No guessing, only counting

Newly introduced randomized variants also pose a good question: how much there are different initial setups? Let's calculate!

Note that randomized Classical Chess variants and randomized Croatian Ties variants of the same size has the same number of possible initial setups. In Croatian Ties variants Knights are replaced by Pegasuses, but otherwise number of different pieces and their count is the same. Pawns in all variants are irrelevant, since they can be arranged in only one way. So, the only difference is the order of figures arranged in the first and last rank. Both mirrored and symmetrical flavors of randomized variants has the same number of combinations, so it's enough to calculate only once, then double the result.

First, a very brief introduction to binomial coefficients:

$$c=\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!\cdot (n-k)!}$$

where $n\ge k\ge 0$, and $n!=n\cdot (n-1)\cdot (n-2)\cdot ...\cdot 2\cdot 1$.

Our factor $c$ is number of combinations when placing $k$ pieces of the same kind (e.g. 4 Rooks) onto $n$ available fields (e.g. 10 empty fields).

So, we'll start with randomized Classical Chess 14, and with Bishops, since we have to make sure we have the same number of Bishops on light and dark fields. There are 4 Bishops in this variant, so that would be 2 Bishops on light and other 2 Bishops on dark fields. In these variants there are 14 fields at each rank (surprising, I know!), so that gives us 7 light and 7 dark fields. Obviously, each set of light and dark fields is completely separated from each other, and so numbers of possible arrangements of Bishops on different fields are the same:

$$c_{\mathrm{light}}=c_{\mathrm{dark}}=\left(\genfrac{}{}{0ex}{}{7}{2}\right)=\frac{7!}{2!\cdot 5!}=21$$

where $7$ is the number of either light or dark fields; and $2$ is count of Bishops on those fields. So, our factor $c_{\mathrm{light}}$ is number of combinations of 2 Bishops arranged over 7 light fields; and $c_{\mathrm{dark}}$ is the same over dark fields.

Next, we have to arrange 4 Rooks over $14-4=10$ fields, where 14 is size of a variant, and 4 is number of fields already occupied by Bishops:

$$c_{\mathrm{Rooks}}=\left(\genfrac{}{}{0ex}{}{10}{4}\right)=\frac{10!}{4!\cdot 6!}=210$$

Next, we have to arrange 4 Knights over $14-8=6$ fields, where 14 is still size of a variant, and 8 is number of fields already occupied by Bishops and Rooks:

$$c_{\mathrm{Knights}}=\left(\genfrac{}{}{0ex}{}{6}{4}\right)=\frac{6!}{4!\cdot 2!}=15$$

Queen can now pick between two remaining fields (ladies are choosers! 😎):

$$c_{\mathrm{Queen}}=2$$

King will have to be satisfied with whatever field is left empty:

$$c_{\mathrm{King}}=1$$

So, our number of combinations for randomized Classical Chess 14 variants is:

$$c_{\mathrm{14}}=c_{\mathrm{light}}\cdot c_{\mathrm{dark}}\cdot c_{\mathrm{Rooks}}\cdot c_{\mathrm{Knights}}\cdot c_{\mathrm{Queen}}\cdot c_{\mathrm{King}}=\mathrm{2,778,300}$$

that is for either mirrored or symmetrical flavor. If flavor is also randomized choice, total number of initial setups is doubled:

$$c_{\sum \mathrm{14}}=c_{\mathrm{14}}\cdot 2=\mathrm{2,778,300}\cdot 2=\mathrm{5,556,600}$$

both those numbers also apply to randomized Croatian Ties 14 variant(s).

Calculation of number of different initial setups for randomized Classical Chess 20 variant(s) -and, by extension, Croatian Ties 20 variant(s)- is the same as above, but numbers are slightly bigger. Size of a chessboard is now 20, there are 10 light fields, and 10 dark fields, there are 6 Rooks, 6 Knights and 6 Bishops (of which 3 are on light fields, and other 3 on dark ones). Taking all into account gives the number of initial setups as:

$$c_{\mathrm{20}}=\mathrm{2,421,619,200}$$

that is for either mirrored or symmetrical flavor. If flavor is also randomized choice, total number of initial setups is doubled:

$$c_{\sum \mathrm{20}}=c_{\mathrm{20}}\cdot 2=\mathrm{2,421,619,200}\cdot 2=\mathrm{4,843,238,400}$$

both those numbers also apply to randomized Croatian Ties 20 variant(s).

