Ganjifa revisited

For a long time I’ve been fascinated by the Indian playing cards known as Ganjifa, which represent a different branch of the history of playing cards from the more familiar 52-card deck. In 2011 I acquired my own Ganjifa deck and wrote a couple of blog posts on the subject, including my own idea for a game. In 2015 I perfected the rules for this game and rewrote the blog post, giving my creation the title Kendra Kari (केंद्र कड़ी).

In response to that post I was contacted by Sunish Chabba, who was developing a new deck of Ganjifa cards under the brand “Guru Ganjifa”. This began a fruitful correspondence in which I helped Sunish with his project in various ways, and learned a great deal more about Ganjifa along the way. Sunish ran a regrettably unsuccessful kickstarter for the project, but lessons were learned and he plans to relaunch when the time is right.

Ganjifa cards can be looked upon as objects of play, as objects of art, or as objects of history, and there is something to be learnt from each perspective. In their heyday they were popular across most of India and proliferated into various forms, chief of which was the Dashavatara deck, with ten suits each representing an avatar of the god Vishnu. The deck developed by Sunish is of this type, as opposed to the eight-suited Mughal deck featured in my earlier posts.

(For more information I recommend this video if you generally enjoy history/culture documentaries, or if you’d like to explore even deeper you may find something of value in this freely-available research paper. If you’re in a less intellectual mood, perhaps you’d prefer to sit back and watch cards being painted. Other links can be found elsewhere in this post, or by searching. No source can be assumed accurate in all respects, as the sparseness of information makes fact-checking difficult.)

One of the challenges in producing a modern Ganjifa deck is deciding what rules to include for the traditional game. There are three main reasons why one cannot simply reproduce rules from historical sources. (1) As a folk game, Ganjifa spread from player to player over hundreds of years and accumulated innovations in each locality, such that each village evolved its own version of the rules. (2) The few versions that have been preserved in writing — at least in English — are not adequately detailed to serve as a comprehensive guide to play. (3) Any attempt to revive something from the past typically calls not for a faithful facsimile, but for a blend of tradition and innovation that captures the spirit of the original but adapts it to modern needs. Adaptability is, in games as in everything else, the key to survival.

As part of my collaboration with Sunish I drafted a set of rules that will form the basis of those to be published with the Guru Ganjifa deck. The purpose of this blog post is to make these rules available to potential playtesters, to create a forum (in the comments) for discussion of the game, and to provide a platform where I can document rationales for various decisions that I made. Note that my own playtests were limited to simulations in which I acted the part of all 3-4 players; I have not tested these rules under real conditions against live opponents.

Anyone who owns a Ganjifa deck is invited to help, and as there are so few people who do own a deck, my gratitude is all the more sincere. All feedback is welcome, whether positive or critical; whether pertaining to the game itself or to the clarity of the description. Critical comments can be used to improve the game, and positive ones to promote it. The rules herein accommodate groups of three or four players, and if you own a Dashavatara (ten-suited) deck you can play them exactly as intended. If you own a Mughal (eight-suited) deck you can still test the three-player game, but I wouldn’t suggest a four-player one. And if you’ve tried Ganjifa before but been turned off by its dependence on memory or some other quality not to your liking, perhaps you’ll be willing to give it a second chance.


If you’d like to help, please download my version of the rules here.

The rest of this blog post is more academic than usual as it documents the choices I made in composing these rules.
You don’t have to read it. But if you’re curious, read on.


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Links: 2015 — 8 (with a request or two)

I have, regrettably, been too busy to blog lately. But I’m here now.

There are two personal items I’d like to discuss, so I’ll begin with the first, follow it with selected links from around the Web, and finish with the second item. Both involve some kind of appeal for assistance, although one more so than the other.

The first concerns a card game I invented, and that we playtested at my birthday party last month.

[Update: This first request has been answered. I have shortened the text below to reflect this.]

My birthday was June 12, but for reasons pertaining to Dad’s recent retirement, it was one week later that a small celebratory gathering took place. There were five people present: a friend, an uncle, my parents, and myself (unfortunately, the uncle had to leave early). I left most of the organising and decision-making to my parents, but playtesting a card game was my one explicit request.

The game was one that I invented and blogged about in 2011, but at the time it was a work in progress and the rules yet to be fully refined. But recently I took it off the backburner, made a few small adjustments, and updated my original blog post accordingly.

