Harry Potter and the Method of Entropy

by alkjashRadimentary1 min read31st Mar 201812 comments

11

Exercises / Problem-SetsLogic & Mathematics
Personal Blog

I did some Aversion Factoring and found that my main aversion to writing math is the terrible LaTeX support on WordPress. For now, I’ll try writing things up offline and posting PDFs.

HPMoE is a sequence on the entropy method in combinatorics. It should be fun to think about for anyone interested in information theory and probability as well.

HPMoE1

Please post solutions to the exercises in ROT13.

12 comments, sorted by Highlighting new comments since Today at 3:37 AM
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Our LaTeX support isn't amazing, but probably better than Wordpress. Did you try using ours, and if so, did you run into any problems? (I could probably translate the writeup you have into our format in 15 minutes or so, so it seems doable to me)

It took me about 8 minutes by the clock, and here it is:

https://www.lesswrong.com/posts/KJw47Gw7S5XWduKNW/harry-potter-and-the-method-of-entropy-1-lesswrong-version

Even though I just copied it out of the PDF, I could make most LaTeX work almost immediately by just selecting the text and pressing "CMD+4", and mostly just had to insert some underscores and exponentiation symbols. If you copied the text from a LaTeX editor, it probably would be less than 5 minutes of effort.

The LW LaTeX support does seem to be much better, but it's still quite annoying. For example you'll notice if you copy from PDF the logs get messed up, and worse would happen with original LaTeX (\section, \begin{equation}, etc. all need to get converted).

There's moderately annoying additional inconvenience because I have to convert LyX to LaTeX and probably scrap a bunch of other LaTeX code but I expect it takes > 15 minutes.

I would be happy to take a bet on whether it would take me or you > 15 minutes, given that the PDF version only took me around 8. But ~10 minutes is still a pretty significant inconvenience, though there are probably few things we can do to get the effort of translating below that.

Gunaxf sbe guvf! Vg jnf ernyyl sha. V'z rkpvgrq sbe gur arkg bar. V'ir hfrq jbeqf n ybg va zl nafjref gb cerfreir ebg13-vat. Fb vg'f zvtug abg or nf fanccl nf vg pbhyq or

1.2

  1. Ol Fgveyvat'f Nccebkvzngvba
  2. Fhccbfr Uneel vf jrvtuvat fhofrgf bs jnaqf. Ur jrvtuf gb fhofrgf p_v naq pw ntnvafg rnpu bgure. Gur nafjre yrgf uvz qvfgvathvfu orgjrra gur frg bs jbeyqf F< va juvpu p_v < pw naq F> (juvpu vf gur vairefr). Ohg ur znl trg hayhpxl, or yrsg jvgu gur ynetre bar, guhf erqhpvat ahzore bs cbffvoyr jbeyqf ol ng zbfg snpgbe bs gjb. Naq fb jr trg gur fnzr erfhyg.
  3. svir snpgbevny vf bar uhaqerq naq gjragl. Gur prvyvat bs gur ybt bs gung vf frira.
  4. V qba'g unir n snibhevgr, ohg yrg'f fnl dhvpxfbeg. Abeznyyl bar cvpxf n enaqbz cvibg, fb vg frrzf lbh'er rdhnyyl yvxryl gb trg nal bs gur cnegvgvba fvmrf (nffhzvat ab qhcyvpngrf). Gur zber rdhny-fvmrq gur erphefvir fhoyvfgf, gur zber jbeyqf jrer ryvzvangrq. Vg frrzf lbh'er whfg ng gur zrepl bs enaqbzarff sbe dhvpxfbeg.

2.2

  1. Rahzrengr gur cbffvoyr fpurzrf. Nfx "vf vg va gur ybjre unys?" ercrngrqyl. Lbh pna ebhaq qbja sbe unyirf
  2. V pbhyqa'g pbzr hc jvgu orggre guna ybt_2(a-fdhnerq). Bar jnl bs trggvat gung vf whfg nfxvat rirelguvat gjvpr. Nqzvggrqyl, lbh pna fxvc qbhoyvat hc nsgre lbh'ir qrgrpgrq gur yvr
  3. Hfvat zl nobir fpurzr, ybt_2(a-gb-gur-cbjre-bs-x)

3.1

  1. Vs a vf vasvavgl, jr trg rknpgyl 2. Fb jr pna gnxr njnl gur raq bs gur frdhrapr nf 2 * 2 gb gur zvahf a, juvpu vf 2 gb gur zvahf (a zvahf bar). Gnxvat gung njnl sebz 2, jr trg 1 naq (2 gb gur zvahf (a zvahf bar) zvahf bar)/(2 gb gur zvahf (a zvahf bar))
  2. Xrrc nfxvat "vf vg gur uvturfg cebonovyvgl bs gur erznvavat pnaqvqngrf?" hagvy gurer'f bayl bar bcgvba yrsg. Gur fhz bs yvxryvarff_enax(v) * c_v npebff 1 gb a. Abg fher jung zber V pna fnl jvgubhg xabjvat gur cf.