Number of different initial setups for randomized Classical Chess 26 variant(s) is calculated the same as above, but numbers are bigger still. Size of a chessboard is now 26, there are 13 light fields, and 13 dark fields, there are 8 Rooks, 8 Knights and 8 Bishops (of which 4 are on light fields, and other 4 on dark ones). So, number of initial setups is:

$$c_{\mathrm{26}}=\mathrm{2,013,316,519,500}$$

that is for either mirrored or symmetrical flavor. If flavor is also randomized choice, total number of initial setups is doubled:

$$c_{\sum \mathrm{26}}=c_{\mathrm{26}}\cdot 2=\mathrm{2,013,316,519,500}\cdot 2=\mathrm{4,026,633,039,000}$$

both those numbers also apply to randomized Croatian Ties 26 variant(s).

As an interesting tidbit, if we try to calculate number of initial setups for randomized Classical Chess, we get:

$$c_8=\mathrm{2,880}$$

that is for either mirrored or symmetrical flavor. If flavor is also randomized choice, total number of initial setups is doubled:

$$c_{\sum 8}=c_8\cdot 2=\mathrm{2,880}\cdot 2=\mathrm{5,760}$$

Obviously, these numbers are somewhat bigger than Fischer Random Chess would have it (i.e. 960), since it also imposes an additional constraint on how initial setups are generated; namely, "the King must be placed on a square between the Rooks", and does not support symmetrical flavor, only mirrored.

Even more guessing!

Previously, I wrote down my guesstimates how much One variant is more complex than Classical Chess, turns out a lot. Now, it's time to do the same for new, simplified variants. I'll do just Classical Chess 26, smaller variants are of lesser interest here. As for Croatian Ties variants, I haven't yet got around how to calculate complexity factor due to increased mobility; so, using previous method Croatian Ties variants would have exactly the same complexity as their Classical Chess counterparts, which doesn't feel right, but it is what it is.

If you haven't already, read blog post linked above; it explains how complexity factors are calculated, and why. So, let's start with size:

$${\mathrm{cf}}_{\mathrm{size}}=1+\ln \left(\frac{{26}^2}{8^2}\right)=1+\ln \left(\frac{676}{64}\right)=3.357310$$

Next, it's total number of pieces on the chessboards:

$${\mathrm{cf}}_{\mathrm{pieces}}=1+\ln \left(\frac{104}{32}\right)=2.178654996$$

There are no new pieces, so complexity factor for different types of pieces is one:

$${\mathrm{cf}}_{\mathrm{types}}=1+\ln \left(\frac{6}{6}\right)=1.0$$

The same applies to Croatian Ties variants, since every Knight is replaced by Pegasus, so there are the same number of different types of pieces.

There are also no new interactions, so complexity factor also goes to one (this one too applies to Croatian Ties variants):

$${\mathrm{cf}}_{\mathrm{interactions}}=1+\ln \left(\frac{3}{3}\right)=1.0$$

Our complexity $c$ is then defined as a product of all factors calculated above:

$$c_{\mathrm{regular}}={\mathrm{cf}}_{\mathrm{size}}\cdot {\mathrm{cf}}_{\mathrm{pieces}}\cdot {\mathrm{cf}}_{\mathrm{types}}\cdot {\mathrm{cf}}_{\mathrm{interactions}}=7.314420189$$

This is complexity scaling factor of regular games from Classical Chess into Classical Chess 26 variant, i.e. all games should be 7.314 times longer. For instance, the longest recorded tournament game was 538 moves (269 FIDE moves, aka cycles), which turns into 3835 moves (1968 cycles) for Classical Chess 26 variant. Average on-line match lasts about 80 moves (40 cycles), in Classical Chess 26 variant that would become 585 moves (292.5 cycles). Average tournament match lasts about 88 moves (44 cycles), which becomes 644 moves (322 cycles).

The same, however, does not apply when calculating maximal possible game length, because players will try to maximize each and every metrics available to prolong the game. So, for maximum game length we have to calculate linear scaling factors; for chessboard sizes factor becomes:

$${\mathrm{cf}}_{\mathrm{size}}=\frac{{26}^2}{8^2}=\frac{676}{64}=10.562500$$

Next, for total number of pieces on the chessboards we have:

$${\mathrm{cf}}_{\mathrm{pieces}}=\frac{104}{32}=3.25$$

Complexity number for different types of pieces is still one:

$${\mathrm{cf}}_{\mathrm{types}}=\frac{6}{6}=1.0$$

Finally, complexity factor for number of different interactions is also one:

$${\mathrm{cf}}_{\mathrm{interactions}}=\frac{3}{3}=1.0$$

Taken together, our complexity $c$ becomes:

$$c_{\mathrm{longest}}={\mathrm{cf}}_{\mathrm{size}}\cdot {\mathrm{cf}}_{\mathrm{pieces}}\cdot {\mathrm{cf}}_{\mathrm{types}}\cdot {\mathrm{cf}}_{\mathrm{interactions}}=34.328125$$

