Longest Rectangular PP2 Pathology Puzzles

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Spencer Spenst edited this page Oct 2, 2024 · 16 revisions

This wiki page documents the longest known PP2/Pathology puzzles which fit within small $M \times N$ rectangles.

To preview and play any of these levels, just copy the level data into the Pathology Level Creator

In PP2/Pathology, all types of boxes are allowed and there can be holes.

	2	3	4	5	6	7	8	9	10	11	12	13	14	15	$\cdots$	N
1	1	2	3	4	5	6	7	8	9	10	11	12	13	14		$N - 1$
2	2	4	6	10	16	19	24	35	40	53	60	72	81	91		$\Theta(N^2)$
3		10	16	32	49	83	94	111	?	136	?	?	?	?		$\geq N^2 + 3N - 17$
4			32	70	152	210	237	?	?	?	?	?	?	?		?
5				146	240	308	?	?	?	?	?	?	?	?		?
6					338	?	?	?	?	?	?	?	?	?		?
7						570	?	?	?	?	?	?	?	?		?
8							846	?	?	?	?	?	?	?		?

$2 \times N$

The longest $2 \times 2$ and $2 \times 3$ match the results for PP1.

The longest $2 \times 4$, 5, and 6, were found by LifereaperX by hand: LifereaperX's 2x4, LifereaperX's 2x5, LifereaperX's 2x6

sspenst provides a general construction for an $2 \times N$ with $N > 5$ which is suboptimal but seemingly close to optimal:

If $N = 0 \mod 3$ then we can achieve $\frac{N^2 + 2N}{3}$ steps

For $N = 6$ we have:

310000
552024

To scale this puzzle, insert $3 \times 2$ segments of the form:

000
520

Between the last hole and the first box, ex.

310000 -> 310000000 -> 310000000000
552024 -> 555202024 -> 555520202024

If $N = 1 \mod 3$ then we can achieve $\frac{N^2 + 2N - 6}{3}$

For $N = 7$ we have:

3100000
5502024

The construction is similar to that of $N = 0 \mod 3$, but now there is a space between the boxes and the holes

If $N = 2 \mod 3$ then we can achieve $\frac{N^2 + 2N - 14}{3}$

For $N = 8$ we have:

31020050
55042020

This construction is also similar.

$2 \times 9$

Found by mathmasterzach by computer search

550856064
315600000

$2 \times 10$

Found by mathmasterzach by computer search

3100560000
5508506064

$2 \times 11$

Found by mathmasterzach by computer search

00800805513
48080056055

$2 \times 12$

Found by mathmasterzach by computer search

550805600064
315500600600

$2 \times 13$

Found by mathmasterzach by computer search

5508560000060
3155560060064

$2 \times 14$

Found by davidspencer6174 and FlashBack by computer search

4I0002020555F3
00020B5H05A055

$2 \times 15$

Found by FlashBack by computer search

550805000060064
315556060006000

$3 \times 3$

LifereaperX's 3x3 (by hand)

060
370
140

$3 \times 4$

LifereaperX's 3x4 (by hand)

0413
0210
0255

$3 \times 5$

Created by hand by davidspencer6174 and mathmasterzach

08555
0BH29
04535

$3 \times 6$

Found by davidspencer6174 via computer search

000085
0CBA01
004153

$3 \times 7$

mathmasterzach's Analog Watch (via computer search)

0000000
0H20HG0
0041035

$3 \times 8$

FlashBack's Insular Dwarfism (via computer search)

35114000
0F0F2GG0
0H000000

$3 \times 9$

Found by mathmasterzach via computer search

50H000005
90B2HA9DJ
357014000

$3 \times 11$

Found by mathmasterzach via computer search

08050000801
0HA97979753
00560000041

$4 \times 4$

RisingStar111's 4x4 (by hand)

$4 \times 5$

davidspencer6174's Beyond Measure (via computer search)

52020
0F4E3
CG625
585B5

$4 \times 6$

davidspencer6174's Break Infinity (via computer search)

$4 \times 7$

mathmasterzach's Cassette Rewind (via computer search)

$4 \times 8$

davidspencer6174's Polar Night (via computer search)

$5 \times 5$

davidspencer6174's Imperial Gallon (via computer search)

$5 \times 6$

Variant of davidspencer6174's Bigfoot (via computer search)

000E05
FE0060
5BE0I0
0J4025
536J50

$5 \times 7$

davidspencer6174's Barn's Broadside (via computer search)

$6 \times 6$

Reach and davidspencer6174's Crownbreaker (with computer aid)

$7 \times 7$

By davidspencer6174 (via computer search)

$7 \times 8$

By mathmasterzach (via computer search)

$8 \times 8$

By qqwref