The twist is that this is not a game you can play with the familiar deck of hearts, diamonds, spades and clubs. Rather it is designed for Ganjifa — traditional playing cards from India that are typically circular rather than rectangular with at least 96 cards per deck. Regrettably they are used for only a handful of games and even those are dying out. I don’t care too much about the traditional games, but it’s a pity that the cards themselves lack a living tradition of new games being invented. The world could do with more Ganjifa games, and the game I’ve invented is my contribution to that end.

Here’s where my appeal to the Internet comes in — because the one thing I have not managed to do is decide on a name for the game. Since it is played with Indian cards, I feel that giving it an Indian name would be an appropriate tribute to the culture that created them, and have been appealing for help from people who know Indian languages and are interested in helping to name a new card game.

If you can help, or know someone who probably can, I would greatly appreciate it. I’ve suggested the name केंद्र कड़ी (crudely: Kendra Kari) for reasons outlined in my Google Plus post, but I really require informed human feedback and not just the results of automated translation tools.

[Update: I’ve now received a response to this query and have updated the game.]

Now that you’ve read this far, here are some links:

Now to the second personal item.

A while ago, just for fun, I decided to see if I could guess an approximate formula for time dilation in general relativity. I didn’t expect to be right, but using a combination of high school physics, Ockham’s Razor, and a few elementary facts, I gave it my best shot.

So I was genuinely delighted to discover later that my result checks out against real-world examples!

This page on Quora has a couple of obvious faults (the biggie is that the altitude of the ISS varies between 330 and 435 kilometres according to Wikipedia, so calculating a result to so many significant figures makes no sense), but the relevant formula corresponds closely to mine. Moreover, plugging their result into my formula puts the ISS in the right range — at about 375 kilometres. Another example I found is that time goes faster by 10 nanoseconds per year for every floor you go up in a building; again, my formula checks out, putting the difference between floors at a very reasonable three metres.

I am no genius (trust me on this), but in all the popular science I’ve read, I have never seen anyone mention that an approximate but workable formula for time dilation in general relativity is astonishingly easy to guess. Feeling that I’d stumbled on an angle that popular writers have missed but would be useful to others, I decided to write up my results as an article of my own. Please remember this is an un-factchecked draft, and I am no more an expert than the target reader, but you learn by doing. It’s written in the style of a popular science article, and is 1000 words long.

If any real popular physics writer reads this — I would love to hear your feedback. Also, you are welcome to take inspiration for your own blog posts, and I hope you will be so kind as to send me a link if you do. A tweet to @GoldHoarder will do nicely.

(Incidentally, I used a free PDF converter, so the links don’t work, but since they’re fully spelt out they won’t slow you down very much. Since it’s a draft, I find this acceptable.)

[Update: I linked to my relativity article in a comment here.]

Musical relay game

Here’s something to try. Call it a game, or an experiment, or a project — it’s interesting and fun no matter how you describe it. You need two players, each with a passion for listening to music, reasonably broad tastes, and a completely different selection of albums.

The rules

To begin, send your partner a track from your personal music collection, having agreed beforehand on a protocol for how to do so. Don’t trouble yourself with thoughts of copyright, because this is private correspondence and nobody cares. It’s almost optimised to lead to extra sales.

Your partner should listen to the song you selected (I use the word “song” to avoid the formality of overusing “track”, not to imply that all the tracks should have lyrics) and after careful reflection, send you back a song that, in their opinion, resonates with the one they received. By resonance I mean a subjective sense that the two songs fit together somehow — that they have something in common which unites them — so that a transition from one to the other feels as smooth as possible. It has to be defined intuitively rather than mechanically.

In turn, you should send back a song that resonates with the one you received, and so on, back and forth, until you’ve sent each other six tracks each. That’s enough for a twelve-track album, and a satisfying length for a game.

Some advice: Occasionally a specific song may come to mind straight away, but usually you’ll have to hunt for a bit. Try starting with an album that might provide a match, and listen to a few seconds of each track to refresh your memory. Aim for a track that resembles its predecessor in multiple respects, but if you can’t find a good match, do the best you can. In my experience you can always find something that will fit, even if it’s a song you don’t normally pay attention to because it’s overshadowed by other tracks on the same album.

Different people relate to music differently, and you must accept your partner’s subjective opinion of what resonates with what.

