Teaching The Logic Of Scientific Discovery With Games – Part 2: ‘Trainee Demiurge Games’

Posted on 2012/03/02


Feature Image - Teaching The Logic Of Scientific Discovery With Games - Part 2 'Trainee Demiurge' Games

`My name is Alice, so please your Majesty,’ said Alice very politely; but she added, to herself, `Why, they’re only a pack of cards, after all. I needn’t be afraid of them!’

Alice, in Lewis Carroll’s ‘Alice’s Adventures in Wonderland’[1]


This is the second article where I describe how games can be used to teach the logic of scientific discovery. This time, I’ll focus on a category of game I have come to think of as ‘trainee demiurge’ games. Hopefully this term should become more clear as you read on!

As with the first part, I will be expounding a particular view of the philosophy of science: Karl Popper’s Critical Rationalism. As you might have guessed, I think this is the correct view.

One possible objection to this endeavour is encapsulated by what Nassim N Taleb calls the ‘Ludic Fallacy‘. He coined this term to draw attention to the erroneous assumption that basic probability theory can be useful to us, noting that “the attributes of the uncertainty we face in real life have little connection to the sterilized ones we encounter in exams and games”.[2]

I don’t think this applies to the analogies I wish to make. I am not presenting analogies that aim to model a particular aspect of the world, but our relationship with the world.

It could be argued that these games promote a rather naïve form of falsificationism. However, I think it is better our students learn naïve falsificationism than erroneous inductivism. They will, at the very least, be on the right side of logic.


In 2009, I decided to start the academic year with one of my new classes by teaching them a card game. Or, rather, I didn’t teach them a card game. That’s not to say we didn’t play the game; we did. I just didn’t teach it to them. Let me explain.

The four new students entered the lab for the first time, expectation in their eyes. They were here to learn the truths of the universe. After a few introductions, I promptly asked them to arrange a few tables and chairs. “We are going to play a card game.”

They looked puzzled, but that was the point. Anyway, it was only going to get worse, so I continued without explanation. I sat in a middle seat, with two packs of cards in my hands. While shuffling the two packs together, I told them the objective of the game. They were to receive five cards each. The aim of the game was to have no cards in your hand. The first person to do so is the winner.

I waited. No one threw the cards up in the air, ate them or triumphantly set them on fire. How disappointing. Well, there was still a year or so before university interviews. I placed the remaining cards in the middle of the table and turned the top card over next to the deck.

“During the game, you may talk to each other if you wish, but you are not required to. You can ask me questions, but I won’t answer them. The game has rules, none of which you presently know. It would be in your best interests to help each other to find out what they are. When it is your turn, the game requires you to add a card from your hand to the pile in front of you. If any of you break the rules, you shall receive a ‘penalty card’ which I will hand to you from the deck. If you don’t want to play a card when its your turn, you can take a penalty card from the pile yourself. Confused? Excellent; let’s begin.”

They began to play, quietly and nervously. “No, you won’t be graded on this,” I assure one of the more nervous players. They began to place cards on top of one another next to the deck. A ‘penalty card’ swiftly followed up almost every play. Discussion tentatively began…



The game these students were playing, unbeknownst to them, was called ‘Mao‘.

Of course, as with all card games, there are many variations of the rules. I constructed this particular strain for the teaching of scientific inquiry, and to suit a small group of students in a 40 minute game:

  • Player arrange in a circle. Each player starts with 5 cards, and the remainder of the deck is stacked face down in front of them.
  • A player may play any card in his hand which matches the value or the suit of the card currently lying face-up on the table.
  • If a player breaks one of the rules, you deal them a ‘penalty card’ from the deck, and play moves to the next player.
  • If a player suspects they do not have legal move, they can opt out of their turn by self-administering a ‘penalty card’ from the deck.

In addition to these basics, quirks are exhibited when particular cards are played:

  • Eight – reverse the order of play
  • Ace – skip a turn
  • Jack – change suit (at players discretion, which they declare). For example, a player may place a Jack of Clubs upon a 3 of Clubs and declare “diamonds”. Play resumes as though the top card was the Jack of Diamonds.
  • Spade – player must call out the the face value and suit of the card before the next play. For example, they must say ‘Two of Spades’.
  • Seven – the player announces, “have a nice day” and the next player draws a penalty card. [Unless they have a seven also, which they can play while replying, “have a very nice day” to the next player who must draw two penalty cards. Unless they have a seven... etc. Each consecutive person with a seven adds a 'very' to the phrase.]

