Zach won the rules competitions for part 2 and 3. For part 2 there are 4 heads. In phase 1 the red head looks at the nails, and the yellow head looks for the top and bottom nails. When a nail head is detected and we go into phase 2, the green and yellow heads come running and fix the wrong nails.

A little extra work for part 3. When a nail head has been detected, there’s a marker written ($) for that nail, to keep track of it.

Code .

No. 2 in part 1, rules section, is actually kind of familiar to me in structure.

Do the very elegant solution of hammering the log instead of the nails, and do it from both ends of the log. There’s an extra little dance with the heads, when a nail is all head, no shaft.

Code .

Her er årets nominerede romaner, med samt link til mine anmeldelser:

Death of the Author by Nnedi Okorafor	##-	Forfatterens død
A Drop of Corruption by Robert Jackson Bennett	###	Korruptionsdråben
The Everlasting by Alix E. Harrow	##-	Den evige
The Incandescent by Emily Tesh	###	Den strålende
The Raven Scholar by Antonia Hodgson	##-	Ravnelærd
Shroud by Adrian Tchaikovsky	#–	Slør

Det bliver dråben, der løber med sejren her. Det var bare så rart at tage en tur til i det her univers.

This week the #puzzle is: Can You Tile the Hexagon? #counting #hexagon #tiling #macmahon #coding #memoization

I’m redoing my kitchen floor using rhombus-shaped tiles composed of two congruent equilateral triangles. One such tile is shown in blue below. How many distinct ways can I use these to tile the outlined region below, which consists of 24 equilateral triangles arranged in a regular hexagon?

And for extra credit:

I’m also redoing my patio, using similar rhombus-shaped stones. How many distinct ways can I tile the outlined region below, which consists of 54 equilateral triangles?

Can You Tile the Hexagon?

Solution, possibly incorrect:

Program

Method 1: I quickly whip up a program. And I add memoization to it. And it arrives at a state, where it can find the answer in 3 minutes.

Result: 20.

I also produce a video, documenting all the possible tilings. (Beware: this post has a lot of videos and images.)

And just to go nuts, I produce screenshots from that video. Yes, I know, you don’t have to thank me.

I go through my notes and end up producing small families of solutions, where each family is connected by rotations. Like this one, with 6 rotations of the same solution.

Or this one:

A smaller family.

And an even smaller.

Like this one:

And finally a singleton.

Mirrors just end of producing the same results as rotations.

I will get back to the fiddler again later.

And for extra credit:

Wikipedia YouTube

Method 1: Run the program again. Ehm. That will take too long. Sigh.

Method 2: Search internet. Hey, what’s this?

“the total number ${\displaystyle N_1(r,s,t)}$ of plane partitions that fit in the ${\displaystyle r\times s\times t}$ box ${\displaystyle \mathcal{B}(r,s,t)}$:”

I think the box is if the hexagon tiling is viewed in a 3d way. The “box” has the same side lengths as the hexagon. The plane partitions correspond to the tilings. Yeah, this one confirms it:

And just to make sure, the fiddler has a 2x2x2 box.

Or, from the other formulation of the count:

Awesome! And the video already did the calculation we want. $\underline{N=980}$.

More extra credit later.

The fiddler, again

Studying the sources mentioned above, I find a different method to produce the same solutions, with slightly squinked rhombi:

This set is numbered like xy: x cubes, the yth way.

The 2 sets of solutions correspond. To make me (and probably nobody else) happy, here’s the full correspondence description.

01	1	11	2	21	7	22	4
23	3	31	9	32	8	33	5
41	15	42	10	43	12	44	6
51	16	52	13	53	11	61	18
62	17	63	14	71	19	81	20

And just to show that these are the same solutions, here’s a comparison of 2 versions of the same, 9 and 31:

Or maybe it’s easier to compare these 2 prettier versions of the same solution:

I then proceed to create a new program, to make really pretty versions of these tilings. Here’s you go, fiddler:

And extra credit:

The top “few rules” solution in part 1:

Like other solutions doing well in this category, there’s an element of “write a lot, possibly delete it later”. Elegant. (Zach’s solution to this one is almost identical.) If the (yellow) reading head sees a =, write a wave below and in front. If it sees a #, delete behind, as too much was written.

Code .

In quest 2, Zach won all 3 rules competitions. Let’s first look at part 1. In this gif, I’ve actually added a rule and changed 2 others to keep everything nice and horizontal. Originally the @’s went up and down.