An example

I recently played this with a friend, Dan, and below I’ll describe in detail how this particular experiment went. Dan’s choices will appear in magenta and mine in orange so they stand out. I’ll include commentary, but in most cases won’t spell out what the tracks have in common — the reader is welcome try and identify these as an exercise.

Dan picked the first song. Rather than send tracks (or even links), he opted to confine himself to songs I could find on Youtube and let me search for them. His opening play was Birima by Youssou N’Dour [video].

My method for sending songs was to upload them to Box and send links by email (I deleted each track after Dan acknowledged its receipt). In answer to Dan’s first play I chose Iamagit by Telek. Here’s a thirty-second sample, and here’s a really bad video in which a couple of Russian boys mime rather pathetically to a recording (also, the first few beats are missing). The similarity between the song title — a word in one of the languages of Papua New Guinea — and the English phrase “I am a git” rather amuses me. You may sometimes see the song referred to as Lamagit (due to an ambiguous font on the album), but Iamagit is correct.

Dan struggled to find a match, and after a few days decided to let his subconscious do the work and simply choose the first song that came into his head. He picked Mr Wendal by Arrested Development [video]. This is a very different song from Iamagit, and I struggled to see any similarity. Dan, in retrospect, thinks it was a bad choice too. Eventually I realised the connection must lie in the metre: Iamagit goes something like /x/x x/// on the bass in bars without lyrics, while Mr Wendal goes /x/x x//x on the drums through the whole song.

In reply, I chose On The Edge by Oysterband. Here’s a thirty-second sample, but the only videos I can find are from an inferior live performance that really will not do (it leaves out half of the instruments). Dan says that of all my choices, this is the only one where he struggles to see the connection to its predecessor, but here are some thoughts. First, occasional use of a rather tinkly-sounding guitar or other instrument adds texture to both songs. Second, I don’t listen to hiphop-style songs with lyrics spoken rather than sung, but On The Edge provides some approximation in that there are often several consecutive syllables at the same pitch, and many syllables between pauses. Third, both songs contain a lot of very harsh sounds plus a strong beat, and fourth, both have lyrics about social inequality. Together I think these qualities combine to make it a reasonable match.

Dan responded with Have You Ever Seen The Rain by Creedence Clearwater Revival [video], commenting that he’d settled on the strategy of choosing the first song that comes to mind. This is, of course, by far the most well-known song in our exchange. As a match, I felt it was mediocre — it didn’t clash with On The Edge (as Mr Wendal did with Iamagit), but nothing about it seemed particularly connected, either. The main link I can see is that both songs have a downward-tending riff between parts.

My next choice was Ubeyneihem by Meir Banai [video], a Hebrew song about being content in both urban and rural environments. The chorus (according to my sources), translates as: “And in between them, and in between them, I go and then come back. I have in me some of one and some of the other; I have some of both.” Just for your interest.

Dan then selected King Without A Crown by Matisyahu [video], which has nothing in common with Ubeyneihem except that one is sung in Hebrew and the other has lyrics with Jewish themes. Moreover, Dan wouldn’t have even known that Ubeyneihem was in Hebrew if I hadn’t mentioned it. Talking about this after the game was over, he wrote: “I think up until this point, I was approaching this as a bit of game rather building an album. And it was as much about introducing new songs and bands to each other. And from this point of view a cultural link between two songs is just as valid as a musical link. Even though the cultural link in this example is pretty skin deep too. Thus from this song onwards, when I started to think about the musical connections a bit more carefully, I think the album becomes a lot more coherent.

In my view it is a game, but one in which the challenge lies in finding the best match you can from your own collection of music. If played well, a coherent album emerges as a side effect (in which a listener can experience the stepwise evolution from each song to the next).

Now entering the second half, I picked Tine Lasta by Kíla [video]. (Note: some of the Youtube videos I’m linking to contain someone else’s — e.g. the uploader’s —  theme tune added to the beginning and/or end of the track, and just to warn you, this one is particularly obnoxious. Ignore it.) This was Dan’s favourite of all the tracks I shared, and Kíla by far his favourite artist; despite being Irish himself he had not previously heard of them.

Dan’s next selection was Me Gustas Tu by Manu Chao [video]. By this point he was, as indicated earlier, putting more thought into his choices.