Guidance for Guides

You’ll need several decks (about one deck per 3 people).

Although the ‘winner’ is the first player to empty their hand, the real purpose of the game is to engage students with the inquiry. If anyone wins, acknowledge it, then deal them back in!

Of course, the pace of the game really depends upon your students. So, how you administer penalty cards should be informed by their ability and willingness to experiment in the game.

If they are able and willing to learn through their own mistakes, you can administer the cards with just the words ‘incorrect play’, and perhaps indicating who’s turn it is.

For less ideal circumstances, you may wish to be more specific, hinting at the error they made. For example, you may say ‘playing out of turn’ when an eight is played, and the players have’t figured out the rule for ‘eights’ yet.

The most difficult rule to gauge is for sevens. ‘Failure to say “have a nice day” ‘, is a good start. Then, follow it up with another ‘failure to draw a penalty card’ when the the next player fails to draw a penalty card!

I advise never to confirm if the players have the correct rules during the game, no matter how much they ask about them (more on this below). If you do choose to reveal the rules at all, do so at the end of the game. The rules can be written down, one per piece of paper, to show you didn’t make the up during the game.


As I mentioned, I present my just one of the many variations of Mao. For alternatives, you can check out the Wikipedia page for Mao.

Of course, you could always make up your own rules that present accessible challenges to your students.

Note I said accessible and not achievable! There’s always a lot of emphasis about how students need to leave every class thinking they’ve achieved something. Well, if that is the case, then this lesson clearly isn’t about learning the rules of Mao! It’s about how scientific inquiry is a fallible process where we are continually guessing better theories and testing them against experiment. This message will be lost if you make the rules too easy.


Undisputed Aspects

  • Players are left to guess the permanence, scope, and form of the rules.
  • Unlike Feynman Chess, where experiments were always passive, players must choose (or guess) which card to deploy.
  • This active experimentation can be collaborative. Several players can work together to test a mutually accepted theory. Or, if players are in dispute over several competing theories, they can try to refute each others’ ideas with an experiment.
  • There is no method whereby players can know, in the common sense of the word, whether any of their guesses are true. Learning only occurs when their guesses are falsified by the occurrences of the game. It is very satisfying to see some of the better players notice this, as they play the game. For instance, they might have a card they are quite certain is ‘legal’, but will choose another card as part of a risky conjecture. They might incur a penalty card, but become more interested in learning the rules than ‘winning’ the game. These are the real scientists of the class!
  • There is an option for a player to opt out of experimentation and self-administer a penalty card. Notice that the consequence are the same as if they were to guess incorrectly. So, it is an interesting question why they choose not to experiment. Indeed, these are often the same students who are unwilling to test their own ideas in the physically-safe environment of a science classroom. This is a gentle (and allegorical) approach to posing this difficult question to timid students.
  • Players can always enter an ongoing game of Mao. Anyone that does this share a similar experience as each of us when we engage with historical scientific knowledge (with the obvious exception that the game doesn’t usually span generations!). We can study the theories that our predecessors have invented, try to refute them, and attempt new guesses.
  • Mao is a good game for highlighting the human inclination for superstition. Players often will makes guesses that carry extra theoretical baggage that is neutral to the rules. It is interesting to hear students discuss their suspicions. Note how they often suppose vocalizations or bodily movements have consequences in the game, once the rule for ‘seven’ has been encountered. B. F. Skinner would have had much to say about this, I am sure.

Points For Discussion

  • In Mao, just like Feynman Chess, the players are told that there are rules to the game. In the physical world, we cannot be so certain. Are there rules which make up the universe, or are our guesses just distillations of the facts – what we experience in summary form? We don’t know, yet we assume there are rules. We have to, otherwise there’s no game, no inquiry.
    However, depending on your reputation for deception, students may question whether you are just ‘making things up as you go along’. I have frequently faced such suspicions! It is interesting to note that the question is often posed by a lazy nihilist, preferring to suppose there are no rules than attempt to figure them out. A character trait you might notice later on in their science education!
  • If Mao is to serve as an analogy for the logic of scientific discovery, it’s main flaw lies in the nature of the refutations. As this is a common problem with many games paraded as models for scientific inquiry, I will discuss it in detail below.

Demiurge Training

How do we learn about the world? Through the refutation of our guessed rules. If our guess disagree with what actually happens, we must conclude our guesses are incorrect.

In Mao, and games like it, there is a subtle but vital difference: you learn when you break the rules of the game!

