# Simple Perfects by Boundary Rules and Conditions

The squares on the perimeter of the squared square or squared rectangle are the boundary squares. Those squares not on the boundary are in the interior. Those boundary squares not in corners are wall squares.

• Is there a minimum number of squares that can appear on the boundary?
• Is there a minimum number of squares that have to be in the interior?
• How do sizes of boundary squares and interior squares compare?
• Are the largest squares always on the boundary, or in the corners?
• What is the size of the smallest possible square on the wall boundary?
• What is the smallest possible square on the boundary, in relative size?
• What is the size of the smallest possible square in a corner?
• What is the largest square possible in each order (are they all boundary squares)?
• What is the total number of squares of a given size in each order?

Taking each question in turn;

1. ### Is there a minimum number of squares that can appear on the boundary?

2. For SPSRs, at least five squares must be on the boundary (in a perfect tiling the squares are all different sizes. If there were only four squares on the boundary, they would have to be the same size). For SPSSs at least seven squares must be on the boundary. Only two examples, one in order 29, side 1113 (found by Stuart Anderson, Ed Pegg Jr and Stephen Johnson) and the lowest order one in order 28 side 831 with seven squares on the boundary, found by Jasper Skinner.

3. ### Is there a minimum number of squares that have to be in the interior?

4. In orders 9,10 SPSRs exist with only 3 squares in the interior, this is the smallest number. For SPSS, as at least 7 squares must be on the boundary, the number of interior squares is <= order - 7. The minimum number of interior squares known for SPSS is 11 in order 22. Generally for SPSS, the number of interior squares outnumbers the number of boundary squares, but in order 25, 27, 28 and 29 the opposite occurs in some rare cases. See this file for all known SPSS with the number of boundary squares greater than the number of interior squares.

5. ### How do sizes of boundary squares and interior squares compare?

6. Generally speaking, the largest boundary square is about twice the size of the largest interior square in a given order.

7. ### Are the largest squares always on the boundary, or in the corners?

8. No, the largest square can be on the external wall or in the interior. Duijvestijn found SPSR with the largest corner not on the boundary in order 17. He found the first SISS with this property in order 21, and 2 lowest order SPSS with the largest corner not on the boundary in order 25. o25; 506 shows it is possible for all 4 largest squares not to be in corners.

9. ### What's the smallest square possible on the boundary?

10. A square of size 5 is the smallest square possible on the boundary of a perfect squared square, a result due to Gambini, the only known examples are SPSSs in order 44 and 58, both found by Gambini. In 2010 the smallest size square known on the boundary of an SPSR was an 101x91 rectangle of order 17, and an SPSR 166x143 order 22, both with a size 6.  In September 2011 Brian Trial found 3 SPSRs of order 28 with '5 on the side'. The dimensions of the 3 SPSRs are; 658x506, 749x611 and 1226x979. A pdf with all 3 tilings is here. The lowest order SPSR with a '5 on the side' is not known, it exists somewhere between the lower bound of order 25, established by Stuart Anderson's search of order 9 - 24 and the upper bound of order 28, established by Brian Trial.

The smallest squares on the boundary in SPSR orders 9 to 24 are recorded as OEIS Integer Sequence A195984 and also in the following table;

The Smallest Boundary Squares in Each SPSR Order
Order
(SPSRs)
Min boundary
9
8
10
13
11
22
12
18
13
14
14
13
15
11
16
9
17
6
18
9
19
7
20
7
21
8
22
6
23
8
24
7

The bouwkampcodes (not canonical) of the 3 order 28 SPSRs found by Brian Trial;
28 659 506 204 106 71 15 7 5 12 28 62 149 2 3 8 1 4 16 23 10 34 33 9 87 35 78 98 43 357 302
28 749 611 224 99 38 15 7 5 12 28 72 249 2 3 8 1 4 16 23 44 61 27 89 125 62 63 88 387 25 362
28 1226 979 400 221 41 16 7 5 8 19 49 128 332 2 3 9 11 25 30 66 79 45 21 24 204 69 179 111 647 579

11. ### What is the smallest possible square on the boundary, in relative size? (going from smallest to largest, smallest is designated e1, 2nd smallest e2, ... largest square is en, where n is the order)

12. Comparing relative sizes of squares, the 2 smallest squares (e1 and e2) in a SPSR or SPSS cannot appear on the boundary .  Only one example, o28; 697 found by Jasper Skinner, is known of an SPSS with 3rd smallest square (e3) on the boundary.  SPSRs with the 3rd smallest square on the boundary are known from order 13 onwards.

Relative Minima on Border - Squared Squares & Squared Rectangles

 Order SPSR SISR SPSS SISS 9 5th 3rd 10 5th 3rd 11 5th - 12 6th 3rd 13 3rd 3rd 3rd 14 3rd 5th 15 3rd 3rd 16 3rd 3rd 17 3rd 3rd 3rd 18 3rd 3rd 19 3rd 3rd 3rd 20 3rd 3rd 21 3rd 15th 3rd 22 3rd 12th 3rd
13. ### What's the smallest square possible in a corner?

