Balanced Incomplete Block Designs and Steiner Triple Systems

When I was playing Take It Easy today I realised that the tiles formed a block design. So the thought immediately came to mind: "is there a Steiner Triple System in this game?" At this point you're thinking, "well duh, didn't you know that?" I guess I'm just slow.

A block design is a set of b subsets (called blocks) of a set X, each of size k. Set X has v elements, each of which appears in r blocks. In an incomplete block design, not all k-element subsets of of X are in the list of blocks. In a balanced incomplete block design (BIBD) there is a number λ such that any two elements of X appear in λ blocks together. This is called a (v, b, r, k, λ) BIBD.
Let me clarify. X = {1, 2, 3, 4, 5, 6, 7, 8, 9} so v=9. Each tile is a subset of size 3 of those numbers, so k=3. There are 27 tiles, so b=27. From this we can determine r... there are 27 tiles with 3 numbers each, so there are 81 numbers in all - and each can be one of 9 values, so each must appear 9 times, so r=9. It's true in a block design that v*r = k*b.

What about λ? 7 and 8 appear together on 3 tiles - with 1, 5, and 9. But how many times do 1 and 5 appear together on a tile? Never! So there is no value for λ and the block design is not balanced. That's a pity because I can't show you how λ*(v − 1) = r*(k − 1).

A Steiner triple system is a set of m elements with 3-element subsets such that for any two elements they appear together in exactly one 3-element subset.

Set contains a Steiner triple system - the subsets of size 3 are the sets, and for any two cards there is a unique third card which forms a set. For any two numbers which appear together on a tile, there are 3 such tiles on which they appear (e.g. 178, 578, 978). However it's not the case that on any two tiles there's a unique third tile which makes up the triple... e.g. 123 and 178. Unless we do something really cunning...

Take any two tiles. If they have two numbers in common, the third tile in the triple is the other tile with those two numbers on it. If the tiles have zero numbers in common, e.g. 123 and 578, the third tile is the tile with exactly the numbers that don't appear on either tile, so 946 in this case. That tile is unique. If the tiles have one number in common, then the third tile is the tile with that number and two completely different numbers, e.g. the third tile for 123 and 178 is 146. So now, given any two tiles there's a unique third tile forming the triple (or set). That's a Steiner triple system! Woohoo!

In fact, the simpler rule is that in each of the 3 positions on the tile the numbers must be all the same or all different. There are only 3 dimensions rather than 4, and Take It Easy has only 27 tiles instead of 81.

I decided long ago to collect games with Steiner triple systems so I guess I have to get Take It Easy now and I could use it to play Set. Hmm... could I use Set cards to play Take It Easy?

New Games Played On Holiday

So here's why I've been so quiet since Scrabblette got back: we've been on holiday. We went down to the Blue Mountains to Euroka Homestead for a week with some other gamers. Not to say that Scrabblette is a gamer, but she can approximate one when she needs to. We did a couple of trips into Sydney to visit friends and do some of the tourist things including seeing an exceedingly dull exhibition at the Museum of Contemporary Art. Luckily, I played several new games during the week.

Flaschenteufel - this is a nasty little card game where one player gets screwed each hand. It's fun to plan for it to not be you. I think with experience it would be excellent.

Expedition - ho hum and double ho hum. This game reminds me a lot of Australia which I really don't like. Luckily the theme doesn't suck anywhere near so bad. I dislike both of them because some players can have long turns and get heaps of stuff, often leaving other players to have a boring turn and get nothing. Maybe that's because those players have skill, but maybe it's just because the games suck? Anyway, I wasn't excited by it at all.

Mü - I'd heard good things about this so I was keen to try it, and as we went through the rules it sounded exciting. Soon I realised it sucked. My fundamental problem is that if 1s are trumps I can have a blue 1 and not be allowed to play it on a blue trick. I know why, I know it's the rules, it just sucks. If I wanted to play a game where some trumps changed suit I'd play 500.

Phoenix - I played half a game of Phoenix and quite liked it. I'll have to buy my own copy and play some more.

Carcassonne - Hunters and Gatherers- I like the idea of a different Carcassonne with nicer tiles, but as I played it I realised that the different tiles just made the board look busy and the bonus mechanism, although it encouraged people to complete forests, wasn't very exciting most of the time. The game seems to be a bit more complex than the original with not many advantages to compensate. Ho hum.

Mystery Rummy: Murders in the Rue Morgue - Not really a new game, as the kid and his mum and I have played this many times. However it was new to play 4 players which made the game a very interesting partnership game. We played against our usual Tichu opponents, and I liked it a lot.

Elefantenparade - I like the theme of this game, I liked the simplicity, but it has really failed to deliver both times I've played it. The first time I got by the world's biggest ever kingmaker move, and the second time we got stuck in analysis paralysis trying to figure out how to prevent people from winning. Scrabblette suggested playing in two teams of two so that you can validly choose to help your partner and avoid kingmaking. I hope to try that.

Take it Easy - This is a weird little game which is half solitaire puzzle and half solitaire puzzle. Umm, let's add that up. I don't know if it's actually very interesting.

My Word! - This is the sort of word game I'm very good at. Everyone else played for second place. That means I like it a lot but will rarely get it to the table.

The Battle of Lanka - Scrabblette brought me this from India. It's a simple card game and I was worried that it might not work at all. We played three games with CyberKev and Mrs CyberKev. The first two games were won after one turn when the first player felt they'd got lucky with the deal and were in a strong enough position to win. In both cases they were right. We then decided that it should be compulsory to play all the way through the deck, which rule change caused us to play the game we thought the designer probably wanted to design. It was OK - nothing super, but an interesting experience.

Maharaja: Palace Building in India - I've been looking forward to playing this for a while, but I was amazed as I heard the rules that you could do so little on a turn. It didn't seem right to me. It turned out that the rules work OK but I can't add 12 and 2 and managed to screw up my first turn. I struggled to find a strategy based on screwing up your first turn and lost interest as the players who were doing well took their time making moves. I'd like to play again, but maybe just with Scrabblette for a while. She liked and beat me in the 4 player game.

That's all the new games I played while on holiday, but some other things happened on the way back home. There's a shop called Mal's Toy Store on the highway in Taree so of course we pulled over and had a look. They had a copy of Batik for a good price, which my FLGS has not so far supplied, so I bought that. I was going to buy 50c worth of lollies but the nice lady gave them to me. Then she mentioned she hadn't learned Tantrix so she couldn't sell it, so I showed her how to play the solitaire version.

Later in the day we stopped at The Big Banana. There's a puzzle shop next door, so we took a look. I found two games I'd never heard of called Landlord and Spin and Trap, and it turns out that BGG hadn't heard of them either. So I bought them for $A30 each. Scrabblette and I tried to play Landlord while eating a banana split but we didn't really understand the rules and have to try it again with our improved understanding. At the same shop Scrabblette bought a word game called Snatch which leaves an opening for some pretty appalling double entendres which I won't go into here.

I had a good session of Batik with BIL this evening, and although it's not a brilliant game it works OK. It LOOKS beautiful. I love Gigamic games - I have 3 now and plan to acquire more. After watching BIL and I play, 6yo nephew demanded a game and played against Scrabblette... he kicked her butt. He's a gamer, that lad, you mark my words.

Anyway, that's most of the gaming news from the holidays. Maybe I could tell you the story about... nah, if I even tell you what the story's about Scrabblette will... er, better not say that either.