8 Values: Extremal Functions in Square Packings

• 8 Values refers to the extremal functions f(8) and g(8) that optimize the total edge length of squares in a unit square under different packing and tiling constraints.
• The analysis employs geometric decomposition and the Cauchy–Schwarz inequality to derive g(8) = 13/5 and establish a constructive lower bound f(8) ≥ 8/3.
• The strict inequality f(8) > g(8) highlights that allowing gaps in packings can yield a greater edge sum, offering new insights in discrete geometry.

The problem of maximizing the total edge length of squares placed inside a unit square, under different covering constraints, yields a rich interplay between discrete geometry and extremal function analysis. For $n=8$, this analysis distinguishes between the cases of arbitrary packings and complete tilings, resulting in two distinct extremal functions, $f(8)$ and $g(8)$. The key result is the strict inequality $f(8) > g(8)$, indicating that allowing empty space can enable a strictly larger sum of edges than strict tiling requirements (Praton, 2011).

1. Formal Definitions and Problem Setting

Let $U = [0,1]^2$ denote the unit square. For a sequence of $n$ squares $A_1,\dots,A_n \subset U$ with side lengths $s_1,\dots,s_n$, define the total edge length as

$\Sigma(s_1,\dots,s_n) = \sum_{i=1}^n s_i.$

Two central extremal functions arise:

• Packing-maximum:

$$f(n) = \max\left\{\sum_{i=1}^n s_i : s_i > 0,\, \exists\,\text{pairwise-disjoint squares } A_i \subset U\,\text{ of side }s_i\right\}$$

$f(8)$0 is the maximum total edge length attainable by packing $f(8)$1 non-overlapping squares within $f(8)$2.

• Tiling-maximum:

$f(8)$3

$f(8)$4 imposes the stricter tiling constraint: the $f(8)$5 squares must partition $f(8)$6 exactly, with no gaps or overlaps.

By construction, every tiling is a packing, so $f(8)$7.

2. Computation and Structure of $f(8)$8

The exact value of $f(8)$9 is determined to be

$g(8)$0

This result is obtained through geometric decomposition and application of a Cauchy–Schwarz-based upper bound (Erdős' bound). For any $g(8)$1 squares with combined area $g(8)$2, the sum of side lengths satisfies

$g(8)$3

with equality only for all $g(8)$4 equal (Praton, 2011).

Applying this with $g(8)$5 and $g(8)$6, a preliminary bound $g(8)$7 is established. The proof sharpens this by analyzing possible configurations:

• There must be a “corner-tile” in one corner of side $g(8)$8.
• Three coastal tiles, each of side $g(8)$9, align along the square’s edges.
• Four interior tiles fill the remaining space.

Area accounting and application of the Cauchy–Schwarz inequality to the four interior tiles yield a global upper bound dependent on $f(8) > g(8)$0,

$f(8) > g(8)$1

Calculus shows the maximum occurs at $f(8) > g(8)$2, yielding the maximum sum

$f(8) > g(8)$3

This value is tight, as an explicit geometric tiling attains it (Praton, 2011).

3. Packing Construction and Lower Bound for $f(8) > g(8)$4

A constructive lower bound for $f(8) > g(8)$5 is realized by omitting the central cell from a $f(8) > g(8)$6 grid of $f(8) > g(8)$7 squares. This leaves eight $f(8) > g(8)$8 squares, each entirely contained and disjoint in $f(8) > g(8)$9: $U = [0,1]^2$0 No additional packing with larger total edge length is provided, but this configuration demonstrates that $U = [0,1]^2$1 exceeds $U = [0,1]^2$2.

4. Geometric Distinction and Implications

The geometric constraint of tiling—requiring the union of all $U = [0,1]^2$3 squares to exhaust $U = [0,1]^2$4 without gaps—forces a complex interplay between tile sizes and locations, restricting the extremal sum of side lengths. In contrast, the flexibility of packing allows the exclusion of a central region, enabling eight equally sized squares to achieve a larger total edge length.

The table summarizes the key numerical results:

Extremal Function	Constraint Type	Maximum Total Edge Length
$U = [0,1]^2$5	Packing (with holes)	$U = [0,1]^2$6
$U = [0,1]^2$7	Tiling (no gaps)	$U = [0,1]^2$8

The strict inequality $U = [0,1]^2$9 demonstrates that, for $n$0, maximizing the sum of side lengths is strictly easier when packings are permitted to leave some area uncovered (Praton, 2011).

5. Boundary Analysis and Methodological Notes

For $n$1 tilings, the “coastal vs. inland” analysis is instructive. The result that three edge-adjacent (coastal) tiles must share identical size $n$2 and are combined with a larger corner square of side $n$3 rigidly constrains the solution space. Area constraints force exact distributions, and maximizing the sum of edge lengths reduces to a univariate optimization over $n$4, with the Cauchy–Schwarz inequality bounding contributions from the four smallest, “inland” subsquares. This methodology distinguishes between the strictly greater edge-sum achievable with loosened constraints.

The distinction between packing and tiling is sharpened by these structural arguments: packings can exploit “holes” to allow eight relatively small, equally sized squares, whereas tilings are geometrically constrained to partitions that reduce the maximal sum. This phenomenon highlights the broader fact that, for selected $n$5, packing and tiling extremal functions can diverge.

6. Broader Significance and Open Directions

The result $n$6 directly answers a question posed by Benton and Tyler as to which $n$7 exhibit equality of $n$8 and $n$9, establishing a separation at $A_1,\dots,A_n \subset U$0. This suggests the structural rigidity imposed by tiling can provoke sub-optimality—relative to packing—for maximizing certain additive shape functionals.

A plausible implication is that for other values of $A_1,\dots,A_n \subset U$1, with careful geometric analysis, similar divergences between $A_1,\dots,A_n \subset U$2 and $A_1,\dots,A_n \subset U$3 may be found. Further research may aim to classify exactly the values of $A_1,\dots,A_n \subset U$4 where equality holds, and to generalize such extremal analyses to higher dimensions or other polygonal domains (Praton, 2011).