Trivalency (Solitaire VIII – Pillowbook XIII)

Triv 0

Today we will use the special trillows above to tile the region(or similar ones) below:

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Let’s call for the moment the grayish and greenish trillows simple, and the brown and purple ones special. Here are two puzzles as starters: Can you tile the same region just with the simple trillows? Can you do it using just one special trillow?Triv 3

As an outline what is involved in creating and solving puzzles, let’s overlay the region we want to tile with a graph as above. The purple tiles will later correspond to special trillows, the brown ones to simple trillows. Let’s get rid of most of the purple tiles. We do so by deleting a yellow bridge between two adjacent purple tiles, and replacing them with brown tiles, keeping the remaining yellow edges. This amounts to partially tiling the purple region with diamonds. By now you should have solved the two puzzles at the beginning.

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Now we orient the edges of the yellow graph. This is a bit deliberate. The outer loop we orient counterclockwise, and for the inner graph we make sure that at the two trivalent vertices the not all three edges point in the same direction.

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Now we inflate/deflate all triangles according to the direction of the arrows, and obtain a tiling with just two special trillows.

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We have used three more gray trillows than green trillows. Can you do it with just one more gray trillow? Or with no green trillows a all? More about this next time.

Horizontal and Vertical (Ohio VII)

From last week’s Old Man’s Cave, there is a path that takes you along the stream to two different waterfalls. Following either way becomes a meditation on horizontal and vertical motion.

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To reach the lower falls, you will need to cross yet another bridge.

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And again one has a choice: continue downstream, or climb the steep cliff for an alternate route back to Old Man’s Cave.

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The upper fall is even more spectacular.

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There are more bridges further upstream, including this double bridge at two different levels. This view finally reminds us that we live in a 3-dimensional world. 

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Trillows (Solitaire VII – Pillowbook XII)

Below are the 11 possible triangular pillows, or short trillows (up to motions).


They are obtained by replacing in an equilateral triangle some edges by circular arcs curving either inwards or outwards, The numbers indicate their defect, which is the difference between the number of convex and concave edges. This defect is additive when you tile larger regions. This means that the sum of the defects of the tiles will be equal to the defect of the entire region.

Triangles 01

For instance, suppose we want to tile a straight triangle with the three trillows above. The green one has defect +2, while the red and yellow ones have defect -1.  To see how this becomes useful, let’s suppose in such a tiling there are just  trillows with defect +2 (A many)  and  trillows with defect -1 (B many).

A triangle of edge length 3 requires 9 trillows, so A+B=9. On the other hand, the total defect gives the equation 2A-B=0, because a straight edged triangle has defect 0. Solving this shows that A=3 and B=6, as is indeed the case in the left figure. Now go ahead and try to tile a triangle of edge length 4 with the same type of tiles…

Periodic 2 01I have written about the four purely circular version briefly before. If you want to use them for a periodic tiling of the plane, the total defect of a fundamental piece must be 0. Above and below are examples that use just two trillows.


Every purely circular simply connected region has total defect +6, so if we want to tile a large circular triangle with C cyan, B blue, P purple and G green circular pillows, we have two equations to satisfy: C+B+P+G=22 for the total number of tiles, and3C+B-P-3G=6 for the total defect. Solving this for B and P gives B=14-2C+G and P=8+C-2G. Below you can see the region of values of C and G for which C,B,P,G to be nonnegative. Each dots in this regions represents a pair of integers (G,C) for which there is maybe a solution.



The edges correspond to solutions where only three of the tiles are used, and the vertices to those with only two tiles. The blue and purple lines are examples for which G and C are constant. At their intersection we have G=3 and C=4, which determines P=6 and B=9. Below are two different solutions with these numbers.

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For instance, if we wanted to tile this triangle without blue, the potential solutions are on the top edge. Below are actual solutions for G=0,2,4,6. Is there one for G=8 or even for G=10?

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It would be interesting to find further simple obstructions to the possibility of tiling such shapes.

Old Man’s Cave (Ohio VI)

The most prominent feature of Hocking Hill’s State Park is Old Man’s Cave, reportedly the home of a hermit who lived there in the late 17th century. If not for the visitors, this place comes close to my ideal of a place for contemplation.

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The recess cave itself is very open, like a balcony, and unsuitable for permanent shelter. This is where we stop, free our mind of ourselves, and let the raw landscape take its effect.

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Another view is downstream, towards the bridge. This is the place to contemplate decisions. Three paths meet here. So one has a threefold choice: remain in the cave, or cross the bridge, and then continue either left or right. We will talk about the two latter options next week.

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From the bridge itself, one has the view onto a serene waterfall. This is the place to find focus.

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Finally, if we decide to leave, there is always the look back, turning presence into memory.

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Hexons (Solitaire VI – Pillowbook XI)

Hexons template 01

Take a regular hexagon, and mark points on each edge. Connect these points with the center of the hexagon to obtain six quadrilaterals. If we choose the marked points so that they divide an edge in one of the proportions 1:3, 2:2, or 3:1, there are exactly six such quadrilaterals (not counting mirror symmetric copies), and as you can see, they can tile a hexagon. I will call these hexons, in analogy of a puzzle by Alan Schoen that I will discuss in the near future. Above are mirror images of this solution, print & cut them out, then glue them together back to back.

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Here is a first puzzle: Suppose you flip one of the three pieces over (like here the blue one) that is not mirror symmetric, can you still tile the hexagon?

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More complicated is the challenge to use seven hexons of each type to tile the seven hexagons above so that corners only meet corners, not just edges of neighboring hexons.


