🔵 ⚔️ Death Match (Blue Meadow): Zendo ⚔️ 🔵

Research, experiment, and reveal the absolute truth to claim victory.

Discover a hidden pattern before your opponent does. You get to construct structures to help you figure them out, which will be revealed to obey the hidden pattern or not.

This game first appeared in DM Colosseum (with a few rule changes), and is based on the 2001 board game Zendo by Kory Heath.

ABOUT STRUCTURES

In this game, you will construct structures. Structures in turn are made of one or more pieces.

A piece is the building block of a structure. All pieces have the same color. However, they have different sizes, with a corresponding number of pips on them. There are three kinds of pieces:

• Small-size with 1 pip on it
• Medium-size with 2 pips on it
• Large-size with 3 pips on it

In addition, pieces also have an orientation and can point in one of the four cardinal directions (up, down, left, right). Altogether, this gives 12 different pieces, taking into account their orientation:

A structure is made using these pieces, satisfying these rules:

• There is at least one piece.
• All pieces follow an underlying square grid. Each piece occupies a square of the grid.
• No two pieces overlap each other, i.e. no two pieces are on the same square.
• The bounding box of a structure is the smallest rectangle that is aligned to the grid and fully contains the structure. The bounding box of a structure cannot be larger than 6×6.

In addition, structures are translation-invariant; that is, the entire structure can be moved on the grid without rotating, and it will still be the same structure. However, structures are not rotation-invariant nor reflection-invariant; if a structure needs to be rotated or mirrored to become another structure, the two structures are different.

🎭 The following are several examples of what are valid structures and what are not.

• Everything in the top half are valid structures, disregarding the bounding box.
• Structures are not rotation-invariant; therefore, the first two structures are different.
• However, structures are translation-invariant. Therefore, the last three structures are the same. In fact, the fourth and fifth structures don't have a correct bounding box; the bounding box of that structure is given by the sixth structure.
• Everything in the bottom half are invalid, each breaking a rule for valid structures as described above: first one is empty, second one is not aligned to the grid, third one has multiple pieces in one square, and fourth one has bounding box not within 6x6.

A pattern is a predicate about structures; in other words, something about structures which can either be true or false. Phrasing it differently, a structure can obey a pattern (the pattern is true for the structure), or not. Patterns satisfy the following rules:

• Patterns unambiguously divide all valid structures into those obeying the pattern and those not obeying the pattern.
• Patterns only refer to the pieces in the structure. They do not refer to anything else; for example, they do not refer to other patterns, the time, the players, and so on.
• Since structures are translation-invariant, patterns do not refer to the absolute position of a piece. However, they can refer to the relative position with respect to other pieces.
• Patterns can also refer to the absolute orientation of a piece, since structures are not rotation-invariant.

🎭 The following are some example patterns to give you an idea how patterns look like. Feel free to ask the umpires about some example statements on your own to check if they are valid patterns or not.

• There are exactly three pieces.
• There is at least one small piece.
• All pieces point up.
• The total number of pips is exactly ten.
• There is a unique top-most piece.

For the following structure, with the five example patterns above:

• It does not have exactly three pieces (it has five), so it does not obey the first pattern.
• It has at least one small pieces (it has two), so it does obey the second pattern.
• All pieces in it point up, so it does obey the third pattern.
• The total number of pips is exactly ten (2+1+1+3+3), so it does obey the fourth pattern.
• It does not have a unique top-most piece (there are two top-most pieces, the medium and one of the small), so it does not obey the fifth pattern.

GAME RULES

🀄 At the start of the game, the umpire will select one hidden pattern. This pattern is not known to either player; in fact, the whole goal is to figure out the pattern. The hidden pattern is guaranteed to be reasonable; check the clarifications section below for what it means to be reasonable.

🀄 Also at the start of the game, the umpire will provide two structures. One structure is revealed to obey the hidden pattern, while the other is revealed to not obey the hidden pattern.

🀄 The game also uses guessing tickets. You don't start with any guessing ticket. You will earn guessing tickets by correctly answering challenges, and use them to state guesses for the hidden pattern.

The player with the DM Advantage chooses the starting player. Then players alternate turns until one player wins.

🧠 Your turn consists of two steps:

1. Construct a structure, then either observe or challenge it.
2. Make any number of guesses about the hidden pattern.

📌 Step 1. Construct a structure, then either observe or challenge it.

🧠 During this step of your turn, construct any valid structure. See the rules above to check what makes a structure valid. There is one restriction: you may not construct a structure that has already been given in this game, whether that's because it's one of the initial structures, it has been given by one of the players, or it has been given as a counterexample (see below).

