This is a list of Göbel strip packings, shown alongside their alternative packings or rearrangements. For the main list, see Squares in Squares.

Where the word "alternative" is used, this designates an alternative packing, which is enclosed within the same-sized bounding square as another packing, but cannot be reached by continuously translating and/or rotating the squares in that packing. When the packing can be reached by such continuous transformations, the word "rearrangement" is used.

For more information on each packing, view its SVG's source code.

Generalizing from Frits Göbel's findings, with $a \in \mathbb{Z}^+$ and $b=1\!+\!\lfloor{(a\!-\!1)\sqrt 2}\rfloor$, a Göbel strip consists of $n = (a\!+\!1)a\!+\!2\!+\!b$ unit squares packed inside a square of side $s = a\!+\!1\!+\!{1\over 2}\sqrt 2$. This is accomplished by placing a 1-width strip of $b$ squares at a $45°$ angle in the center, flanked by two unrotated squares at the corners, and sandwiched by two unrotated "staircases" each having $a$ steps. This yields the best known packing for all $a\!\lt\!44$ except for $a\!=\!3$.

Most Göbel strips also have a "simplified" alternative packing or rearrangement in which the $45°$ strip can contain $b\!+\!2$ squares, without any unrotated squares flanking it at the corners. If $(3+2\lfloor{(a-1)\sqrt 2}\rfloor)\sqrt 2-4a > 0$, then this is not possible at all. Otherwise, if $(3+\lfloor{(a-1)\sqrt 2}\rfloor)\sqrt 2-2a-2 > 0$, then the strip can only contain $b\!+\!1$ squares, and needs to be flanked at one corner by an unrotated square.

David W. Cantrell found in 2005 that there are alternative packings with minimal rotated squares for $s(27)$ and later, filling them out up to $s(84)$. David Ellsworth noticed that the $s(52)$ with minimal rotated squares is rigid (which coincides with its Göbel strip counterpart have very little space to slide), and later found that the pattern of rigid packings continues for every $a ≡ 1\!\mod\!5$.

David Ellsworth found in 2024 that whereas the general formula for the number of minimal rotated squares (i.e. number of copies of $s(5)$ in the packing) is $\lfloor{(2a+3)/5}\rfloor$, the number of rotated squares needed to match the efficiency of the Göbel strip is $\lfloor{(a-1)\sqrt 2}\rfloor+2-a$. Up to and including $s(331)$, these two formulae agree with each other (because $\sqrt 2\!-\!1 \approx 0.4$), but at $s(369)$, they differ for the first time. Following that point there are only $26$ more minimal rotated square packings for which they agree (out of $53$), the last one being $s(5213)$. This includes all of the rigid ones, which is not a coincidence – they have the least wasted space. But since Göbel strips are inoptimal starting at $s(2043)$ or possibly earlier, there are actually only at most $9$ alternative packings with minimal rotated squares for Göbel strips that may be optimal that are lost due to the divergence between the two formulae: $s(369)$, $s(586)$, $s(853)$, $s(974)$, $s(1170)$, $s(1311)$, $s(1537)$, $s(1698)$, and $s(1954)$.

Starting at $s(18)$, there is another type of Göbel strip, in which the strip is $b+1$ squares long, and is touched by the two unrotated squares in the corners, not the two "staircases" as the others are. These have $n = (a\!+\!1)a\!+\!3\!+\!b$ and $s = 2\!+\!(b\!+\!1){1\over 2}\sqrt 2$, and have a chance at being optimal when $\{s-{1\over 2}\sqrt 2\}$ is small. For example, with $a=13$, $n=202$ and $\{s-{1\over 2}\sqrt 2\} \approx 0.02081528$ which is indeed small.

For more details, see Erich Friedman's paper on the subject: Packing Unit Squares in Squares: A Survey and New Results (or David Ellsworth's edit).