A System for Sizing Chess Pieces and Boards (long)

Many vendors and organizations publish guidelines for properly scaling chess sets and boards. Some of these are based solely on the height of the king as the determining factor for that selection. Yet it is the diameter of the king, resting on its square, which gives us the best visual indication of scale and proportion. Plus, I’ve seen 4½” kings with diameters everywhere from 1¼“ – 2” wide, so I don’t think a one-size chess board solution will work for all these sets of varying diameters.

Also, you can match a king to its appropriately sized board, but if the pawns are not properly matched as well, the set will look disproportioned. It makes sense considering half the pieces on the board are pawns!

The solution, of course, is to size both the king and the pawn to the same chessboard. In a properly designed chess set, all the remaining pieces – the queen, rooks, bishops, and knights -- should be proportionally scaled to the dimensions of the king.  

The formulas presented here form an integrated system for sizing chess pieces to a given chess board based on the diameters of both the pawn and the king. So matching both is all that’s needed to ensure a well-scaled chess set that’s easy to use and pleasing to the eye.

Starting from accepted guidelines and on what looks pleasing, we will arrive at two separate formulas, one for sizing the king to its square, and the other for sizing the pawn, each with a different scaling factor. As it turns out, those two factors share an interesting mathematical relationship.

Using these two constants, we will calculate:

1. What size king best fits a given chessboard?
2. What size pawn best fits a given chessboard?
3. What size board best fits a given king?
4. What size board best fits a given pawn?

SCALING THE PAWN AND CHESSBOARD

In their regulations, the International Chess Federation (FIDE) suggests that the diameter of a pawn should be about half the width of the square. Using these guidelines, you could fit four pawns inside of a single chess square. To my eye, this leaves you with pawns that are scaled too small for the size of the board, and oftentimes mis-scaled when compared to the remaining chess pieces.

Aesthetically, I feel a pawn best fits a chess square when two pawns, placed diagonally, fit perfectly inside the square. Calculating this dimension, we arrive at a pawn with a diameter equal to 58.6% the size of the square. This is about 17% larger than the FIDE guidelines, but still within regulations for tournament play.

•  Pawn Diameter = Size of the Square (S) times .586 = (.586S)

 SCALING THE KING AND CHESSBOARD

Best practice has placed the ratio of the King Diameter (Kd) to the Square (Kd/S) between 72% and 82%. U.S. Chess Federation specifies a range of between 73-78% for tournament play. Other sources recommend anywhere from a low of 68% to a high of 85%, the median of which is 76.5%.

• King Diameter = Size of Square (S) times .765 = (.765S)

According to these formulas, the ideal King and Pawn for a standard 2.25” tournament size chess board would be:

• King Diameter (Kd) = .765S = .765 x 2.25” = 1.72”

• Pawn Diameter (Pd) = .586S = .586 x 2.25” = 1.32”

In  this arrangement, the king occupies 76.5% of the square's width, and two pawns (placed diagonally) fit exactly inside of the square.

So how are these two constants mathematically related? This is the spooky part, if we take the square root of .586, the scaling factor for the pawn, we get -- .765 -- the scaling factor for the king! Because of this curious relationship, we can now make the same calculations using .765 as our sole constant. 

This makes an ideal King Diameter 76.5% of the square, and the ideal Pawn Diameter 76.5% of the King’s Diameter.

Using same 2.25” chessboard:

• King Diameter (Kd) = .765S = .765 x 2.25” = 1.72”

• Pawn Diameter (Pd) = .765 x Kd = .765 x 1.72” = 1.32”

As a shortcut, we can calculate both in a single operation:

• [Square Size (S) x .765 = Kd x .765 = Pd]

or

• 2.25" x .765 = 1.72" x .765 = 1.32"

In conclusion (finally!), here are the four formulas you can use to ensure your chess pieces and boards are perfectly matched.

 1)  To find the ideal king size for a given chess board:

• (Kd) = (.765) x (S)

2)  To find the ideal pawn size for a given chess board:

• (Pd) = (Kd) x (.765)

3)  To find the ideal chess board for a given king:

• (S) = (Kd) / (.765)

4)  To find the ideal chess board for a given pawn:

• (S) = (Pd) / (.586)

I realize we don’t all share the same sense of beauty, scale, and proportion. As they say, "One man's ceiling is another man's floor." Some may prefer a slightly smaller board, others a slightly larger one. If your preference is the former, then select a board ⅛” smaller than the calculations call for; if your preference is the latter, then select a board ⅛” larger. Either way, your pieces, and pawns will be proportionately scaled to the size of the chessboard you select.