🧠 Along with your structure, choose one of two options: observe or challenge.

📡 If you choose to observe, you will be told whether your structure obeys the hidden pattern or not.

🧠 If you choose to challenge, both players have the opportunity to obtain a guessing ticket. Both players privately answer whether the structure obeys the hidden pattern or not. You may also decide not to answer.

📡 Once both players have answered the challenge, their answers and the correct answer will be revealed. Each player that gave the correct answer receives one guessing ticket.

Note that whether you observe or challenge, it will eventually be revealed whether your structure obeys the hidden pattern or not. The only difference is that when challenging, you have the chance to obtain a guessing ticket, but your opponent will also have the same chance.

🧠 You may opt to not make any structure, but at a significant penalty: your opponent receives one guessing ticket for free.

📌 Step 2. Make any number of guesses about the hidden pattern.

During this step of your turn, you may spend your guessing tickets to guess the hidden pattern.

🧠 To make a guess, pay one guessing ticket, then state a pattern. The pattern must follow the rules of being a pattern (such as being unambiguous and only referring to the pieces), but it does not have to be reasonable; it only needs to be understandable by both the umpire and the other player.

If your guess is not clear enough, the umpire will ask clarifying questions until they are satisfied, and the other player is also permitted to ask clarifying questions with the umpire's approval. Once your guess is deemed sufficiently clear, it is accepted as official. Before this point, you may abort your guess attempt and get your guessing ticket back, and you may make another guess if you wish. Once the guess is official, you can no longer retract it.

🧬 In rare cases, the umpire may outright disallow the guess, if they believe the guess is so contrived that its only purpose is to confuse people and not actually figure out the hidden pattern. The umpire has the final say. Similar to aborting, if your guess is disallowed, you get your guessing ticket back.

After you make your guess, the umpire will construct a counterexample if possible. The counterexample is a structure for which your guess and the hidden pattern give different answers: either the counterexample obeys your guess but not the hidden pattern, or vice versa.

You may make any number of guesses as long as you have the guessing tickets. You will know the result of each guess before you decide whether to make another guess or not.

👑 If the umpire is unable to construct a counterexample to your guess, that means you have guessed the pattern correctly, and you win the game.

ADMINISTRATION

📌 ⏰ Timing

Every turn, you have 3 minutes to complete all steps of your turn. The timer is paused in several cases:

• On a challenge, while both players are answering a challenge.
• After you construct your structure (and both players have answered the challenge, if applicable), while waiting for the umpire to respond with the result of the structure.
• After you make a guess for the pattern, while the umpire constructs a counterexample.

During a challenge, both players receive 30 seconds to answer the challenge.

Whenever you make a guess (and it's accepted as official), you get an additional 30 seconds for the current turn.

The umpire has the final say on timing matters. The umpire is allowed to pause the timer for any other reason, if reasonable.

If you run out of time, depending on what part of the turn you are in, the default case differs:

• If you run out of time in step 1 (construct a structure), you are assumed to not construct any structure and not make any guesses. Note that as you fail to construct a structure, your opponent gets one guessing ticket.
• If you run out of time in step 2 (make guesses), you are assumed to not make any more guesses. All previous guesses you've made stand.
• If you run out of time while answering a challenge, you are assumed not to answer, which means you don't get any guessing ticket.

📌 🧠 Submissions and allowed tools

Your submissions are done in the public game room. We will take the first legal submission you make, unless you promptly fix the submission (e.g. in case of making a typo or sending a move too early). There is one exception: when answering a challenge, your answer is made in your table channel.

In step 1, your submission consists of:

• Your structure, in any reasonable way. See below for a suggested method.
• A choice of ""observe"" or ""challenge"".

If you don't specify whether to observe or to challenge, it is assumed that you will observe.

Writing down a structure is not easy, and it is recommended that you set aside some time to write out your submission. You can use any reasonable way to submit your structure. If you can generate your own image, you can use that; if not, you can submit via text.

If you submit a structure via text, the recommended way is to use code blocks (```text```). Each cell within the bounding box of your structure is written using two characters. Cells in the same row are separated by spaces; rows are separated by newlines. The cells are written as follows:

• The first character is 1, 2, or 3, representing its number of pips (and hence its size).
• The second character is ^, >, v, or <, representing its pointing direction.
• If a cell is empty, use .. (two periods).