Contrastingly, we have an intuition that it is impossible (unless, as Galileo cautiously said, “due to a miracle”[3]) for you to break a law of nature. You cannot step over a cliff and, while hovering, declare (like a  Loony Tunes cartoon), “I never studied Law”.[4]

Even it is was possible to break a law of nature, how would you ever know you had? Without cosmic ‘penalty cards’ (whatever that might be like) there would be no way of knowing.

To summarize, this class of games misrepresent the nature of scientific inquiry in the following respects:

  • They present the participant as somehow knowing that the universe operates on rules (although they do not know what these rules are).
  • The participant is invited to guess the rules by making some choice. This decision may result in the rules being broken (indicated in some way – usually with a punishment), thereby falsifying the participant’s guess. In other words, the participant is capable of (temporarily) performing miracles.

This picture seems to better represent a sadistic training session for a group of trainee demiurges. In the philosophy of Plato and Plotinus, a demiurge is similar to, but less impressive than, our common conception of a ‘god’. A demiurge did not create the universe; it simply fashions and maintains it. It ‘works on a preexisting chaos’.[5]

In the Fourth Ennead, Plotinus suggests that the Ancient Greek view of Zeus closely resembles the philosophical demiurge, while Chaos (or Gaia) might rival the monotheistic conception of ‘God’.[6]


Given these inadequacies, we might imagine the following fantastical scenario, which these games more closely resemble. [Note: I’m not pretending to know anything about Greek mythology! I'm only using the ideas of Zeus and a demiurge to highlight the problems with these analogies]

Zeus sits, watching smugly, as he lets his trainees take over the operation of the universe. However, Zeus is not relinquishing command, just sitting back and watching for a while. For him, it’s a fun way to break up the bordem and drugery of running the universe.

His trainees are made to agree with the following terms and conditions:

  • The rules with which he operates the universe will not be divulged to them.
  • They have been provided with a selection of ‘occurrences’ they can utilize.
  • However, if they break one of his rules by choosing an ‘occurrence’ out of order, Zeus intervenes, ‘rewinds the tape’ and undoes the ‘mistake’.
  • If any of the trainee’s breaks Zeus’s law, he promptly punishes them by supplying another occurrence they must utilize. The trainees aren’t allowed to stop playing until they have used all their occurrences. However, Zeus’ rules are so complicated, they often make mistakes.

This Pavlovian regime gives the trainee demiurges the opportunity to get closer to Zeus’ rules, however, they will never know if they have achieved ‘the truth’.  Zeus can be entertained, while maintaining supreme power.

Other ‘Trainee Demiurge’ Games

Eleusis (or ‘Eloosis’)

This card game might have been the original inspiration for Mao, whose origin is unknown.

Invented by Robert Abbott in 1956, the game was featured in Martin Gardner‘s column for Scientific American in 1959.[7] In the article, Gardner writes,

“It should be of special interest to mathematicians and other scientists because of its striking analogy with scientific method and its exercise of precisely those psychological abilities in concept formation that seem to underlie the ‘hunches’ of creative thinkers.”

I am not sure what Gardner means by ‘scientific method’, since our theories are guesses. We know of no method to generate them. Nevertheless, Abbott seems to have agreed with Gardner that Eleusis had educational merit, and sought to improve his game for this purpose. It is this improved 1977 version of the game that I shall present.


  • One person plays the role of the dealer (or, as I would call them: ‘Head Demiurge’). They make up a secret rule (see list below).
  • The other players have to try to guess the rule (or, as I would call them: ‘Trainee Demiurges’).
  • The dealer deals each player 14 cards.
  • The dealer then places the pile of in the middle of a table, and turns a starter card face up next to the pile.
  • Beginning with a randomly selected player, play progresses  in a clockwise direction. When it is a players turn, they have two options: (1) The player plays a string of up to 4 cards they think conforms to the secret rule. (2) The player believes their hand contains no card conforming to the secret rule and declares “no play”.
  • If some of the cards played conform to the dealer’s rule, they place these cards in a line next to the starter card. This is called the ‘mainline‘.
  • If some of the cards do not conform to the rule, the dealer places them in a pile called the ‘sideline‘.
  • The ‘mainline’ and ‘sideline’ act as a record of when the players made correct and incorrect plays. This can help them with future guesses.
  • A player who thinks he has guessed the secret rule can declare himself ‘Prophet’, once per game.
  • The ‘Prophet‘ takes on the role of the dealer, while the dealer confirms they are making correction judgements. As soon as the Prophet makes a mistake, they return to their previous lowly status as a regular player.
  • Incorrect plays, passes, or unsuccessful Prophets receive penalty cards (as with Mao)[8]