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PICTURE #1 (below) shows an overview of a set with a 1.5" king on a 2" chessboard (75% scaling factor) with FIDE recommended 1" pawns (50% scaling factor)

PICTURE #2 shows how four pawns from this set fit snugly inside the square.

PICTURE #3  shows an overview of the same set with 1.125" pawns (56.25% scaling factor).

PICTURE #4  shows how two pawns from this set fit diagonally inside the square.

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PICTURE #1 (Kd = 75% - Pd = 50%)

PICTURE #2 (notice how 4 pawns fit into the square)

PICTURE #3 (Same set with larger pawn: Kd = 75% - Pd = 56.25%)

PICTURE #4 (notice how 2 pawns fit into the square)

nice  ,,  thanks for ur advice  

Excellent work Lou.  Maybe you want to even submit that to the USCF for publication in one of their upcming magazines. 

Problem is with a set where you only have a choice of the K looking overcrowded or the pawn looks lonely.  The 1849 set coming from CB would have been nicer if the pawn was a pinch wider but then it wouldn't be an "exact" replica. So in this case you will have no ideal board. Now it becomes a matter of choice. I myself like the pawns looking a bit lonely vs. the pieces looking overcrowded.  I don't have the measurements just Alan's you tube video. He prefers the K being a bit overcrowded vs the paws looking good and I can see the reasoning as there are 8 pawns and only 1 king. But I like it just the opposite so I will go for a baord that is 1/8" to 1/4" larger squares.

Mike, as often happens with a set where the pieces and pawns are not properly scaled, I try and select a board that offers the most pleasing compromise. Depending on the design of the pieces (tall and thin, short and stocky, etc.) I'll opt for right-sizing the pawns at the expense of a slightly crowded king. But in sets where the king and queen have the same diameter (or almost), I'd probably use a slightly larger board.

Yes I know your preference. I myself feel the set Alan was playing in the park with on that board was way too small. I would rather have the pawns for that set get lost a bit than to crowd the pieces.  This will be a problem with the CB 1849 set since the pawn and king base were not taken into consideration 165 years ago for that particular set. It's a matter of what you like etc.  

Thanks OP for the mathy and helpful post. I have a 2.125 board I bought to use at home for my 3.6 inch Dubrovnik set (1.6 inch diameter base on the king). I just found out that it is...perfect! I knew it looked good to me but now I know why. In a plug for Dubrovies, I think the proportions of this set are also perfect. 

Kimbacal,

Nice set, and it looks great on that board. What's the diameter of the pawns?

lou

P.S. I see you're a Capablanca fan -- me too. Personally, I think if he was alive today as 25 years old contemporary of Carlsen, Caruana, Aronian, et. al., with all the advatanges of growing up with computer engines and 75 years of advanced chess theory, that he'd be untouchable, with a probably ELO rating between 2950-3000!

Alekhine, an egomaniac who was not given to effusive praise toward anyone other than himself, once said that he had known many chess greats, but only one chess genius -- Capablanca. His vision of the board was such that nobody could touch him in blitz or lightning chess.

Thanks! Yes I like to play through classic games. I am currently doing Capablanca (100 games) and Fisher (60 games) books.

The Dubie pawns are 1.2 inches and two fit exactly across a square on the diagonal. Here's a closer up pic of these little guys.

I just got a new plastic set with folding 2-3/8" board. I was pleased to find that the dimensions of this set are close to my calculations above.

The 4.25" tall king has a diameter of 1-7/8". The pawn is 2-3/8" tall with a 1-3/8" diameter. Making the scaling factors:

King scaling factor - 79%

Pawn scaling factor - 58%

Super nice looking plastic set. I can even see it has artificial wood grain. Which one is it ?

http://www.ebay.com/itm/390607428215?_trksid=p2059210.m2749.l2649&ssPageName=STRK%3AMEBIDX%3AIT

 Positves: Unique heavy polymer design, extra heavy weighting (68 oz.), wood grown in America from renewable sources, leather pads.

Negatives: (see below)

I found this set on eBay. Made in China. Pieces were nicely molded, and (surprise, surprise!) they have leather (pleather) pads. Two of the pieces rattled a bit. When I removed the pads I found steel discs securely affixed and flush with the bottom of the pieces. So whatever was rattling was underneath. After I removed the old adhesive I tried removing the discs using a super powerful magnet -- they didn't even budge! Since the rattle wasn't a physical problem -- it wasn't endangering the pads, and the pieces didn't wobble -- I chose to ignore it. It's not like they're rattling around while sitting on the board.