I then chose Aguas Claras by Surkuy. I can’t find a video or sample of Surkuy’s version, but here is someone else’s cover of the same song, which I assume is traditional.

Dan’s final pick was Hora Zero by Rodrigo y Gabriela [video].

We’d agreed to exchange six tracks each, and now it was now my turn to pick the final song. I had wanted to avoid including more than one track by the same artist, but the only match that felt right to me was Ríl A Do by Kíla [video]. At least it’s from a different album.

Our post-experiment discussion covered topics like which connections were least obvious, what we thought of each other’s songs in their own right, whether we were content with the variety covered, and so on. Regarding the variety, my only caveat was that we’d exchanged nothing that was carried more by the melody than by the beat, since Dan tended to choose songs with a very strong beat which I had to match. It can be interesting to observe how different people relate to music, and Dan clearly pays more attention to the drumbeat of a song than I do (an observation he confirmed).


It’s a fun exercise to try, and for friends reading this blog post, I am definitely open to doing a musical relay with you at some point. Not in the immediate future — I need a breather — but let me know if you’re interested in doing so at some point.

I also welcome comments of all sorts, as ever. Feel free to add any perspective you have, from a whimsical reply to something I mentioned in passing to a thoughtful analysis of our playlist.

Update: I played this again with @fuddlemark on Twitter in September 2013. My choices were: (2) An Tiománai by Kíla; (4) Nervous Doll Dancing by Nervous Doll Dancing (clip); (6) She’s Moved On by Oysterband (clip); (8) Sweet Comeraghs performed by Karan Casey (this isn’t actually in my music collection; I only know it from Youtube); (10) C’hoant Dimein performed by Cecile Corbel; (12) Night Ride Across the Caucasus by Loreena McKennitt. I’ll let the reader speculate on what Mark’s intervening choices might have been, but this was track #5.

Two nodable challenges

Here are two challenges that are completely unrelated except that both involve nodes in some way.

The first challenge is different for every reader, and is one example of a whole family of challenges you can invent for yourself.

Sometimes, in idle moments, my mind is drawn to the geometic layout of buttons on remote control devices. Considering the buttons simply as nodes and abstracting away all extraneous detail, you can invent puzzles for yourself that involve partitioning these nodes according to set rules.

For example: see if you can partition the remote control into multiple rectangular grids, such that there’s a grid containing one button, a grid containing two buttons, a grid containing three buttons, and so on, until there aren’t enough buttons left for another grid. A grid is an n×m rectangle with a node at every point. It’s most elegant if you make each grid as “square” as possible (i.e. better 2×2 than 1×4, better 2×3 than 1×6, etc). Are yours solveable?

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Playing with Ganjifa (Part II)

[Note: This blog post was originally published in 2011, but has been extensively rewritten several times. Particularly in 2015 when the game that it describes was revised, playtested and named.]

This is the second of two blog posts about Ganjifa cards. In the first I showed sample cards from my own deck (which I bought from a playing card museum in Germany), suggested suitable English names for the eight suits, and shared some thoughts on how to shuffle them. In this sequel, I’ll describe my own idea for a game that could be played with the cards.


In contrast to Western playing cards — which are used for hundreds of games of many different types, the accumulated inventions of many generations — the traditional playing cards of India are used for only a small number of endangered games, and there is no active tradition of inventing new ones. This is not a difference in inherent worth but an accident of history, and in my opinion regrettable. To realise the potential of Ganjifa and revitalise them for today’s world, I propose we invent new games for them, perhaps by adapting elements of familiar Western games to take advantage of cards that are circular rather than rectangular and come in twice as many suits. Let us also feel free to build on each other’s inventions to create families of related games.

Not all Ganjifa are circular, but the fact that most are provides a point of contrast with Western cards, which in my opinion is part of the appeal. Herein, circular cards are assumed and indeed required. Also, there are several types of Ganjifa, but the type intended below is the Mughal variety, which consists of 96 cards in twelve ranks and eight suits. Some other types may be usable but I know less about them. See the end of the post for notes on obtaining a deck.

Introduction to the game

The game described below was unnamed until mid-July 2015, but I have now settled on an Indian name to go with the Indian cards. That name is केंद्र कड़ी, which is Hindi for Central Link and in English can be spelt either Kendra Kari or Kendra Kadi. Credit to Kishor Dabke for confirming that the name is appropriate and means what I thought it did.