Features:

1. There are 5 heads. 1 to read the string, 2 to write @’s for A pairs and 2 to write @’s for B pairs.
2. When there’s an A in the string, an @ is written above the string and an @ might be deleted below the string. If there are 2 A’s in a row, the written @ for the 1st A is left alone. And vice versa.

Part 2 is similar. There’s an extra rule to go back to the beginning of a line.

For part 3, again there are new features. Going down and up a column are very different. There are different mechanisms for the horizontal and vertical pairs.

Truth be told, I haven’t quite wrapped my head around part 2 and 3 here. But I see the general features.

Code .

No. 1 in quest 1, part 3, rules, is Zach. Let’s take a look.

Several interesting features:

1. The character > is used to keep track of an intermediate stage. The dark red head stays on the string and the >, the yellow head moves up and down.
2. There’s a blue head at the end of the string. Sometimes a character is added there at the end. Like, do some processing now, delay some processing until the end. (C at the start of the string adds a B at the end, D at the start adds a C at the end.)
3. Sometimes P’s are written and then deleted. Right when a > is deleted and both dark red and yellow and one more move right, then 2 P’s are deleted. In general, P’s are written by green and purple and deleted by blue and red. Writing left and up.

Code .

Anmeldelse af Shroud, af #AdrianTchaikovsky. Roman. 2025. Hugo-finalist.

Nebula nominees . Hugo finalists .

Skitse: Et stort firma bedriver udforskning af rummet ved at sende frossent personale ud til lovende exoplaneter. Efter behov bliver de tøet op. Således bliver det her lille hold vækket, fordi en måne i det her solsystem sender usædvanligt mange radiosignaler. Undersøg det, måske er der liv her.

Er det science fiction? Jeps.

Temaer: Alting handler om penge. Hver eneste handling bliver målt på, om den nu også er pengene (til mad, husly, udstyr osv.) værd, og hvis ikke, så har vi en fryser klar til dig. Argh. Argumentet er, at firmaer i sin tid reddede Jorden fra klimakatastroferne, og at de derfor må være gode til at lede og fordele.

Månen er ikke meget for at fortælle, hvad der foregår. Det tager mange forsøg at få ganske få sekunders video tilbage til holdet. Det er svært, når ens udstyr er baseret på radio, og der allerede er så mange radiobølger i luften. Desuden er der stor tyngdekraft og stort tryk ved overfladen. Der muligvis er flydende. Bøvl. Ved et uheld havner hovedpersonen nede på månen, og ja, der er lokalt liv, og ja, det giver yderligere problemer. Her har vi i øvrigt små bidder, der viser mødet med de fremmede fra de fremmedes synspunkt. De fremmede er mærkelige.

Missionen til det her solsystem er del af en større plan. Gode planeter og måner skal laves om til tankstationer, så senere missioner kan komme endnu længere væk. Så det er lidt i vejen, hvis en stor bunke ressourcer ikke kan bruges, fordi her er lokalt liv.

“Devilment in an engineer is a terrible thing.”

“My thoughts measure its exterior…”

“I reach within myself and tap the encrypted energy, the inflammatory well of stored fats, and I am quick. My segments blaze, and the swift thrash of the tubeworm slows for me until it is leisurely while I am like the wind.”

Er det godt? Hm. Mødet med de fremmede og deres omgivelser er klassisk udforskning og ikke ret meget andet. Og det er ikke lige min stil. #–

No. 2 in the quest 1, part 1, rules competition was gapples2. I’d just like to gush over the code.

There are 5 heads in use.

1. If the 1st character is an A, the 1st head (yellow) will read the rest of the line. Similar for B / 2nd head (red) and C / 3rd head (green).
2. For the rest of the program, there are different rules depending on which character was the 1st one. Essentially, if the 1st character was an A, only the rules connected with that situation are used. Instead of 1 big general program, there are 3 small programs, and only 1 of them will be run in any given situation. This cuts down on the number of rules used in each of the cases.
3. Similarly:
   • If the 1st head is reading, the 2nd and 3rd head will write extra P’s.
   • If the 2nd head is reading, the 3rd and 4th head will write extra P’s.
   • If the 3rd head is reading, the 1st and 2nd head will write extra P’s.
4. If we’re in a situation, where the 1st character was a C:
   • The writing of the extra 2 P’s for the 1st C will be delayed until the program ends. Elegant, as “program ended” = “there’s empty space to write extra P’s”.
   • When a C is read, the situation is partially dealt with and then the C is replaced with a B to deal with the rest. Again, I think this is elegant, as the B related rule was probably already run.
5. The program only uses 2 states.
6. I’m not quite sure what the 5th head is doing, but maybe it’s just there because it will be needed in part 2.

