Zendo: Realizations

[Simon]

Realizations about Zendo

I will write several posts about the tabletop game Zendo. Background knowledge:
Rules of Zendo, a game of inductive logic.

Part 1: Different piece sets (this post)
Part 2: Downsides of letter strings
Part 3: Geometrical games feel nicer than sequence games
Part 4: Link dump
Part 5: Bongard problems
Part 6: Zendo 1.0 or Zendo 2.0


Part 1: Different piece sets

In Zendo, you build structures of pieces. Any source of pieces is eligible, but ideally you have many copies of each piece and can compose pieces in many different ways. The structures should be easy to grasp by looking.

Example piece sets



The 2017 Zendo release ("Zendo 2.0") has 3 shapes in 3 colors each. Everything is the same size, there are no pips on any piece.



Icehouse pieces. Everything is a pyramid. The 2001 Zendo boxed set has 4 colors of pyramids in 3 sizes each. Large pyramids had 3 pips, medium pyramids had 2 pips, small pyramids had 1 pip.



Digits, and structures are then natural numbers.



Letters from the alphabet. We played Zendo on Lemmings Forums where structures were finite-length strings over A-Z.

Good piece sets

Above, I've already sorted the piece sets according to how good the resulting game feels. The Zendo 2.0 pieces are probably the best -- objectively. I have a deep soft spot for the Icehouse pieces, and prefer them slightly over the 2.0 pieces. But what's much more important: Either of these are geometric shapes, and either makes for a great sets. Geometric shapes and colors fit the game much better than letters or numbers. Visual pattern recognition is a deep part of Zendo.

Reason of my soft spot for the Icehouse pieces over the 2.0 pieces: The Icehouse pyramids come in green, the table looks more colorful with red/yellow/green/blue pieces, and still the number of colors is small enough. Icehouse pyramids have a single clearly-defined tip, this can be used for rules.

I've always thought our Lemmings Forums A-Z game produced insidious problems with harder rules. That will warrant an extra post these days. Until then!

-- Simon

[Simon]

Part 2: Downsides of letter strings

Consider Zendo with strings of letters A-Z and Zendo with strings of digits 0-9.

Letters have some nice properties: Some are vovels, they have an alphabetical ordering, some have straight lines, some have only round strokes, some letters enclose hollow areas (A, B, D, O, P, Q, R), some form Roman numerals.

The problem with letters is that, no matter what rule you play, the set is too big at 26 different letters A-Z.

For example, the letters A-Z have an inherent ordering (A, B, C, ..., Z) and this order requires mental energy. Can you tell at a glance whether KLMUTW is alphabetically sorted? We had rules like "the first letter comes alphabetically earlier than any other" and, for this particular game, the set of letters was not optimal. It would have been easier to play this rule with digits as pieces. Digits require little mental energy to order. The rule works perfectly well with 10 game pieces instead of 26.

Consider the rule "contains at least one Roman numeral, and no misformed Roman numerals". This would have been easier to play with far fewer noise pieces. All of A, B, E, F, G, H, J, K, N, O, P, Q, R, S, T, U, W, Y, Z are pure separator pieces. The rule was hard, and it didn't help that there were 19 pieces of noise when a single separator would have been enough. Similary, the rule "must contain at least one of AKHTBN and all such letters must appear sorted like this among the other 20 noise letters" was very hard. At least, both of these rules generated helpful single-letter koans, pointing exactly to the important few letters. But will you always analyze all 26 single-letter koans? Both of these rules would have been easier with fewer noise letters.

Even with 3-dimensional pieces, 3 different colors make the game very interesting already with 99 % of all rules. 4 different colors aren't needed, they make the game more colorful and I like the 4th color, but it won't enrich the game much. If the 4th color is already expendable, surely some of the 19 or 20 noise letters were expendable.

The set of letters is okay if you absolutely require the exact same set for all games, but even then, the digits 0-9 seem preferable with fewer pieces and a much clearer ordering.

Next part: 3-dimensional pieces make for better games than any string game (letters, digits, or emoticons).

-- Simon

[grams88]

Just a quick thought

Actually could you use Lemming levels in a game of zendo as you could say that Mayhem levels have the buddha nature or whatever else you can think of. (etc)

[Simon]

Right, you can play Zendo with Lemmings levels as structures. Tiles would be the individual building blocks, and entire levels would be be koans. To show koans to other players, one could post screenshots or the level files, or merely name the level if it's a well-known level from DMA's games.

The downside is that it takes long to build new levels. When your building blocks are letters or 3-dimensional blocks in real life, you can easily combine them and finish a new koan in seconds. That's important to test new theories quickly.

You could also restrict the game to level titles from DMA's games, then you wouldn't need screenshots or self-bulit levels. But then you could theoretically win merely by testing all of the finitely many levels.

-- Simon

[Simon]

Part 3: Geometrical games feel nicer than sequence games

By sequence Zendo, I mean Zendo where the koans are numbers, or strings of letters, or sequences of emoticons, ...

