Napoleonic & ECW wargaming, with a load of old Hooptedoodle on this & that

Saturday, 25 May 2013

The Man Who Killed Pythagoras – and other tales

Pythagoras, indicating the tricky diagonal
I’ve recently been involved in a number of Real Life issues which have left me very little time for any hobby-related activities, but I have managed to spend the odd moment reading other people’s rules, and scribbling and pondering ideas – some of them very old ideas, it has to be said.

I have banged on about hexagons and their pros and cons at great length in the past, and am confident that I don’t have very much more to say on the matter – you would think...

In one of my odd moments, the other day, I was recalling a series of debates that I had with a few friends – a very long time ago now – in which we considered gridded miniatures games and their advantages, but which mostly served as an excuse to drink beer. We agreed, very early in our discussions, that the most innocent comment any of us had made on the topic to date was credited to our resident optimist, Alan Low, and it went along the lines of:

“It is much easier to consider the merits of hexes if you can rise above all the prejudices and sacred cows which they seem to upset.”

Yes, Alan, we said – but you can’t really separate these things – the problem is that the biggest single disadvantage of hexes is that people hate them. Whether that is justifiable or even fair is beside the point – if HG Wells had been pictured with hexagons scribed on his floor then no-one would worry about it. As things stand, hexes are an affront to everything which is cherished in miniatures gaming. Worst of all, they are associated with BOARD GAMES, which are the greatest affront of all. We are, after all, talking of orthodox religion here.

We also agreed that the only acceptable plea we might make on their behalf was that Joe Morschauser was famous for gridded games – though I believe that at the time Joe was regarded as less Old School than he is now. His game was generally seen as a harmless eccentricity, and not proper wargaming.

Morschauser, of course, used squares. Squares are easy to draw, and have an ancient precedent in the chessboard, but for wargames they have some inherent snags, the very largest of which is Pythagoras. Orthogonal moves of 1 square are fine, but a diagonal move of 1 square is 1.4142135 (etc etc) times as far. Some games get round this by prohibiting diagonal moves or combats – somehow, units which are adjacent to each other along a diagonal cannot see each other – or, as in the De Gre/Sweet game, the square root of two is taken to be a rather more convenient 1.5. That certainly helps.

A to B is easily seen to be 4 hexes
Hexes are not easy to draw at all – even with an accurate template, you can get a gradual drift with accumulated small errors, so it is necessary when marking out a hex table to have copious guidelines and preliminary sketchings. They do have the advantage of six-fold symmetry, and they get rid of Pythagoras, but many gamers object to the fact that they distort straight lines. You can either lay out your hexes so that there are straight columns going across the table (like my own hex-based games) – in which case your units may advance in a straightforward manner (literally), but do not line up side by side very neatly – or you may have the straight columns running sideways across the table (like Commands & Colors) – in which case you may form exemplary lines of battle, but your units advance in a rather odd zig-zag.

In fact both these issues can be solved visually at a stroke by having the hexes a good bit larger than the units, so that you can place the units off-centre and smooth out the battle lines and the marches.

We rambled around this subject through many beers, enjoying the scenery but not really deciding anything, and then one Sunday morning Pat Timmins rang me and announced:

“I may have just killed Pythagoras.”

Pat had been applying square vinyl tiles to his kitchen floor – in a very bold combination of navy blue and white. His wife objected to the basic chequer-board configuration because, she said, it “gave her the buzzings” and seemed likely to promote epilepsy. He had tried various alternatives, and at one point experimented with alternate rows offset by half a tile, like this:

He realised that such an arrangement on a wargames table would allow movement in six directions, and was in fact a sort of hexagonal arrangement without the hexagons. Judging distances, for example A to B in the illustration, was not quite as intuitive as with hexes, but was still possible with a bit of methodology (I reckon AB is 5 squares distant).

We were unreasonably enthusiastic about this – perhaps we could pass off our offset squares (or “squexes”, as Pat called them) as a sort of logical descendant of Morschauser’s game, and overcome some prejudices. The next non-development was that someone suggested that the squares should not be squares but rectangles with sides in the proportions of √3 to 2, which would even up the six-fold symmetry so that it was a proper 60 degrees all round. It made the table layout closer to natural hexes, but made the board look even more distorted – at this point, we actually preferred normal hexagons, which put us back where we started. So we eventually decided that squexes had had their brief moment, and resigned ourselves to being outcasts in the wargaming fraternity with our conventional hexes.