This is scaling factor for the longest possible games, i.e. the longest games should be 34.33 times longer in Classical Chess 26 variant compared to Classical Chess. For instance, previously mentioned 11797 moves (5898.5 cycles) game as the longest possible with 50-cycle rule in Classical Chess 26 variant becomes 404,968 moves (202,484 cycles) game. Even longer 17697 moves (8848.5 cycles) game with 75-cycle rule in Classical Chess 26 variant turns into 607,505 moves (303,752.5 cycles) game.

These doesn't appear to be large numbers, if you recall estimates for One variant, but should not be underestimated; even "just" 7.314 times increase in complexity results in some prolonged games, especially if increase in complexity is allowed to also translate into longer time allowance per turn. So, 15 seconds per player's turn in bullet game now becomes approx. 110 seconds per turn; given that average Classical Chess 26 game length would be 585 turns, it would last for approx. 17.834 hours of gameplay time; or, 2.23 days if we assume 8-hour gameplay in a day. If we don't increase time allowance per turn, 15 second per turn bullet game would take approx. 2.438 hours of gameplay time.

In short, new Classical Chess variants does pose a challange, even if only just by scaling up. Croatian Ties variants builds more on top of that challenge, by replacing Knights with more mobile siblings, and also allowing Pawns to move sideways, which throws off known gameplay patterns, tactics.

Guessing game

Merging pieces and tags mentioned in a previous post have gone much smoother than I expected, and it's finished now. I also wrote in that post my estimates how long a game in the largest variant could take. In the meantime I have revised my estimates, and so I present you with bigger, better numbers. Again, these are very rough guesstimates, it's difficult to even assess how much numbers presented here could deviate from real-world matches; in short, those shouldn't be taken too seriously.

Estimates here are based on the fact that most large, complex systems settle its metrics somewhat in the middle, since most extremes tend to cancel each other out. For this analysis we'll be comparing easy to calculate metrics such as:

• size of a chessboards,
• count of all pieces in a variant,
• number of different types of pieces,
• number of possible interactions,

and will be comparing One variant against Classical Chess. For each metrics we'll calculate its contribution to overall complexity, then multiply them all together, since they are all (mostly) independent; although, total number of pieces is always larger than number of different piece types.

All complexity factors have the same form, so let's define generic complexity factor $\mathrm{cf}$ as a simple ratio between new ($n$) and old ($m$) metrics, like so:

$$\mathrm{cf}=\frac{n}{m}$$

Edit: with provision that $n\ge m$.

This definition is all well and good, but applies only if corresponding metrics contribute to system's complexity directly, in a linear fashion. Most of the time, this is not the case. For instance, adding two ranks and files to a classical chessboard is a much larger change (as a percentage) then adding the same to the second largest variant, Discovery. So, each increase in metrics yields diminishing increase in complexity; for such a non-linear increase there is a function which sets limits to growth, and that's natural logarithm ($\ln$):

$$\mathrm{cf}=\ln \left(\frac{n}{m}\right)$$

There is still a small issue to solve here, before calculations can take place. Observe what happens if we compare e.g. Classical Chess with its own self:

$$\mathrm{cf}=\ln \left(\frac{m}{m}\right)=\ln \left(1\right)=0$$

Our complexity factor $\mathrm{cf}$ gets to 0. This is actually fine, all that calculation is showing us is that there is no additional complexity when comparing a variant to itself. Still, we'd like to multiply our complexity factors, as independent variables should be. So, we'll add 1 to formulae, like so:

$$\mathrm{cf}=1+\ln \left(\frac{n}{m}\right)$$

Now that we have generic formulae for complexity factors sorted out, we can actually calculate something; lets start by comparing sizes of chessboards:

$${\mathrm{cf}}_{\mathrm{size}}=1+\ln \left(\frac{{26}^2}{8^2}\right)=1+\ln \left(\frac{676}{64}\right)=3.357310$$