We have often played Zendo without turns, as single-team puzzle: Everybody proposes koans and guesses the rule whenever they like. In our games on Lemmings Forums with strings of A-Z, I've often wanted to write a loop. E.g., I wanted to test all strings of length 5 that start with A, and see if they're all white. Or I want to test ZA, ZAA, ZAAA, ZAAAA, ..., up to length 100.

Since everybody is allowed to ask koans at any time, asking all koans in a loop would be legal. Because we shun bureaucracy, the loop itself should be legal. If you tested only a few possibilities with sequence koans, you always believe you're still lacking so much. Testing only a few sequences doesn't feel satisfying. You wouldn't accidentally run into important features to which the loop might lead you.



Compare this with Zendo with 3-dimensional shapes, such as Zendo 2.0 blocks (left) or differently-sized pyramids (right).

With 3-dimensional shapes, it becomes less tempting to test in a loop. There are so many variations already with 2 or 3 pieces in a koan. You can test for color, certain sizes, or other relations, but these tests will rarely require a loop over all possibilities. For most properties, it's easy to select a few most important cases to test with 3-dimensional shapes. This feels satisfying.

When some 3-dimensional koans are built during the early game, many fundamental physical properties arise naturally from the test cases even though you haven't tested explicitly for them: All pieces must relate to all other pieces in some way, some must touch the table, they all must have corners, sides, at least one piece has to be the highest, ... It is natural to develop a surprisingly reliable, though vague feeling for the rule merely from these seemingly-random physical properties exhibited by completely unrelated test cases.

Maybe the most important reason for 3-dimensional shapes is human pattern recognition. We love to find geometrical relationships. Sequence Zendo lacks this visual pattern recognition.

-- Simon

[Simon]

I still want to write a comparison between Zendo 1.0 (with pyramids in 4 colors and 3 sizes) and Zendo 2.0 (with 3 geometric shapes in 3 colors). That's for sometime else.

Part 4: Link dump

I'm interested in game design in general, be it computer games or tabletop games. Here are some collected links.

Zendo homepage by Kory Heath. Scroll down for the links for further reading. I recommend:

• Design History of Zendo, an amazing essay full of lovely details ,
  
• Tips for the Student, and
  
• Tips for the Master.
  

Journal of Elegant Game Design by Kory Heath (designer of Zendo) from 2001 to 2003, on archive.org. The best piece is the article about coercive rules: Rules that don't feel natural in a game, but rather feel tacked-on to force desirable player behavior.

Old Zendo homepage by Kory Heath. Most is already on the current homepage, see first link in this list. But the old homepage has variations for two players (one should probably only play the simple variant, i.e., treat Zendo as a singleplayer puzzle without scoring), and other variations (handicapping).

Elegantly Wasted by Nick Bentley, a list of game design goals. I agree with nearly everything, I'm sure not everybody else will agree. Lovely reading.

Andy's Eleven Principles of Game Design by Andrew Looney, the publisher of Zendo 2.0, with a focus on the practical side: Playtesting, maintaining a gaming/testing group, and getting honest criticism on your designs.

Archive of The Games Journal by various authors, with hundres of lovely articles on tabletop game design. To get truly lost in the vortex that is late-night quality internet reading.

-- Simon

[Simon]

Part 5: Bongard Problems



The six structures on the left page satisfy a secret rule.
The six structures on the right page do not follow that rule.
The task is to formulate the secret rule.

Because it's singleplayer and a static problem, the author can't provide extra structures during your solution attempt. It's the responsibility of the problem author to provide a sufficient variety of examples on both sides.

The secret rule is typically satisfying to discover. You will feel that it's the right answer once you have it.

List of 394 Bongard problems -- can you solve most of them? The first 100 are Bongard's classic set. I warmly recommend to start with these first 100 before tackling later problems.

Harry Foundalis's research into artificial intelligence that solves Bongard problems.

The On-Line Encyclopedia of Bongard Problems -- Welcome
The On-Line Encyclopedia of Bongard Problems -- Solve Problems

-- Simon

[Simon]

Part 6: Zendo 1.0 or Zendo 2.0

From inception of this post to finishing writing, it took nearly 2 years...

I have argued that Zendo with colorful 3-D pieces feels better than sequence Zendo. The most popular piece sets are Zendo 1.0 with pyramids in three sizes, and Zendo 2.0 with three shapes.

Both are great. Which is better? It's subjective.

Here are some desirable properties of the sets (you can debate what's really desirable) with explanations of which set is better at each properies (you can debate again) sorted from important at the top to less important at the bottom.



Ease of aquisition. Zendo 2.0 is easiest to get: The 2017 Zendo 2.0 release costs $40 USD + shipping from the US. It comes with marking stones and rule cards to help beginners with mastering rules of appropriate difficulty.

Zendo 1.0 must be constructed manually from buying several sets of pyramids, e.g., from 2 (minimum), 3 (deluxe) or 4 (insane) Homeworlds sets for $20 each. In addition, you need marking stones from a craft store, or repurpose Go stones.