The following example shows a structure and how you would write it with this text method:

```

3> .. .. 2^ .. 3v

.. 1< 1<

```

To answer a challenge, simply state ""obey"" or ""not"", or even a simple ""yes"" or ""no"" (to the question ""does this structure obey the hidden pattern?"").

In step 2 of your turn, when making a guess, your submission consists simply of your guess. It can be stated in any way as long as it is understandable for all parties; the umpire will ask clarifying questions as needed.

You may use any available tools for note-taking. However, you may not talk to other people or use any automated programs.

📌 📡 Information

Pretty much all information is public. The only hidden pieces of information are the hidden pattern itself, and the players' answers to an ongoing challenge.

More specifically, this game has the following pieces of information, along with how and when they are revealed to the players, if at all. If a piece of information is not in this list, it is likely public information, but feel free to clarify.

🔺 Publicly announced

• At the start of the game: What the two initial structures are, together with which one obeys the hidden pattern and which one doesn't.
• At the start of steps 1 and 2 of a turn, as well as after each guess: The number of guessing tickets each player has.
• After a player submits a structure: What the submitted structure looks like, in an image form.
• At the end of a challenge: What answers the two players gave.
• At the end of step 1 of a turn: Whether the submitted structure obeys the hidden pattern or not.
• After a player makes an official guess: What the guess is, and what counterexample is given for the guess.

🔻 Explicitly not known to anyone, before the end of the game

• What the hidden pattern is.

CLARIFICATION

📌 Game inspiration

This game is nearly identical to the actual board game Zendo, released in 2001 (and later re-released in 2018). The main difference is on what structures are allowed.. The 2001 version of Zendo (what I consider as the definitive one) uses Icehouse pieces: colorful pyramids in three sizes and four colors that can be placed freely on a table. This game, to adapt to the digital format, restricts the pieces onto a grid and with only one color, but also introduces absolute orientation of pieces. Other than that, the game is nearly identical in every way; the only additional changes should be handling if a player misses a submission due to time-out. Although the rules are pretty much identical, the theme of the game has been completely changed.

The designer's website can be found here: http://www.koryheath.com/zendo/

📌 Hidden pattern

The hidden pattern of a Zendo game practically makes or breaks the game. Patterns that are unreasonable lead to a grueling, unfun experience, and it's definitely not what we are looking for. In fact, I don't even trust the umpires fully to come up with reasonable patterns. Therefore the following system is used:

I (chaotic_iak) have a small pool of possible hidden patterns. Only the hosts know this pool. I wrote every single pattern in the pool. I allowed other hosts and spectators to give some pattern ideas, but they had to go through me, so I could decide if they would be acceptable. When hosting this game, the umpire chooses one pattern they like from the pool as the hidden pattern.

Along with each pattern in the pool are two structures; these are the recommended initial structures. However, initial structures are much less problematic, and umpires are free to design their own if they wish.

Patterns are guaranteed to be reasonable; however, ""reasonable"" is a very subjective term. All the example patterns given in the rules are reasonable, although they have varying difficulty. Some good rules of thumb for what makes a reasonable pattern:

• It does not have many elements to it. Stringing together a lot of ""and""s and ""or""s, or using complicated terms like ""top-most small piece facing horizontally"", make for poor patterns.
• It is often short to write down. In DM Colosseum, I promised hidden patterns could be written in at most 10 words. While the same guarantee doesn't exist here, patterns will have similarly short lengths.

If you come up with your own patterns, feel free to ask me if they will be reasonable under my view.

📌 Counterexamples

Unlike most DMs, the umpire plays a large, active role in this game. There are two big places where the umpire plays a role: choosing the hidden pattern and constructing counterexamples. The hidden pattern is something I can watch over, and thus described above. The other part, constructing counterexamples, is described below.

Constructing counterexamples is a very tough job. Part of it is because it is very restricting, but the other part is because it is very free.

It is very restricting because the umpire needs to understand both the hidden pattern and the guess, and recognize where they differ. This is not at all an easy job to do. The umpire may ask other people for aid if they cannot find a counterexample.

It is very free because other than the above restriction, there is complete freedom on the umpire's part. They can construct a simple counterexample, which often is illuminating and helps steering players away from false paths. They can instead construct a complicated counterexample, which does not help much.

The umpire is free to act as they wish. The only guarantee is that the umpire will be impartial and fair; if they go with one style (e.g. always simple counterexamples), they will stick with that style for both players.

📌 Clarification: Starting player is chosen by the player with the DM Advantage (for Death Matches, this is the DMO).