Glinski's game - note the 3 bishops
Also on the topic of hexes, I invented hexagonal chess in about 1970. My excitement was tempered more than somewhat when I discovered that there were already in existence a number of varieties of hexagonal chess, and that my own new game had been previously invented by a man named Glinski. This was useful, since it allowed me to drop the idea and move on to dabble with something else. I expanded Glinski’s game into a 3-sided version. There are 3-sided chess games now, but mine used a board with a full hexagonal grid (most of the available games now use distorted squares) – the board was a little larger than the normal (normal?) Glinski board, and the 3 sets of pieces set up in alternate corners.

It looked spectacular, but it didn’t work very well. Early experiments revealed that a game of this type for 3 players brings some interesting problems. The first is order of turns – if red plays white into check then white has to respond immediately, which reverses the turn cycle if it was in fact black’s move next.

More fundamentally troubling is the very nature of 3-player strategy. It is very difficult to have a game in which each of the players is attacking both of the others – it makes more sense to have two gang up on the third, and then double-cross each other at the end, which gets you into all sorts of negotiation, time-outs for diplomacy and other stuff which we decided it was simpler to just ban. No chat, we said – no sign language, no secret notes left in the bathroom. This left us with a game in which the only possible recommended strategy was a passive opening - allow the other two players to attack each other and weaken each other. If all 3 players adopt the same strategy, of course, you get a very strange non-game. You may feel free to draw your own parallels from history on any or all of these.

A very smart looking 3-sided chess set - mine was different from this,
since it used the Glinski layout of pieces, and  the playing board was
a rather larger version of Glinski's
So we gave that one up as well, though we did briefly consider 3-sided soccer on a triangular pitch, but abandoned that very quickly, not least because we could not agree how the offside rule would work. We did, however, think that the winner might be the team which conceded the smallest number of goals.

How very silly.    


  1. Great read Tony - 'squexes' indeed! Enough to give anybody the 'buzzings'. I myself spend far too much time with my head full of hexes, they trouble me, but they work, it's a love- hate relationship. They have me in their grip, I have upstairs wooden hexes in at least 3 different sizes, where will it stop?

    Those chess sets are just insane! I thought the same of the 3 dimensional version played in Star Trek..... but God, they intrigue me!

    1. Hi Lee - the chief reason I mentioned the hex chess games was because I have a photo of my original 3-sided hexagonal game. The bad news is that I can't find it! A pity. If people are commemorated in the long term by unhelpful one-liners (King Alfred famous for burning cakes, Isaac Newton for being hit by a falling apple) then I would not object to being remembered as a man who invented a fantastic variant of chess which didn't work.

      One of my better failures.



  2. Interesting post. Like you I think, I became interested in boardgames because they worked well, which was not always true for miniatures games however much I would like it to have been.

    Peter Gouldesbrough and a couple of other older wargamers I knew in the 1970s were dismissive of boardgames to an extent which I thought was unreasonable. It was once explained to me that a game with free measurement (and incomplete rules?) was a proper pastime for grown men - games on boards with squares and hexes were just that - slick games for kids. The implication, which is still a bit scary, is that miniatures wargames were sort of scaled down real warfare, and thus to be taken seriously. The other implication was that we only played with officers and gentlemen, who would never cheat, thus the weaknesses in the rules were OK.

    I told Peter I was sure that all grown men who played with toy soldiers were guaranteed respect anyway, especially if they carried a ruler, but he didn't see the funny side. For myself I would prefer to play without a grid, but the grids often make the game work better. A game which does not work is not a good game unless you are very lucky.

    Cheers - Lou

    1. Hi Lou - I hadn't thought of it before, but you may well be on to something there - using measuring sticks and so on made wargames a scientific pursuit for gentlemen, not at all like draughts or snakes & ladders.