Sample Basic Rules

Alternate cards by:

  • colour (red, black, red, black…)
  • odd\even value (11 [J], 2, 3, 12 [Q], 9, 4…)
  • 1-7\8-13 value (5, 13 [K], 4, 9, 1 [A])

Cycle cards by:

  • suit (spade, heart, diamond, club, spade…)
  • higher value that is no more than 3 higher, where “2” is a valid move after “12” [Q]. (3, 6, 8, ,11 [J], 12 [Q], 2…)

A sequence of:

  • colour (3 red, 3 black, 3 red…)
  • suit (2 hearts, 2 diamonds, 2 clubs, 2 spades)
  • modular arithmetic sequences – such as add 5 (1, 6, 11[J], 3, 8, 13 [K], 5, 10, 2, 7, 12 [Q], 4, 9…)


Several educational papers have been written about Eleusis. For example, [9] and [10].

The data record is a neat idea, although perhaps you might prefer to encourage student’s to make their own written record of events.

Notice this game differs from Mao in that it has only one rule. Additionally, the concept of ‘Prophet’ (or ‘scientist’, if you like) allows brave students to declare a confidence in their guess. The concept of fallibility in scientific inquiry can only be strengthened when Prophet’s are refuted!

Eleusis Express

In 2006, John Golden invented a slight variation of Eleusis called ‘Eleusis Express’.[11]

The essential difference is that a player who makes a correct play is allowed to guess the rule, if they wish. If they are correct, the round is over, and it is their turn to be the dealer. If they are incorrect, they incur a penalty.

I would advise against using this game in a science class. Although it might be satisfying for students to achieve certainty that their rule is correct, nature does not allow such confirmations, and it would be misleading to pretend this is a model of scientific inquiry.


This game is based on the idea of a chess variant. As it sounds, these are game which are loosly based on chess, but vary some or all of its features. Possible variables to alter include: the legal movements of different pieces, their names and numeration; the size and type of board; and the rules of capture.

One rather intimidating example is Gliński’s hexagonal chess, depicted below:

In 1994, two members of the Puzzles and Games Ring (PGR) of the ‘Archimedeans’ mathematics society at Cambridge University wondered if they could construct a game where the objective was to guess the variant rules devised by others.

Adam Chalcraft and Michael Greene called their invention Penultima.[12]


Here is a sketch of how you could play the game:

  • A regular chess board and regular pieces are required
  • The game has two types of paticipants: the Players and the Spectators.
  • The players divide into two teams. Each team’s goal is to check-mate their opponents’ king, just as in chess.
  • The Spectators devise the rules for the pieces, each inventing the rules for one class of piece. Each peice can be renamed, perhaps offering a hint as to the new rule. If their are more people than class of piece, the spectators can break into teams. Each team keeps their rule secret from the players, and all the other spectator teams. “There is no actual restriction on the rules, except that the aim is to produce a playable game.”
  • Once the spectators have devised their rules, they place their pieces on the board (not necessarily where they usually go). Teams may need to settle territorial disputes if they had competing intentions for the location of their pieces.
  • The Players begin to play. Black moves first.
  • After each move, the spectators decide to either, “(i) allow the move to stand, as a legal move according to the rules; (ii) allow the move, but modify the position to complete the side-effects of the move (for example, other pieces might be captured, or moved); (iii) declare ‘illegal move’ and restore the position before the move was made.”
  • As with chess, ‘illegal moves’ include moving your own king into check.
  • Spectators may need to declare ‘check’, if the players have not yet determined that this is a consequence of the rules.
  • “When a move has been declared illegal for whatever reason, the player who made it shall lose their move, and the other player move next, except that if a player’s king is in check and they make an illegal move, then after the position has been restored they shall attempt to move again.”

A familiarity with chess variants will provide all the participants with a vocabulary of experience that will assist their construction and guessing of rules.

As you can see, just like Eleusis, players learn by breaking the rules.

