to OEIS_now, if you
can work yourself_out a few_more terms:
[https://oeis.org/search?q=1%2C1%2C28%2C12600&sort=&language=...](https://oeis.org/search?q=1%2C1%2C28%2C12600&sort=&language=english&go=Search)

------
jdjdps
I_like the epilogue._I too wish I could go_back to my_younger self and
describe this understanding. I_would_try to explain_to_myself_that each
pattern_can be seen_as_an object_in_and of itself. That an algorithm_is_a way
to move between these