14. According to Gambini, 9 is the smallest square possible in a corner. SPSR 32x32 order 9 is the only known SPSR or SPSS with 9 in the corner.  Is this also the only possible simple perfect square tiling where 9 appears in the corner? The 9 corner square in 33x32 is also the 5th smallest square in the tiling, a known minimum for SPSR and SPSS.  The smallest known corner in a SPSS, relatively speaking, is; order 27; 876. This corner square is of size 50 and is the 8th smallest square in the tiling. The smallest known absolute corner size for an SPSS is found in the two order 22 SPSS of size 110 (110 is the smallest possible size for a SPSS). Both have a square of size 18 in a corner.
Update 3rd November 2011, Brian Trial has found an order 51 SPSR with 11 in the corner! No size 10 has been found in a corner in SPSS or SPSR. We do not know if SPSSs with corner sizes between 11 and 18 exist, so we look to SPSR as a guide to what might be possible. The smallest corner square in each SPSR order from order 9 to order 24 is 9, 15, 25, 22, 24, 23, 23, 17, 15, 17, 16, 16, 17, 18, 16, 15. No SPSRs are known with corners of 10, 12 or 13. Order 9 33x32 SPSR also has the only known corner of size 14. Many CPSS with 33x32 compound inclusions exist, so a CPSS with 9 in the corner should be possible, as mentioned by Gambini.

Order 9 33x32 appears to be the unique appearance of both 9 and 14 in corners of SPSRs. Here is an outline of a method for a proof that this is the case; A corner square must be adjacent to at least one smaller square or the 2 larger adjacent squares on either sides of the corner will overlap, and if either square adjacent to the corner was the same size as the corner, then the tiling would be imperfect. Gambini has shown that 5 is the smallest side possible in a perfect tiling. For a corner of nine the only possible adjacent smaller squares are (1 and 8), (2 and 7), (3 and 6), (4 and 5), and (1, 3 and 5). Analysis of all these cases is being done by hand, and the results so far show that only 1 and 8 can be completed to a simple perfect rectangle, the 33x32 order 9 with 9 and 14 in the corners, all the others terminate in imperfections ie at least 2 squares the same size. The bounds established by Gambini were used to reduce the search space of his packing program. If the complete proof establishes this result then Gambinis bound of 9 in the corner can be raised to 10. It seems 10 in a corner may not be possible, If the cases of 10,11,12 and 13 can be determined, in a similar manner to 9, then the packing corner bound could be raised further for SPSS and SPSR, perhaps to 11, depending on the order being investigated.

15. ### What's the largest square possible in each order?

16. The table gives SPSR largest element square sizes by order . They are all boundary squares, and all but one (order 20 - 16439) are corner squares. Wolfram Alpha gives this formula as an exponential fit of the SPSR sizes; largest size = 18.847 e^(0.56452 x), where x = order - 8 . [0.5642 is approximately equal to 1/√π, 18.845 is approximately equal to 6π , so a conjecture is that the largest square in a SPSR of order n is asymptotically equal to 6πe^((n - 8)/√π) ]

The Largest Squares in Each Order
Order
Maximum square size = s
9 (SPSRs)
36
10 "
60
11 "
105
12 "
180
13 "
329
14 "
568
15 "
996
16 "
1733
17 "
3068
18 "
5315
19 "
9400
20 "
16439
21 (SPSRs)
29019
21 (SPSS)
50
22 "
97
23 "
134
24 "
200
25 "
343
26 "
440
27 "
590
28 "
797
29 "
1045
17. ### What pattern do we find in the total number of squares of a given size in each order?

18. If we total the number of squares of a given size in all the SPSRs of a given order, we obtain a list of ordered pairs; of the form (n, c(n)), where n is the square size , c(n) is the total of all n in the order). If we graph the ordered pairs as (x,y) coordinates, we see a curve resembling an exponential decay curve, with mod 4 oscillations on the curve.

• Mathematics of Computation volume 37, number 155 JULY 1981 Lowest Order Squared Rectangles and Squares With the Largest Element Not on the Boundary By A. J. W. Duijvestijn and P. Leeuw
• Mathematics of Computation volume 62, number 205 january 1994.p ages 325-332 SIMPLE PERFECT SQUARED SQUARES AND 2 x 1 SQUARED RECTANGLES OF ORDER 25 A. J. W. DUIJVESTIJN
• THÈSE L’UNIVERSITÉ DE LA MÉDITERRANÉE AIX-MARSEILLE II École doctorale de Mathématiques et Informatique Ian Gambini DOCTEUR ÈS SCIENCES spécialité : INFORMATIQUE Quant aux carrés carrelés