Triangle 01

Likewise, can you properly tile the above shape, so that the vertices of the tiles only meet at other vertices?

After playing with these hexons for a while, you will find it tempting and useful to group three of them together along so that they meet at their 120º-vertex. This can be done in 11 possible ways:



You will get inflated equilateral triangles, where an edge is either straight, or has a kink inside or outside. I have discussed some of these triangles before, and the square shaped analogies (which I called pillows) in a long sequence of blog posts. Understanding how these 11 triangles can fit together will help to create and solve many more puzzles for the hexons. We will begin this next week.

In Memoriam (Solitaire V)

I just learned with sadness that John Conway has died two days ago, on April 11, of Covid-19. I am in no position to write an obituary. His playful creativity led to a synthesis of simplicity and depth that I will always gratefully admire. 

I like puzzles with few pieces, as the readers of this blog know.Pieces

Here is such a puzzle with just one puzzle piece, which comes with its mirror image. I suggest to print the template below, cut along the fat lines, and fold & glue the diamonds into triangles. This will give you six identical triangles that you can flip over to get the mirror symmetric version. I will explain later how this puzzle piece came into existence.



We are going to use this tile according to timely rules: We may put two triangles together along an entire edge if the colors nowhere match along the edge:Rules

This is a strong limitation, but we can still use this puzzle piece to tile larger triangles, like so:


Do you think you can tile even larger triangles? Try it!

Below is a more interesting puzzle. I have been trying to tile the hexagonal ring, but failed, as there is no way to place the puzzle piece into the remaining gap. Can you find a solution that closes up and doesn’t violate the rules?



Then you can make your own challenges: Print a game board with triangles as below, and place randomly two of the puzzle pieces onto it. Can you form a chain that (legally and ethically) connects the two triangles? Below is a very simple example with solution.


Here are two more puzzles. At least one of them is harder than you probably like. But who knows.


A slightly cryptic hint about what is going on here is below. Stay safe.



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When it’s cold and cloudy outside, it’s time for a little introspection.The pictures here were taken with Laowa’s 2.5-5X Ultra Macro Lens, at or near the maximal magnification. DSC 5757

Focussing gets hairy, literally. What you see here are mosses, with some morning dew. Below is another variety.

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All this is in reality just a few millimeters across. This makes it hard to look for motives, because when you are standing up, you can’t tell what is at your feet.

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Below is an algae. The fine hairy threads are quite something up close. 

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The problem with this perspective is: If you have seen that much, you want to see more. 

Presents to Self (Solitaire IV)

A birthday in quarantine is a limited experience.

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Above is the 2020 Glenburn Moonshine Elite, the only Darjeeling that has made it to me this year so far. An amazing tea.

Below are some Pu-Erh cakes that will hopefully last a year.

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More food, for heart and brain:

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Of course there needs to be a puzzle. Let’s call it Quarantine. You have to go on an errand, visiting all twenty vertices of the map below (it will take a while to walk this, I hope).Errands 01

But there is a curfew. This means that you can only visit each of the twenty places once. If you are seen somewhere a second time, you become suspicious and will be eaten by a grue. You also need to end up where you started — sleeping in someone else’s home is suspicious.

Finally, traveling from place to place requires a special permit in the color of the edge along which you travel. So you will need to carry a few permits (being eaten by a grue is unpleasant). For instance, an orange permit will allow you to use any of the orange edges as often as you dare, but only those. There are six different colors, and hence six different permits.

Permits are expensive. What is the smallest number of permits that allows you to visit each place exactly once, returning to your starting point at the end?


Poly-Worms (Solitaire III)

I have occasionally written about polyforms before. These are shapes obtained by putting simple shapes (like squares) together to form more complicated shapes. In the case of two squares, you ged dominoes, and more generally polyominoes. If you use other shapes, you get general polyforms.



If we, in the insatiable desire for more, allow the shapes to change size, we get even more general polyforms. The ones we study today I will call poly-worms. We start with an isosceles right triangle, halve it, and attach the smaller copy to the larger, edge-to-edge. Then we keep going, halving and attaching restlessly. Above you see the first four generations, giving us eight 4-worms, which come in mirror symmetric pairs.Template 01

Above is a template that allows you to print all eight 4-worms at once. Growing the polyworms further leads to problems with self-overlapping, but also to the tantalizing possibility of having polyworms with infinitely many sides. Maybe more about this in the finite future.

Polyworms 01

Let’s practice tiling with 4-worms. Below are combinations of 4-worms that can be used to tile the plane periodically by translating them. There are also two 4-worms that tile the plane by themselves. It should be amusing to study this for general polyworms.

TwoTiles 01

Here are two puzzles, hopefully not too easy. The goal is to tile the left one with 8 and the right one with 16 4-worms.  You will need the same number of each kind.

Puzzles0 01

The Personal Cave (Ohio V)

Part of Hocking Hills State Park is Rock House, with two short trails and one main feature, a preview of which you can see en miniature below:

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A cave from the outside can be a foreboding place. Do we dare to enter?

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The pattern of a cave per se doesn’t appear in Christopher Alexander’s Pattern Language, but there are a Child’s Cave and Secret Place, which relate to it.

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We can uncover the secret of a place only by having the courage to step into it. This is like entering somebody’s private space: When inside, we see the world from a new perspective.

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After a while, the darkness dissipates, and we feel simultaneously protected and protecting.

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After leaving a cave, something has changed. We will not be afraid anymore.

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