## Black and White Are Not Colors

For over two months now, I have been walking Pete Hollows Trail almost daily.

I have made a few friends, I hope, saying hi while cautiously getting around each other on the narrow trail. Nobody meant any harm. Thank you.

I have met the trail maker. He told me that this trail is his masterpiece. I think I start to understand what he meant by this. Thank you.

I also met the Hermit. A friendly looking shy guy wearing an NRA cap, and camping out in all kinds of weather away from the rest of us. He took care of the trail by cutting down fallen logs, too. Thank you.

When I started this, it was still winter, and every little drizzle would soak you. Now, the tree leaves provide shelter enough. Thank you.

There are secret spots, too, that provide inspiration, where you have to step off the trail a bit. Thank you.

Then there are seven streams to cross. They help keeping track of time. Thank you.

Oh, there is light and dark, separate and together since the beginning.

All this makes a place, and forms its character, and builds it a home, slowly, for some.

And then there is Cassiopeia. This is her home.

She taught me that it is not us who own a place. It is not a question of ownership, but of belonging. Thank you.

We have no right to remove her, or anybody, from where they belong.

In Memoriam George Floyd, May 25, 2020.

## Trinity (Solitaire XI)

The puzzle from over a month ago is based on a curios 3-coloring of the edges of the dodecahedral graph:

Whenever you delete the edges of one color, the remaining edges form a Hamiltonian cycle. Incidentally, on the dodecahedral graph there are two different such paths, up to symmetries of the graph. The question arises whether there are more such graphs.

One such example is the Heawood graph drawn on a torus above (i.e. you identify opposite edges of the big shaded hexagon like in hexagonal astroids). There is a very symmetrical (and boring) edge coloring that colors all parallel edges the same way that is triply Hamiltonian, but there is a different one as well. You can think of this as living on the vertices of a map drawn inside a big hexagon, so that you can travel across a hexagon edge by continuing at the same position of the opposite edge. There are three types of roads, color coded, and each inhabitant of your country is issued two colored passes allowing them to travel on roads of those two colors only. Luckily, the colorings above allow each inhabitant to still travel everywhere, regardless of the passes they have been issued.

To turn this into a puzzle, we use the dual tiling by triangles, that gives us a single triangular puzzle piece (and its mirror) to place on the vertices of the graphs. For instance, below we place one triangle (with its translated copies) on three of hexagon vertices, choose another triangle elsewhere, and are tasked to complete this to a triply Hamiltonian tiling. The solution on the right corresponds to the first coloring of the Heawood graph up above.

For a decent puzzle, the Heawood graph is a little bit too small. Now, Hamiltonian cycles on trivalent graphs have been extensively studied (partially because of Tait’s conjecture, which turned out to be incorrect), so I suspect this phenomenon is known. There is a list of small symmetric cubic graphs, the Foster census, and it is no hard to write code that searches for more examples. Below is thus today’s puzzle, on Foster26A, also a map on a larger hexagonal torus. Complete it to make it triply Hamiltonian.

## Nekyia (1947)

In their gray was a memory of all the colors that didn’t exist anymore.

The title Nekyia of this blog post refers to a Greek rite of necromancy, and it is also the title of a little book written by Hans Erich Nossack which appeared 1947 in Germany, just after the war. The quotes are from this book.

It takes place in an unnamed city which has been drained of all color and which represents the negative space of our existence:

Don’t you realize that I am talking about the life span between death and birth? A span of which we know that it stretches across far wider spaces, and about which we remain silent only because it cannot be measured by numbers.

The book thus reverses time. The narrator seeks his mother, in order to be born:

It is possible that I had been forgotten to be born, and the people didn’t like to be reminded of it.

During his search, the narrator meets different people from his past, among them his teacher:

“Why does he tremble?”, these eyes that held and probed into me asked. I didn’t realize that they meant me. “It is not fear,” answered my teacher next to me (…), “it is the trembling of the leaves at nightfall. It is the uncertainty of a being that doesn’t know his mother.”

