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.