Also, two pads were glued off-center, but it was easy enough to remove and reglue them using a dollop of "Amazing Goop."

The board is a very nice maple and walnut inlay. Though not a perfect set, at this price ($100) I found these slight imperfections acceptable.

Good buy! The board alone is probably worth that.

   Zigmund, that's a good point. I recognize there's a vast difference in the design of chess pieces. That's why the basis of my system is based on my aesthetic sense -- what looks good to me. Being an engineer, these formulas are a way of expressing my opinion mathematically. For those who share this view -- great, for those that don't -- that's great too.

   Even with a set of ideal dimensions, we must take into consideration the mass of the piece -- how much space it occupies in three dimensions. There are a lot of commemorative sets (Romans, Civil War soldiers, etc.) where the upper parts of the pieces take up more space than a typical Staunton style set. Here's where one has to be flexible. Starting with the 'ideal' board based on the formulas and upsizing (or downsizing) the board in small increments until you reach a happy medium that looks good to you. Adjustments must also be made where sets are taller or shorter than average.

   I've seen pleny of these decorative sets sold with boards that barely large enough to contain the pieces. I suspect that these sets are sold more for show than go.

   I agree, these formulas work best with Staunton style sets and that adjustments must be made for different sytles, such as the commemorative sets, or the tall, thin pre-Stauton designs. In which case, this system can give the collector a point of departure, a place from which to begin.

Zigmund - Thanks for mentioning that. From my original posting #1:

   "I realize we all don’t share the same sense of beauty, scale and proportion. As they say, "One man's ceiling is another man's floor." Some may prefer a slightly smaller board, others a slightly larger one. If your preference is the former, then select a board ⅛” smaller than the calculations call for; if your preference is the latter, then select a board ⅛” larger. Either way, your pieces and pawns will be proportionately scaled to the size of the chess board."

  So the system is actually recommending three board sizes -- one based on the calculations, one 1/8" smaller, or a board 1/8" larger, whichever you prefer. Like Goldilocks and the three bears.

  Since you find the 51mm (2") too crowded, as per my guidelines, you simply move up 1/8" (4mm) to a 2.125" (54mm) board, as you have suggested. And you bring up a good point, two sets can have the similar diameters for both the king and pawn, and yet, because of their design differences, look best on two different size boards.

  One of the main points of this system is emphasizing the necessity of properly scaling the king and pawn diameters, so that regardless of which size board you prefer, the proportions of the pawns and back row pieces are balanced. Again, all of this is based on my own preference, with a little wiggle room thrown in for good measure.

  Some people may prefer a crowded back row and undersized pawns (or vice cersa), in which case, this system won't be to their liking.

  This system can be used for more than just sizing chess pieces and boards. It can also be used as a guideline when purchasing new and used chess sets. Had I been aware of this last year, I would not have purchased a set with a 1.625" diameter king with 1" diameter pawns. I would have continued my search, 'til I found an acceptable set with a 1.25" pawns instead. Because I didn't, now I have a set where the king wants a 2-1/8" board and pawns a 1-3/4" board. My solution was to use a 2" board and split the difference -- but it still looks a bit imbalanced.

I have always found this topic way more important than people give it credit for being.  I recently have done some research into what I consider the most pleasing proportions.  76.5% is on the high side of what I like.  My sweet spot is 72-76% king base to square size ratio.

I have noted that one of the most commonly used ratios is the standard tournament set with the standard vinyl tournament boards.  Those are very often 1.5" king base and 2.25" squares.  This works out to 67%.  Definitely below the sweet spot, though I've found that this ratio is certainly playable for me, probably because I've done it so often.  Any smaller of a ratio than this would almost certainly be unplayable to me, though.  That said, anything higher than 76% would feel unplayable to me, as well.

So, my personal preference is as follows:

Acceptable/playable range: 67% - 76%

Sweet spot for me: 72-74%

This topic is also very important for 2D computer chess boards, to me, at least.  I feel the optimum king width (not the base, the full piece) is more like 80-85% of the square, though, again, I prefer the low side of this.  81 or 82% feels just right to me.

Below is an example:

The forums will scale the image down, but the ratios should stay the same.  In the original, the king width is 86 pixels and the square width is 106 pixels, making the ratio 81%.

 Gee, thanks so much for posting that link and pictures and many thanks to pawpatrol for digging up the 3 month old thread .... Just what I needed, a really nice looking set to spend more money on.

That makes two things now that pawpatrol has made me buy .... a very nice wooden board and now this set.

Thanks guys, my starving family appreciates you! 

lol ;)

"Blame it on my OCD."