(In case you’re curious I’ve added a footnote on pronunciation — but this is a card game, not a linguistics lesson, so it doesn’t really matter.)

As for the game itself, it is inspired by the Eights family, though it lacks the wild cards that give that family of games its name. It is a true Ganjifa game in the sense that it could not easily be adapted to non-circular cards. The game is reasonably simple and should be suitable for players who are new to Ganjifa.

The rules

Preliminaries and Deal

The game is suitable for 3-6 players. I will assume that play proceeds clockwise. (Card games in India are traditionally played anticlockwise, so if that is the custom of your group, replace “clockwise” with “anticlockwise” throughout.)

Throughout the game, cards are played to seven distinct positions on the table. Six of these positions form a ring (we’ll call them Positions 1 to 6, proceeding clockwise), and the seventh is in the centre of that ring (we’ll call this Position 7).

Six cards are dealt to each player, and another card is dealt face-up to Position 7. The remaining cards form the stock.

Normal Play

The first player, if able, selects from their hand a card with either the same suit or the same rank as the card on Position 7, and places that card face-up on Position 1.

In the normal course of the game, each player who can plays a card with either the same suit or the same rank as the most recently played card, and does so by placing it on the next position around from that previous card. So if the previous card was played on Position 1, you play on Position 2. If the previous card was played on Position 2, you play on Position 3. Position 7 is not ordinarily played to, so if the previous card was played on Position 6, you play on Position 1, and round it goes.

A player must play if they have a playable card.

A player who is unable to play picks up a card from the stock. If the card drawn is playable it must be played immediately, but cannot be used to build a bridge (see Special Play below). Otherwise it is added to the player’s hand. Play then proceeds to the next player.

Special Play

So far, I have addressed only what happens in the normal course of the game, but we must now discuss special moves that players can perform when the opportunity arises.

First I must make it clear that when I refer to two cards as directly opposite each other, I refer simply to the layout of the six outer positions. The card currently sitting on top of Position 1 is directly opposite the card on Position 4, and vice versa. The card on Position 2 is directly opposite the card on Position 5, and vice versa. The card on Position 3 is directly opposite the card on Position 6, and vice versa.

A player may play a card to Position 7 if that card not only shares its suit or its rank with the most recently played card, but also shares its suit or its rank with the card directly opposite the most recently played card. Such a move I will refer to as building a bridge, because it connects two opposing positions, metaphorically speaking.

Moreover, a player may play two cards in the same turn if the second one builds a bridge, the first having been played from their hand in the usual way. (Be sure to plan both moves before performing the first, as failure to do so is at least a breach of courtesy.)

After a bridge is built, all cards played so far are gathered up and taken out of the game. The player who built the bridge begins a new phase by playing any card to Position 7, and may, if able, play a second card with the same suit or rank to Position 1. (The first card to be laid down is deemed to be the one at Position 7.) This means it’s possible to play four cards in a single turn: two to build a bridge and two to start the next phase.

Normal play resumes with the next player’s turn.


A player wins by getting rid of all cards in their hand. Specific ways to win include:

  • Using up your last card during normal play.
  • Using your last card to build a bridge.
  • Building a bridge with one card remaining (as this would be used to start the next phase).
  • Building a bridge with two cards remaining that share a rank or suit (as these would be used to start the next phase).


Notes on play

The question may arise of what to do if the stock is entirely depleted. There are two obvious possibilities: either declare the player with the fewest cards to have won, or else create a new stock by shuffling all the cards that have been captured by building bridges. I suspect most people will opt for the former, rather than allow the game to go on indefinitely. However, in my experience this is extremely unlikely to happen if everyone plays well: if the stock does run out before someone wins, it’s probably a sign that players have yet to master the strategy and are missing opportunities to play cards.

Describing a game is hard — it’s often like describing a tower or sculpture that looks simple and elegant when you can see the big picture, but complicated and arbitrary when you can see only one piece at a time. Here are some crib notes that might help when explaining the game verbally to others.

1. Match suit/rank. Play to next position. Aim: get rid of cards.
2. If cannot play, pick up card. Play it immediately if can.
3. Build a Bridge = play to centre position.
4. Can play 2 cards if latter builds a bridge.
5. Cannot do (3) or (4) with card just picked up.
6. After building bridge, begin new phase with 1 or 2 cards.

