Longest Pathology Puzzles By Size

This wiki page documents the longest known PP1 and PP2 puzzles which fit within small $M \times N$ rectangles.

Outside of trivial cases, these numbers should be considered lower bounds for the longest possible puzzles of various sizes.

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

PP1

In PP1, all boxes can be pushed from all sides and there are no holes.

We have the following results for small PP1 puzzles:

	2	3	4	5	6	7	8	N
1	1	2	3	4	5	6	7	$N - 1$
2	2	4	5	7	9	10	12	$\lfloor\frac{3N}{2}\rfloor$
3		8	11	14	17	20	23	$3 N - 1$
4			14	18	26	30	34	$4 (N + \lfloor\frac{N}{3}\rfloor) - 6$
5				34	40	46	54	?
6					49	59	65	?
7						67	78	?
8							90	?

The results for rectangular boards with at most 25 spaces have been confirmed optimal by mathmasterzach by exhaustive computer search.

$1 \times N$

Trivial construction of the form: 40...03

$2 \times 2$

40
03

$2 \times 3$

413
000

$2 \times 4$

4113
0000

$2 \times N$

For odd $N \geq 5$, the optimal is a zig-zag pattern.

41000
00013

For even $N > 5$, the optimal is a zig-zag pattern with a delaying block at the start.

041000
020003

$3 \times N$

A lock with a long handle.

00...020
41...123
00...020

$4 \times 4$

Two very different solutions, found by Flashback and dgriff24 respectively.

$4 \times 5$

Expanded version of dgriff24's $4 \times 4$

3 More designs created by hand by davidspencer6174

$4 \times 6$

Created by hand by davidspencer6174

$4 \times 7$

Expanded version of davidspencer6174's $4 \times 6$

$4 \times 8$

Expanded version of davidspencer6174's $4 \times 6$

$4 \times N$

The best we know so far is a continuation of the above pattern with "wiggles": Ex.

000000100000130
011110001110422
010001110001200
000100000100201

$5 \times 5$

Hem's hemlock (by hand)

$5 \times 6$

Created by hand by davidspencer6174

$5 \times 7$

Created by hand by davidspencer6174

$5 \times 8$

Created by hand by davidspencer6174

$6 \times 6$

Created by hand by Hem

$6 \times 7$

Created by hand by Hem

$6 \times 8$

Created by hand by Hem

$7 \times 7$

Created by hand by Hem

$7 \times 8$

Created by hand by Hem

$8 \times 8$

Hem's Boundary Unstable (by hand)

PP2/Pathology

We have the following results for small PP2 puzzles:

	2	3	4	5	6	7	8	9	10	11	12	13	14	15	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	80	89	$\Theta(N^2)$
3		10	16	32	49	83	94	111	?	?	?	?	?	?	$\geq N^2 + 3N - 17$
4			32	70	152	210	232	?	?	?	?	?	?	?	?
5				132	221	?	?	?	?	?	?	?	?	?	?
6					300	?	?	?	?	?	?	?	?	?	?
7						?	386	?	?	?	?	?	?	?	?
8							504?	?	?	?	?	?	?	?	?

$2 \times N$

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.