      Peter was an amiable chap - I liked him a lot, and learned a lot from him - but could be argumentative and very single minded. I once helped him to redraft his Napoleonic rules for 5mm Minfigs blocks, and he refused to correct a couple of arithmetical errors I found in them. Someone once said to me that he never was defeated in an argument, precisely because he didn't listen. If it was you that said this, please excuse me.



    2. By the way - did you ever see Peter without a tie? I can remember him wearing a tweed sports jacket once, for a Saturday morning wargames convention/demo in Adam House, Edinburgh, but as far as I recall otherwise he always wore a suit - even at home. He was definitely a gentleman though.


  3. I had a couple of interesting emails in response to this post, for which I am grateful.

    Martin sent some tales of Pythagoras, who seems to have been more odd than I thought. Apart from the well-known story of the Squaw on the Hippopotamus, it gave some details of Pythagoras life - it is widely believed that he died during a rebellion at the city of Croton in about 500 BC, when he was caught by the rebels as a result of his refusal to cross a field of beans, a vegetable to which he had a religious objection. Seems perfectly sensible to me.

    The worthy Prof De Vries did a very nice little treatise on the shortcomings of 3-sided soccer. He points out that a triangular pitch is tricky, since every throw-in would be someone else's corner, and it would be hard to decide to whom it should be awarded. Thus he suggests (you guessed) a hexagonal pitch, with touchlines alternating with goal lines. He agrees with the idea of winning by conceding the fewest goals - if a shot from a red forward brushed against a blue player's jersey on the way into the green team's goal, the only fact that we cannot argue about is that the score was AGAINST green. There is a fair amount more of the same, much of which I hadn't thought of, and it adds up to the fact that we are not likely to see triangular football in the Olympics anytime soon.

    The Prof mentions that there are some interesting features of strategy in games (or wars) with more than two participants, and it may be that a defensive posture is the best one. He proposes to read further on this - he thinks there may even be stuff in Napoleon's Maxims on it. I wouldn't hold your breath for further developments - I'm pretty sure that someone out there already understands all this anyway. There is probably a degree course available at the Open University.

    Regards - MSF

    PS - I am kind of diappointed that Pythagoras didn't escape by taking a diagonal path across the bean field, and thus becoming a Stealth Pythagoras

  4. Don't know how I missed this post? I started to design a triangular Chess game a couple of years ago (2008/9), but unlike the one pictured I set my 'start-lines' two rows off the pictured board, so they were two conventional lines of squares, the middle was similar to the one illustrated but the bit in-between was a little different, I keep meaning to finish it, it was as far as I know unique, in that I had Google to reveal all the other designs! Not something available in 1970!

    The main fault with 'set-back' start lines seems to be getting pieces 'trapped' in corners in a way you can't with a conventional or the the posted version? Siege warfare!

    I'm also working on a four-times Scrabble board, which involves buying all the boards I can find at car-boots or charity shops, then cutting the margins off two sides of four of them, re-mounting them together, then steaming the playing areas of a few more and pasting various individual squares onto the bigger board until the ratio of pink and blue squares equates to the original board...then getting sued by Hasbro!


    1. Four-times Scrabble is very good. Like it a lot.

      I was making this morning's coffee when I started thinking about the board, and I have to ask. I'm so ignorant about Scrabble I don't know how many squares there are on the official board, but I do recall that it must be an odd number each way, because there is a "star" square in the centre for starting the game - first tiles laid must include one on the star (and, I remember, I'd been playing for years before I discovered that the star, being pink, is a double-word bonus square).

      A straight 2 x 2 giant board will give an even number, and thus no central square. How do you start the game?

      Now look what you've done!

      I'm still hoping to find my old photo of the hexagonal tri-chess from the 1970s, but I guess it may have gone for good. I could bash up a diagram though...

      You must promise to post a photo of your Big Scrabble when it's ready. It will also give Hasbro some hard evidence for the court case.

      Oversized games are super - I've always loved those giant chess games in the public parks. I've also seen giant draughts (checkers) in Princes Park, Liverpool when I was a nipper - 1950s, I guess. I have long fancied the idea of putting a big Nine Men's Morris game in the garden - (aka Merrelles, Mühle, Mill etc...). I believe they had them on village greens in Tudor times?

      Thanks, Hugh - inspirational comment!