(You can build your own set from other components, often for much cheaper. But this post is only about the 1.0 and 2.0 sets.)

Supply. Zendo 2.0 comes with 9 copies of each of the 9 different pieces, for $40 plus shipping. This is a high-quality standalone set.

Zendo 1.0, historically, came only with 5 or 6 of each of the 12 pyramids, depending on how you purchased it, for roughly the same $40 to $60. Nowadays, with the Zendo 1.0 pyramids, you can buy more pieces standalone. If you want a deluxe supply of 9 or even 12 copies of each pyramid, you must invest $60 or $80 plus shipping.

Building. Zendo 2.0 pieces are much easier to put together into 3-D structures. Whenever the supply runs out of any shape on a harder rule, it's usually the cuboids (blocks in Zendo 2.0 parlance) because they're the most regular and stable piece in stacks.



Look on the table. Zendo 1.0 comes in 4 colors (red, yellow, green, blue) and I find that the green makes it look more balanced on the table. The above photo uses pre-2016 G2 pyramids, these have pointier tips than G3 pyramids or 2.0 pieces.

I really like the green, and I really like having at least 4 colors. It may be subjective nostalgia, though. Either 3 or 4 colors are considered ideal.

Sometimes people mourn the "loss" of green in Zendo 2.0, but it only ever surfaces in comparison with Zendo 1.0. Zendo 2.0 in 3 colors still looks great on the table. Many color rules are excellect with 3 colors. Rarely, you wish for more colors when playing Zendo 2.0, and they would only be useful for rules à la "number of red plus yellow pyramids is equal to the number of green plus blue pyramids".



^ G3 pyramids on the left (what's shipped nowadays), G2 pyramids on the right.

Pointing. In Zendo 1.0, pointing is natural: The pyramids, especially the pre-2016 ("G2") pyramids, have sharp tips and the pointing direction is obvious. Even the post-2016 ("G3") pyramids, whose tips are roughly 2.5 times as fat as G2 pyramid tips, point reasonably clearly.

All Zendo 2.0 also point; pyramids emit a pointing ray as in 1.0, blocks emit a pointing polyhedron, etc. But I believe the 2.0 terminology defines pointing for all pieces merely because of tradition -- because all pieces were pointing in 1.0, so all should be pointing in 2.0. Pointing with the short end of blocks feels particularly arbitrary, and on top of that, only the closed of the two short ends points.

The upshot is that pointing isn't that important a feature anyway. With 2.0, you will naturally drift to fewer rules about pointing. Not a large loss.



Avoiding Number theory. Zendo 2.0 fares better here; 2.0 has no pips on the pieces, and thus doesn't encourage rules such as "the total number of pips is prime" or "the number of red pips is a perfect square". Zendo 1.0 has one, two, or three pips on each piece. The mere presence of the pips encourages rules about the total number of pips.

I deem rules from number theory less interesting, and fiddly to test. Such rules are fine once in a blue moon, but they would be much more suitable to number Zendo, not to the 3-D blocks that invite geometrical pattern recognition.

From experience, in 1.0, the rule "the total number of pips is even" was often guessed correctly as "the number of small pieces plus the number of large pieces is even". This shows that pip count, while perfectly fine a feature, pales compared with a straightforward piece count.

Number theory is best restrained to its basics: addition, subtraction, odd/even, multiplying/dividing by 2 or 3, working with numbers in the range 0 through 10.



Geometry, graph theory, topology. A wash. Zendo 2.0 allows easier building of various geometrical ideas, but 1.0 has the clearer pointing for graph theory. Depending on the group of players, these areas of mathematics lead to hard and exciting Zendo rules. They can be highly rewarding, but these rules run the biggest risk of being too tough to solve. Use cautiously.

Examples for geometrical rules: "there is a plane of symmetry", "no two pieces have their highest point(s) on the same height", "there is a line in 3-D space that intersects all pieces". As said: These are hard, don't play them during your first couple sessions.

Clarity of structures' features. Zendo 1.0 is better here. The pyramids have an obvious orientation by their tip and base. The worst that can happen is that you can nest a small inside a medium inside a large, then the color of the small inside isn't easy to tell.

Biggest visibility problem in Zendo 2.0: Cuboids (blocks) have an open and a closed square-shaped end. Often, the ends are not interchangable, e.g., by the standard definition of pointing, blocks point only with the closed end. The ends aren't always clear to tell apart: Even for a single standing block, you must look conciously to see whether it's upright or upside down. It's worse for blocks deeply dug into a multi-piece structure.

Another, smaller visibility problem of 2.0: From certain directions, wedges look like pyramids, and you have to move your head to see the difference clearly.

For completeness, Zendo 1.0 with G2 pyramids again is minimally better than 1.0 with G3 pyramids because the G2 pyramids have pointier tips. Pointy tips clarify the pointing direction.

Summary. It's subjective. I recommend Zendo 2.0 for new players, it's a quality standalone set despite its visibility issues. I have a nostalgic weakness towards the 1.0 set, but always have to tell people to avoid number theory in their 1.0 rules.

-- Simon