6cm per flag
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Nation, flags and memory card games
One of the “portable” games offered by Flying Tiger is a memory card game with national flags as a theme. The game has $68$ cards ($34$ national flags) and I like it not only because it's one of the largest memory games I've seen, if not the largest (about $50$ cards or fewer being the norm, for what I can see), but also because of its educational sideeffect, as each card features not only the flag, but also (in the “native” language as well as in English) the name of the nation and the name of the capital city.
After the first play, my first consideration was that Mexico was
missing from the flags/
So obviously, my next thought was: how large would such a memory card game be? $190$ states (at least) means $190$ pairs, or $380$ cards. The Flying Tiger cards are squares (with rounded corners) with a side length of $55$mm, and I don't think it's reasonable to go much smaller. In fact, considering the padding between cards that is needed for practical reasons when laying them down for playing, we can assume square tiles with a $6$cm side. That's $36$cm², or $0.0036$m² per tile. $380$ such tiles would cover an area of $1.368$m².
Curiously, $380$ is very close ${19.56}^{2}$, which means that if the $190$ pairs were laid out in a square, they would take up a square area with a side of almost exactly $117$cm, or $1.17$m. Even I would have troubles reaching the cards on the opposite side of the table. Of course, you can't actually do that, because $380$ is not a perfect square, so you cannot tile the cards on a square: you would have to either have to do a $19\times 20$ rectangle (with sides of length $1.14$m and $1.2$m respectively), or a $20\times 20$ square with an empty diagonal.
The issue of placing the memory cards in a pleasing pattern has always fascinated me. My daughter has a $48$cards game, which I like because the card can be laid out in a $7\times 7$ square with a hole in the center. My son has a $32$cards game that can be laid out as a $6\times 6$ square with missing corners.
In fact, the $68$cards game that spurred these consideration is a bit annoying because $68=8\times 8+4$, but the 4 extra cards cannot be placed in a nice symmetric way with good alignment because the sides of the square are even, not odd. Possible placemets are the “wheel” pattern (extending each of the sides with one card), the “shifted” pattern (place each extra card on the middle of each side, spanning the gap between the two central cards of the side), or the “versus” pattern, good when playing 1v1 games, with two extra cards on each of two opposing sides.
Obviously, the next question is: how many nations would you have to include to be able to tile out the cards in a “good” pattern?
 Perfect squares

this solution can be achieved with $20\times 20=400$ cards, or $200$ states (note that without holes or extra cards, the square must be even); problem is, Wikipedia lists $206$ states when including the ones with disputed sovereignity; we'd have to cherrypick some, leaving out six of them;
 The hole

an odd square can be made even by taking out a central hole;

a possible solution would be $19\times 191=360$, but this would require removing $10$ nations from the “uncontroversial” list;

the next one would be $21\times 211=440$, which would require $220$ states: can we find $14$ other territories to add to the $206$ states currently listed by Wikipedia?
 Give or take more

good patterns can be obtained by removing the 4 corners, or by having 4 extra cards —particularly with an odd square;

$20\times 204=396$ would require $198$ states, which could be assembled from the $193$ UN member states, the $2$ observer states, plus more (decisions, decisions);

$21\times 2114=436$ would require $218$ states: we'd only need to find $12$ more than the ones listed by Wikipedia.

$19\times 191+4=364$ would require $182$ states: still too little;
As it turns out, $380$ isn't even that bad: since $19\times 191+4\times 5=3611+20=380$, a decent tiling can be found with a $19\times 19$ square, without the center, plus $5$ cards centered on each side.
And a long stick to pick the cards on the other side of the table.
But I want more
Revision 6.0 of the Unicode standard introduced Regional Indicator Symbols and their ligatures mapping to «emoji flag sequences». Of the $26\times 26=676$ possible combinations, only $270$ are considered valid, and if we ignore the $12$ «deprecated region sequences», this leaves us with $256+2=258$ “current” region sequences. We could use this as the basis for our memory card game!
With $258$ regions we have $258\times 2=516$ cards. What would be a good pattern? We have $516=22\times 22+32=22\times 22+4\times 8$ which actually lends itself to a decent layout: a $22\times 22$ square, plus $8$ cards centered on each side.
How large would such a memory game be? At $6$cm per tile, the main square wold be $1.32$m wide. The extra cards on each side would make it $1.44$m wide, which is actually a pretty nice number1 —if unwieldy in practice. Still, if we're basing our choice on what Unicode supports, why not build such a game around computers? You'd need lots of players to make sense of it anyway.
anybody familiar with T_{E}X would recognize $1.44$ as “magnification step 2”, per Knuth's recommendation of scaling sizes by a geometric progression of ratio $1.2$. ↩