Linguistic footnote

The name केंद्र कड़ी consists of two Hindi words — केंद्र (Kendra) means centre, and कड़ी (Kari or Kadi) means link. It refers to the card in the centre that builds a bridge by linking two opposing positions. I hope the alliteration makes the name punchy and memorable.

The spelling Kari more accurately conveys the pronunciation in English, but there are two distinct sounds in Hindi that the English letter ‘r’ can represent, and if you use the wrong one you’ll be saying Curry, not Link. The alternative spelling Kadi reflects the fact in Hindi the sound is related to a d-like sound, which the Devanagari spelling also reflects. Please see this video and this link for more information about this sound. (For linguists: the ‘r’ in Kendra is an alveolar trill [r] while the ‘r’ in Kari is a retroflex flap [ɽ].)

None of this matters unless you want it to. You can pronounce Kendra Kari any way you like and the game will be exactly the same.

Obtaining Ganjifa

For readers who are interested in obtaining a Mughal Ganjifa deck, here are the sources I’m aware of:

  • I have a copy of the deck produced by Grubbe. It is apparently available online, but the site is in German (which may or may not be an obstacle for you). I can vouch for the durability of the cards, but because some of them are visually confusing it takes time to learn how to identify the suits and ranks. Happily, the deck comes with a reference chart. Another negative is that the box they come in is very bulky.
  • Tab Creations, an American company, has been producing a deck since 2013, and it is available on Amazon. I have not seen these cards, and feel duty-bound to point out that customer reviews (both on Amazon and on the original Kickstarter) have reported that the cards are not sturdy enough to withstand repeated shuffling. On the other hand, the artwork looks simpler, reducing the learning curve for first-time players, and depending on shipping costs in your location this may be a more accessible option for you.

Naturally, I would like to see more options become available in the future. I will add more if I become aware of them, and I’m open to including varieties other than Mughal as long as they’re fit for the same purpose.

Playing with Ganjifa (Part I)

One of the souvineers I acquired in Germany was a pack of Ganjifa cards: circular playing cards traditionally used in India. In this post, I’d like to talk some more about the cards in general. In a sequel, I’d like to write about my attempt to invent an original game that uses them.

There are eight suits, each of which contains numbered cards from one to ten plus two court cards, those being a Minister (think of it as a Jack) and a King. Suit symbols vary from one deck to another, and some decks have more than eight suits. Here are some sample cards from the deck I bought.

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Interesting stuff: Early September 2010

Some interesting links, with a bonus one after the bullets:

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Conversation cards

During the long drives on our recent holiday, we entertained ourselves in various ways. Music from various CDs, a few episodes of Astronomy Cast on another CD, and a game or two of The Art Of Conversation. That last item gives me an idea for a blog post.

The Art of Conversation consists of a hundred cards that, between them, contain 300 conversation ideas (yeah, you’re all thinking Monty Python did it first in Meaning of Life, but this is genuine). To play, you leaf through the cards until you find a question you want to ask, then everyone else answers that question, and then it’s the next player’s turn. Concepts of winning, losing, and competing do not exist in any form.

Below is a photograph of four cards from the game. [Update 2014: I’ve taken a new photograph to replace the out-of-focus original.] Underneath I’ll blog in the spirit of the game, by selecting topics from these cards and writing either about that or about something closely related.


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New web space, plus TV upgrade

I have a new website! I recently registered the domain, which will replace my old site at NetYP once I finish transferring files. My solitaire game is already transferred (and working again).

In unrelated news, I installed a new television last week, replacing an analogue TV with a digital one that also has an inbuilt DVD player. Pictures below: old one left; new one right.

[Update: These items were originally posted separately, but I have decided to abbreviate and merge them.]

Sequel to the Christmas parlour games post

In my previous post (the final for 2009), I described a rotary drawing game from the book Parlour Games for Modern Players by Myfanwy Jones and Spiri Tsintziras.

We’ve also tried out some other games. A favourite of mine was Who’s Closest? (page 229) — essentially the parlour game version of cold reading. Another interesting one is Mafia (page 118), a game of psychology and murder invented in 1986.

But back to the drawing game: we played another round on New Year’s Eve, and my favourite excerpt is reproduced below. (I wrote the caption about mathematical symbols.) Following that is an excerpt from yet another round that we played in the new year.

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