Avpr gevrf, n srj pbzzragf:

1.2.1 Vs lbh qba'g xabj Fgveyvat'f nccebkvzngvba, guvf pna or qbar whfg ol vagrtengvat ybtk sebz 0 gb a.

1.2.3 Gur ybjre obhaq jr cebirq vf 7. Pna lbh npghnyyl rkuvovg na nytbevguz juvpu gnxrf 7 pbzcnevfbaf gubhtu?

2.2 V npghnyyl qba'g xabj gur rknpg nafjre, ohg lbh pna qb orggre guna xybt_2(a). Urer'f nabgure jnl gb guvax nobhg gur ceboyrz jvgu 1 yvr: vavgvnyyl, lbh unir a puvcf ng cbfvgvba 2 ba gur ahzore yvar. Rnpu dhrfgvba, lbh trg gb cvpx n fhofrg F bs gur puvcf naq Serq zbirf rvgure F be F_pbzcyrzrag yrsg ol 1. Puvcf snyy bss gur obneq jura gurl ernpu 0. Ubj znal zbirf qbrf vg gnxr gb rafher bayl bar puvc erznvaf? Sbe rknzcyr, lbh fubhyq or noyr gb jva gur tnzr ba 32 cbffvovyvgvrf naq 1 yvr va 9 zbirf.

3.2 Gur fgengrtl lbh qrfpevorq vf ernyyl onq jura nyy gur c_v ner rdhny gb 1/a; lbh'yy arrq a/2 zbirf ba nirentr vafgrnq bs gur jbefg pnfr ybt_a jr unq orsber.

Gunaxf sbe gur cbvagref. Jvyy erguvax

1.2.3 Fbeg guerr jnaqf. Vs lbh'er yhpxl guvf pbhyq gnxr 2 pbzcnevfbaf. Vs lbh'er hayhpxl, 3. Jrvtu gur zvqqyr jrvtug ntnvafg gur bgure gjb. Guvf gnxrf lbh gb 4 be 5. Vs lbh unir gjb cvyrf bs gjb, cyhf gur zvqqyr, fbeg rnpu cvyr (6 be 7). Vs lbh unir n cvyr bs 3, fbeg vg (8, 9 be 10), naq gur bgure vf qrtrarengryl fbegrq. Fb rvtug gb gra

2.2 BX, V qvqa'g trg na rknpg erfhyg, ohg zl nytbevguz vf gnxr n fhofrg jvgu unys bs gur puvcf ng 1 naq 2 erphefviryl. Fb nsgre lbhe svefg dhrfgvba, lbh unir n unys gung vg'f va hayrff ur yvrq, naq unys gung vg'f va vs ur yvrq. Gura nfx fbzrguvat gung unf unys bs rnpu bs gubfr.

3.2.2 Ur fubhyq perngr gur fznyyrfg pneqvanyvgl fhofrg bs fpurzrf jvgu cebonovyvgl 1/2 naq nfx vs vg'f va gung. Erphefr. Vs ur pna'g trg rknpgyl gb bar unys, cevbevgvfr pybfrarff gb bar-unys (fb gur fhofrgf ner snveyl rdhny).

1.2.3 Gurer rkvfgf n fgenvtug naq aneebj cngu gb fbegvat 5 ryrzragf va rknpgyl 7 zbirf. Xrrc ybbxvat sbe pbzcnevfbaf gung qvivqr gur ahzore bs cbffvoyr jbeyqf nf pybfr gb rirayl nf cbffvoyr.

2.2 Gung'f ebhtuyl evtug, tbbq wbo! Sbe shegure qrgnvyf ba guvf ceboyrz, frr gur "Yvne Tnzr" va Nyba&Fcrapre Pu 14.

3.2.2 Rknpgyl! V'yy fnl zber nobhg guvf arkg gvzr ohg lbh'ir ebhtuyl erperngrq gur vqrn bs Uhsszna pbqrf.

Gunaxf sbe gur ercrngrq srrqonpx!

1.2.3. Guvax V'ir tbg vg! Svefg, pbzcner gjb cnvef, gb trg n < p naq o < q. Gura pbzcner n naq o. Pbf V'ir abg fnvq nalguvat gb qvfgvathvfu gurz, V pna fgvpx jvgu avpr abgngvba naq unir n < o. Abj pbzcner r gb o. Jr'er hc gb 4 pbzcnevfbaf.

Vs vg'f fznyyre jr unir n-r-o, o-q naq n-p. Jr pna pbzcner q gb p. Vs q'f yrff, gura jr'er qbar (va 5!) naq n-r-o-q-p. Vs p jnf yrff gura jr unir n-r-o, o-q naq n-p-q. Pbzcner p naq r. Vs vg'f yrff jr'er qbar (va 6!). Vs p vf terngre pbzcner vg gb o, naq gura jr'er qbar.

Vs r'f ynetre jr unir n-o-r, o-q naq n-p. Pbzcner r naq q. Jr gura unir n-o-r-q be n-o-q-r. Gura pbzcner p gb juvpurire vf abg gur gbc. Gura pbzcner p gb rvgure gur frpbaq be gur gbc, qrcraqvat ba gur erfhyg.

V guvax

Gur svefg unys vf evtug, gur xrl vqrn vf gb pbzcner r gb o, naq purpx gung guvf oernxf n havirefr bs 15 cbffvovyvgvrf vagb 7 naq 8. Nsgre gung lbh unir gb or n ovg zber pnershy nobhg znxvat gur evtug pbzcnevfbaf, sbe rknzcyr vg ybbxf yvxr lbh'er nffhzvat n<r<o whfg orpnhfr r<o, jura vg pbhyq nyfb ybbx yvxr r<n<o.

Another alternative you might consider is using my Equation Editor to generate SVG-rendered formulas, which you can then insert into your posts as images. (Being vector images, they’ll be high-quality, scalable, etc.)