Next, we can compare total number of pieces on the chessboards:

$${\mathrm{cf}}_{\mathrm{pieces}}=1+\ln \left(\frac{190}{32}\right)=2.781288$$

Another comparison is between number of different types of pieces:

$${\mathrm{cf}}_{\mathrm{types}}=1+\ln \left(\frac{18}{6}\right)=2.098612$$

Finally, we can compare number of different interactions:

$${\mathrm{cf}}_{\mathrm{interactions}}=1+\ln \left(\frac{19}{3}\right)=2.845827$$

Our complexity $c$ is then defined as a product of all factors calculated above:

$$c_{\mathrm{regular}}={\mathrm{cf}}_{\mathrm{size}}\cdot {\mathrm{cf}}_{\mathrm{pieces}}\cdot {\mathrm{cf}}_{\mathrm{types}}\cdot {\mathrm{cf}}_{\mathrm{interactions}}=55.767104$$

This is actual length scaling factor of regular games from Classical Chess into One variant, i.e. all games should be 55.767 times longer. For instance, the longest recorded tournament game was 538 moves (269 FIDE moves, aka cycles), which turns into 30002 moves (15001 cycles) for One variant. Average on-line match lasts about 80 moves (40 cycles), in One variant that would become 4461 moves (2230.5 cycles). Average tournament match lasts about 88 moves (44 cycles), which becomes 4907 moves (2453.5 cycles).

The same, however, does not apply when calculating maximal possible game length, because players will try to maximize each and every metrics available to prolong the game. This can be seen in Classical Chess games alone; the longest possible game with 50-cycle rule is 11797 moves (5898.5 cycles), while with 75-cycle rule it's 17697 moves (8848.5 cycles). If we calculate ratio between the two rules:

$$\frac{75}{50}=1.5$$

and game lengths, we can see that contribution to game length by rules extension was almost perfectly linear (50% increase in movement rule resulted in 50% longer game):

$$\frac{17697}{11797}=1.500127151$$

So, for maximum game length we have to calculate linear scaling factors; for chessboard sizes factor becomes:

$${\mathrm{cf}}_{\mathrm{size}}=\frac{{26}^2}{8^2}=\frac{676}{64}=10.562500$$

Next, for total number of pieces on the chessboards we have:

$${\mathrm{cf}}_{\mathrm{pieces}}=\frac{190}{32}=5.937500$$

For number of different types of pieces we get:

$${\mathrm{cf}}_{\mathrm{types}}=\frac{18}{6}=3.000000$$

Finally, we can calculate factor for number of different interactions:

$${\mathrm{cf}}_{\mathrm{interactions}}=\frac{19}{3}=6.333333$$

Taken together, our complexity $c$ becomes:

$$c_{\mathrm{longest}}={\mathrm{cf}}_{\mathrm{size}}\cdot {\mathrm{cf}}_{\mathrm{pieces}}\cdot {\mathrm{cf}}_{\mathrm{types}}\cdot {\mathrm{cf}}_{\mathrm{interactions}}=1191.582031$$

This is scaling factor for the longest possible games, i.e. the longest games should be 1191.58 times longer in One variant compared to Classical Chess. For instance, previously mentioned 11797 moves (5898.5 cycles) game as the longest possible with 50-cycle rule in One variant becomes 14,057,093 moves (7,028,546.5 cycles) game. Even longer 17697 moves (8848.5 cycles) game with 75-cycle rule in One variant turns into 21,087,427 moves (10,543,713.5 cycles) game.
Edit: all this without resurrections, with those used it's -for all we know- infinite.

These are all very large numbers, even "just" 55.767 times increase in complexity results in some ludicrous (as in, almost completely impractical) estimates. For instance, increase in complexity should also translate into longer time allowance per turn, simply because there is so much more stuff a player has to handle. So, 15 seconds per player's turn in bullet game (10 minutes, spread over average of 40 turns per match, see https://en.wikipedia.org/wiki/Time_control#Classification) now becomes approx. 13.941776 minutes per turn; given that average One game length would be 4461 turns, it would last for approx. 1036.5710456 hours of gameplay time; or, 129.5713807 days if we assume 8-hour gameplay in a day. If we don't increase time allowance per turn, 15 second per turn bullet game would take on average 18.5875 hours of gameplay time, or 2.3234375 days with the same 8 hours of gameplay per day.

And that's just bullet, lets not talk about classical time controls here, those numbers would be depressingly huge. One thing that is sorely missing from complexity estimate is mobility, and associated with it, piece powers. These are not easily estimated; also, increase in mobility alone does not add to complexity, rather it's ratio between mobility and available space (i.e. chessboard size), and I'm not sure that ratio has been increased by much. Anyway, this post is already too long, so I'll leave it for some future post.