Squared squares with some interesting visual properties are presented.

How do you represent a particular squared square uniquely? First, you present it in its canonical orientation, that is, with the biggest corner square at top left and the biggest of the two neighbouring squares on the top edge. That gets rid of rotations and reflections. Then for each horizontal edge in the squared square, starting at the top and running downwards, you list the sizes of the subsquares below that edge from left to right. Where more than one edge are at the same height, you list them from left to right. Finally you prepend the size of whole square. Thus a readable representation (called the Bouwkamp description) of the square used on the front page of this site is

    192 (86,49,57)(41,8)(28,37)(19,9)(47,35,4)(31,14)(10,36)(17,26)(12,71)(62)(59) 
    

Of course, the brackets and commas are just visual aids, and a computer will be perfectly happy with

    192 86 49 57 41 8 28 37 19 9 47 35 4 31 14 10 36 17 26 12 71 62 59 
    

Now we have a standard way to list1 the squared squares of a given order. Just list the Bouwkamp descriptions in ascending numeric order on the first field (the size), within the same value of the first field on the second, within the same value of the second field on the third and so on. The length of the list grows with the order, at first somewhat jerkily from one of order 21 to 160 of order 25. Then the increase suddenly becomes quite accurately exponential. There are 441 of order 26 and, as far as 33 at least, each further order has, to one decimal place, 2.6 times as many solutions as its predecessor.

Simple perfect squared squares are given identifying names like this: <order>:<size><letter>. "size" is that of the whole construction, and "letter" starts from A to identify those which have the same order and size. Thus the square used for the front page is 22:192A.

When colouring a squared square for display, it would be nice to have the most equitable colouring, that is, the colouring with the smallest ratio between the areas belonging to the colour with the most area and the colour with the least. That means checking a lot of different2 colourings... no solution has fewer than 1752 colourings, and by order 33 some have millions. Unless otherwise stated, the squares shown below are coloured as equitably as possible.

The one with the biggest smallest square

There are a total of 1152 different squared squares of order 27 and five of them are of size 1032. 27:1032E is distinguished by having a freakishly big smallest component square (relative to its overall size). At 48, its side length is nearly 5% of that of the whole solution. This is the biggest such ratio for any squared square below order 34 at least.

This square could be of interest in physical constructions, whenever the technique used involves difficulties in rendering the smallest squares.

There are 28488 different ways of colouring this "map" with four colours. The (unique) colouring above has the most equitable distribution of colour areas with the biggest colour, orange, having just 0.76% more than the other three, which in turn differ among themselves by barely one part in a thousand. Fewer than one percent of the squares below order 30 can be coloured as evenly as this.

The one with the smallest biggest square

What about turning the above condition around, and looking for the solution with the smallest biggest square relative to its overall size? This constraint makes for a lot of medium-sized squares, as can be seen in 33:1107A, where the biggest square is 27.100% of the size of the whole solution. 25% represents a kind of barrier here, since below that figure the solution would have to be everywhere at least five squares deep horizontally and vertically, tough to achieve in an order as low as 33.

There are 585744 different ways of colouring this solution. The (unique) colouring shown has the most equitable distribution with the biggest colour (green) having just 1.47% more than the smallest (yellow).

The one with the biggest square

If we are looking for the biggest component square, it is always going to be in a corner, since there it constrains the other squares least. 33:2704EM holds the record below order 34; its humongous corner square takes up more than 45% of the whole area. The colouring is the most equitable possible, but that's not saying much in this case!

One needing only three colours

We know that we can colour any squared square with four colours, but what about three? There is a theorem by Saaty and Kainen3 to the effect that for maps where only three countries meet at a point, a necessary and sufficient condition for three-colourability is that all countries have an even number of neighbours. Most squared square "maps" are of this kind; solutions where four squares meet at a cross are quite uncommon. Thus we only have to search up the lists of increasing order, looking for solutions where all the internal squares have an even number of neighbours. Well, they are remarkably rare. We have processed 145786 solutions when we finally find the first one, 32:1776A, and there is only one other to the end of order 33, a grand total of more than 610,000 solutions.

Notes:
  1. See Stuart Anderson's site for exhaustive lists and diagrams of all squared squares up to order 33 (at the time of writing), plus partial lists for order 34 and beyond.  ↑
  2. Different here means really different, not just reachable one from the other by swapping colours around. When checking each possible colour for each square, we avoid counting colour permutations (and save a lot of time) by picking three mutually touching squares and assigning them the first three colours. This naturally fixes the fourth colour as well, and, as any permutation would inevitably alter the colour of one of the three squares, it never gets counted.  ↑
  3. Saaty, Thomas L.; Kainen, Paul C. The four-color problem. Assaults and conquest. Second edition. New York: Dover Publications, Inc. 1986.  ↑