So my motherless brother took me to my mother. How could I have guessed that he knew her?

His mother tells him the story of his past, a story of war and murder, borrowing from Aeschylus’s Oresteia. But the hardest part lies ahead: The separation from the mother without forgetting the past.

Is this too high a price to pay in order to have a chance for a future?

Nossack’s publisher had wanted a love story to satisfy popular demand. Unable to satisfy the request, Nossack stayed silent for six years.

Most things we were quite certain of couldn’t withstand his piercing eyes. They just disappeared, at first leaving an ice cold emptiness around him.

## Trions (Solitaire X – From the Pillowbook X)

A trion is obtained by taking an equilateral triangle, dividing the edges into n segments of equal length, and cutting from the center of the triangle to two subdivision points on different edges. This will give a particular quadrilateral. If you divide the edges into just 3 segments, there are three different trions, which fit nicely into a single triangle:

This is the single puzzle piece from a previous post. We have also seen this mechanism (explained to me by Alan Schoen) to produce what I called hexons. Today we will look what trions we get when we divide the triangle sides into four segments.

There are six such trions, which fit nicely into two triangles. They can, as we did with the hexons, also be arranged in groups of six around a (former) triangle vertex, to create hexagonal pillows, i.e. hexagons whose edges can remain straight or possess inward or outward kinks.

There are too many of those for my taste, but there is only one (not counting its mirror) that uses each trion exactly once, namely the  one to the right. In the spirit of perfect solitairity, this makes an engaging single puzzle piece.

Can you extend the tiling above so that it tiles the plane? Using it as is gets a bit dizzying (but notice the triangle pattern on the left), so I have replaced it by a simpler version that contains all the essentials.

Two hexagons match along an edge if either both sides have no arrow, or you can keep following the arrow, as in the example. Below are two simple examples of periodic tilings:

There is much more one can do with this piece, but for now let’s end with a homework puzzle: Can you fill the board below so that everything matches?

## Dr Jekyll and Mr Hyde (Ohio IX)

Conkles Hollow is a separate small nature preserve belonging to Hocking Hills State Park, featuring two very different trails. These are like two (very) different aspects of the same person.

One leads inside a deep and narrow gorge, and is as wild as it gets. Violence and darkness abound.

The other trail leads up and around, with views of the cliff faces.

During late winter/early spring, this becomes a study in black, white, and green. This is peace and serenity.

Here, the trees seem to mirror the dramatic dance of the rocks below like dreams, occasionally joined by a counterpoint in red.

Is this just one place?

## Domilogue (Solitaire IX)

Another favorite game/puzzle of mine are domino-variations. For the moment, we will allow to lay out the dominoes only as horizontal strips, and, in contrast to traditional rules, we allow non-matching pieces to be adjacent. The two half-dominoes at a connection we call a link, and as two half dominoes can be combined to a new domino as shown in the second row below, the links become new dominoes in their own right.

Today’s first game I call Domilogue, and it is a 2-person game. By now all practicing solitaire players will have acquired the ability to impersonate a second player, so this should not be a problem. Beginning with this game instead of with the puzzle below makes it easier to explain the mechanics. Domilogue is cooperative. Both players need one set of six dominoes containing all dominoes with one to three eyes as shown on top. The game begins with each player selecting from their set one domino that contains a 1, and placing it in front of themselves. We assume they are seated opposite each other, at distance of 6 feet, wearing masks and gloves.

For the following moves, there is only one rule: The last played domino of one player must represent the link for the next domino the other player plays. To complete a move, they choose again a domino from their pile and add in on either side of the strip so that the link with the piece they connect to is the domino that the other player played in the previous move. Above you see the first two moves, with numbers reminding us when a domino was played, and arrows indicating which domino is responsible for which link.

I hope you find this rule sufficiently mind bending.