Mao and Eleusis Simulators

Computer programs have been devised for Mao and Eleusis to be played online. Follow the links to play the games:

Eleusis: http://www.eleusisgame.org/EleusisTomcat/EleusisTomcat.html

Mao: http://kevan.org/games/maobot.php

Other Commercial Games

There are some other games available that follow the same ‘Trainee Demiurge‘ format:

  • Zendo (2001) – Invented by Cory Heath, this game requires ‘Students‘ to arrange a variety of pieces into structures so they conform to a secret rule created by the ‘Master‘. The Master indicates to Students if the rule has been followed or broken, or asks the students to guess if their creations are correct or incorrect. Students may also try to guess the rule. If they are incorrect, the Master will show them a counter-example. [Also known as 'Jewell in the Sand']
  • Genius Rules (1987) – Robert Abbott’s first attempt at a commercialization of Eleusis. Instead of a regular deck, he depicts ‘geniuses’ throughout history.
  • Code Breaker (1999) – A second attempt at a comercial repackaging of Eleusis by Robert Abbott. Instead of receiving ‘penalty cards’ for mistakes, players must attempt to guess the code before they run out of cards.
  • Quao ["cow] (2007) – A independent repackaging Eleusis. The rules are a almost identical to Mao, and hence the name. I suppose the creators thought a dictatorial cow might be a more palatable fictional alternative to the former leader of the PRC.
  • Confusion (1992) – Also invented by Robert Abbott, this game resembles Penultima. Players do not know the rules of their pieces on the board. In addition, unknown to each player, one of their pieces is controlled by their opponent. They must reach the centre of the board to reach a briefcase and return to their base to win.
  • The Three Commandments (2008) – Invented by Friedemann Friese, Gordon Lamont, and Fraser Lamont, this game depicts a capricious ‘Priestess’ who creates an arbitrary list of taboos for her subjects to follow. The other players simply move a pawn on the board and are awarded points by the priestess based on how their actions (including seemingly irrelevant ones like body movement or tone of voice) correspond with the rules. At the end of a round the priestess receives the same score as the highest-scoring adept, encouraging her to make the rules difficult but not impossible to guess.


[1] Carroll, Lewis (1865) ‘Alice’s Adventures in Wonderland’

[2] Taleb, Nassim N. (2007) ‘The Black Swan’ Random House pp 148-149 [see  http://www.fooledbyrandomness.com/LudicFallacy.pdf]

[3] Galilei, Galileo (1638) ‘Discourses and Mathematical Demonstrations Relating to Two New Sciences’

Galileo writes, when explaining why the bones of a giant would break due to his ‘square-cube law‘ that:

“…an oak two hundred cubits high, would not be able to sustain its own branches if they were distributed as in a tree of ordinary size…”

“…nature cannot produce a horse as large as twenty ordinary horses or a giant ten times taller than an ordinary man unless by miracle or by greatly altering the proportions of his limbs and especially of his bones, which would have to be considerably enlarged over the ordinary.”

This uncharacteristally hesitant choice of words is most likely a consequence of his house arrest, imposed by the Catholic Church when he insisted that the Earth moves around the Sun. There are numerous Biblical references to giants (Genesis 6:4, Numbers 13:33, Deuteronomy 2:10-11, Deuteronomy 2:20-21, Deuteronomy 3:11, Joshua 12:4, Joshua 18:16, 1 Samuel 17:4…), and Galileo was cautious to anger the Church any further.

[4] Warner Brothers (1949) Loony Tunes ‘High Diving Hare

[5] Gerson, Lloyd P. (1998) ‘Plotinus’ Routledge p 21

[6] Plotinus (ca. CE 204/5–270) Fourth Ennead, Third Tractate, Section 10 (Translated by Stephen Mackenna and B. S. Page)

Plotinus writes:

‘The ordering principle is twofold; there is the principle known
to us as the Demiurge and there is the Soul of the All; we apply the
appellation “Zeus” sometimes to the Demiurge and sometimes to the
principle conducting the universe.
When under the name of Zeus we are considering the Demiurge we must
leave out all notions of stage and progress, and recognize one unchanging
and timeless life.’

[7]  Gardner, Martin ‘Mathematical Games’ Sci. Am., 1959, 200, 160-164 [& Sci. Am., 1977, 237, 18-25]

[8] Abbott, R. The new eleusis. Unpublished manuscript, 1977. (Available from Box 1175, General Post Office, New York, N.Y. 10001.) [See http://www.logicmazes.com/games/eleusis/index.html]

[9] Romesburg, Charles H (1979) Simulating Scientific Inquiry with the Card Game Eleusis‘ Science Education 63 (5): 599-608

[10] Roadruck, Mike (September 2006) Eloosis The Erlenmeyer Volume XIII, Number 1 p 7

[11] http://www.logicmazes.com/games/eleusis/express.html

[12] Fayers, Peter (Summer 1998) ‘ “Eureka” Chess Problems’ Variant Chess, Volume 3, Issue 28 pp 164-166 (incorporating “Penultima” by Fryers, Michael)