Code . (Find gapples2 in the 2nd position and click the scroll.)

Anmeldelse af The Raven Scholar, af #AntoniaHodgson. Roman. 2025. Hugo-finalist.

Nebula nominees . Hugo finalists .

Skitse: Her til lands finder man den næste kejser via en konkurrence. De 8 huse deltager i konkurrencen, de 7 ved at stille med en deltager, det 8. sådan lidt mere bare. Når der er fundet en vinder, så kan vedkommende herske i fred op til 24 år. Men den her gang gik der lidt kage i det. Andren Valit tabte konkurrencen, men efter nogle år kunne han ikke tage det længere og prøvede at snigmyrde kejseren (bjørne-huset). Det mislykkedes, han døde, og hans familie blev straffet. Hans tvillingebørn er nu blevet gamle nok til at man kan stille dem fornuftige spørgsmål (16), og selvfølgelig bliver de hentet om natten af uhøflige vagter. Det store spørgsmål: Vidste I, hvad jeres far pønsede på? Yana og Ruko Valit får en lang nat …

Er det science fiction? Nix. Fantasy, komplet med magi, omend fortælleren i starten mener, at alt sådan noget jo bare er metaforer og hypnose.

Temaer: Lidt overraskende er det her med Yana og Ruko bare en slags forspil. Fortælleren er nemlig Neema, der er en lærd i ravne-huset. Hun er også “en almindelig borger”, hvilket giver visse problemer med stort set alle de andre studerende og senere kolleger. Da der skulle uddeles straf til Yana og/eller Ruko, var det Neema, der skrev selve dokumentet. (De har en klam tradition her, hvor en kopi af dokumentet bliver syet fast til den straffedes brystkasse.) Senere fik Neema et tæt samarbejde med kejseren, fordi hun er en rigtig god forsker i myter og sange fra hans hjemegn.

Neema var rigtig god venner med Cain (ræve-huset). En ny “vind et kejserdømme”-konkurrence er på vej, og favoritterne er Cain og Ruko (tiger). Neema kan ikke lide den situation. Hendes eget hus, ravnene, bliver repræsenteret af Gaida, der ikke kan komme over klasse-skellet ned til Neema og faktisk er forbavsende ond i den anledning. (I det hele taget er det forbløffende, hvor meget mobning man kan slippe afsted med. Øv.)

Magt. Hoffet. Og alle de intriger og al den politik, sådan noget fører med sig. Officielt må kejseren ikke blande sig i konkurrencen, men … Officielt må der ikke længere kigges på folks klasse og baggrund, når der uddeles jobs, men … Faktisk var jeg lige på nippet til at holde op med bogen, fordi der var for stor en klump af den slags. Men så drejede det lidt.

Kejseren viser sig at være lidt for glad for magt. Og på det sidste har noget flyttet sig. Der bliver installeret nye vagter, der tager sig meget større friheder. Det er blevet mere farligt at have frie meninger. Og hvem er det nu, det minder mig om? Det bliver pudsigt om 20 år at læse bøger fra den her periode, hvor man enten brækker sig over, ja, det er en dårlig idé at hente folk ud af deres hjem midt om natten og fængsle dem under meget dårlige vilkår, det har det altid været, det var det også i USA lige den periode, suk; eller også er der behov for fodnoter.

Konkurrencen minder selvfølgelig den oplyste læser om Hunger Games. Omend udgangspunktet er, at det kun var i gamle dage, at deltagerne kunne dø undervejs.

Tværs gennem det hele er der også en konkurrerende fortæller. En ravn? En bog? Noget, der holder øje med Neema? Blander sig?

“1. No ruler shall govern for more than twenty-four years.
2. Rulers may not choose their own successor.
3. No child of an emperor or empress may take the throne.
4. Instead: seven contenders will compete in a series of Trials; champions from each of the anats. The Dragons, who desire not the throne, shall send a proxy in their name.
5. The winner shall rule without exception, and all shall bow before them.”

‘If this is imperial business why did they send you?’ she demanded. ‘Where are the Imperial Hounds? You can’t just drag someone away like this.’

Er det godt? Som nævnt ovenfor, så havde jeg nogle forbehold undervejs. Det er første bind i en serie, men der er både en afsluttet historie her og nogle gode tråde, der nok vil komme tilbage senere. ##-