After three more moves, both players in my sample game have one domino left and are stuck. We see in these example moves that the orientation of a domino can differ from the orientation of the link it is responsible for. The goal is of course to get rid of both piles. Grab a partner or your other self, and give it a try.

After learning the mechanics, above is a solution to a solitaire variation. In it, you are asked to arrange two sets of all six dominoes in concentric circles as above so that dominoes in the outer (inner) circle are the links for the two dominoes they touch in the inner (outer) circle. This allows for plenty of puzzles: For instance, start with one domino each in the inner and outer circle, and try to complete the circles. Or, make larger circles, either by taking two sets of the six dominoes, or one set of the 10 dominoes that also have squares with four eyes, as in the template below.

## The Passing of Time (Ohio VIII)

The longest trail in Hocking Hills State Park is nicknamed Grandma’s Trail, and it’s an 11 mile roundtrip.

It leads to rather remote regions of the park, making it ideal for self-isolation. The six hours it takes to hike it is an opportunity to contemplate the passing of time, both our own, and the inherent time of the landscape.

Spatial and temporal distance merge in rare views like the one above (from a fire tower).

Then there is an abundance of waterfalls and rock faces: Do we want change, or do we want permanence?

Spectacular views like the one below are rare, reminding us that there is not only passing time, but also meaning.

The trail ends at Ash Cave, another large recess cave with a waterfall.

The enormous overhang provides shelter, but is also an ominous threat: How long will it hold?

## Deficits (Solitaire VIII – From the Pillowbook IX)

I continue last week’s discussion of tiling hexagons with the four trillows below.

Instead of curved triangles I am using the markings with curved arrows that have to match in direction when the tiles are put together. This creates directed graphs like so.

Last time we saw that one can systematically solve such puzzles by first drawing the undirected graphs on the region to be tiled, and then directing the edges. Today we will use the concept of deficits to look at this more closely. The goal is to solve puzzles that tile the hexagon of edge length two with an equal number of green and gray triangles. Above we see two solutions that use 12 and 6 green and gray triangles, respectively. Let’s try it with 10 each.

For this, we need to leave two purple and brown triangles. A solution is up above to the right. To the left is the undirected graph. Each edge comes with a deficit, which counts how many more left turns one takes than right turns when following that edge. This number is determined up to sign only as long as the edges aren’t oriented. The orientation needs to be chosen so that the deficits cancel, as left turns give not unexpectedly green triangles, and right turns grey ones. Below is another example with the brown purple triangles located differently.

Can we do it with 11 green and gray triangles each? Below is the complete graph, from which purple diamonds have to be removed until only two purple triangles are left. There are, up to symmetry, only two possibilities, and we see that the deficits cannot be combined to give 0, Therefore this version of the puzzle has no solution. Thanks for trying.

This can be continued, of course. Try it with 9 green and gray triangles each. Below are two solutions with six triangles of each kind.

## More Moss (Mosses II)

After using Laowa’s 2.5-5X Ultra Macro lens for extreme moss images, I couldn’t resist to try out the new Lensbaby Velvet 28mm lens on the same subject.

People on forums discussing photography fight bitter wars about lenses as if it was about their salvation. It is certainly true that Lensbaby is making lenses on the other side of the spectrum of what many photographers expect.

These lenses are good at not being sharp.  Even closed down they are a bit blurry, and show strong chromatic aberration. The idea is to shoot them wide open, of course, and enjoy how almost everything gets blurred into oblivion. The sporophytes of the moss above are impossible to get sharp with any lens, so why not make them extra blurry?

The Velvet is a 2:1 macro lens (2cm in reality become 1cm on the 35mm sensor), but the working distance is so small that it will often cast a shadow on the subject. So these pictures where taken in the morning when the sun was low.

An advantage of shooting wide open is that you can do everything handheld — just magnify what you want to shoot in the electronic viewfinder and push the button when it’s (relatively) sharp.