# Minimum lines needed to draw the 12 pentominoes

You need 4 lines to draw the monomino and the same for the domino

You can draw the 2 trominoes with 7 lines

For the five tetrominoes I could do it first with 16 lines

Then 15 lines

And my best try is 14 lines

What is the fewest number of lines needed to draw all 12 pentominoes?

• Here is a related question. My answer there provides a link to 5x12 solution #747, which can be drawn with a total of ## line segments. Dr Xorile's answer shows a few solutions which can be drawn only ## line segments. Answers to this question should improve on these or show that the latter is optimal. Commented Aug 25, 2023 at 2:54
• I have confirmed Dr.Xorile's results, i.e. looking only at the rectangular arrangements, the 6x10 ones he lists indeed have the fewest lines (4 more than the number of cuts mentioned there since we also include the 4 borders as lines in our count). Non-rectangular arrangements are not ruled out however because rot13(n zber pbzcnpg aba-erpgnathyne neenatrzrag znl cebivqr zber fcnpr sbe zber pebffvat snhyg yvarf gb erqhpr gur yvar pbhag). Commented Aug 25, 2023 at 10:33
• If you have a gap in a line, would that could has one line or two? E.g. suppose you had a 21x3 rectangle with 3 1x1 blocks missing and one of those was on an edge. Would that be one line or two? Commented Aug 25, 2023 at 17:48

Just to confirm @Jaap Scherphuis's comment that non-rectangular arrangements may be competitive here are some requiring 33 strokes, one fewer than the best rectangular ones according to Jaap.

For the sake of clarity: By rectangular we mean solid rectangular, no holes, no chamfered corners, etc. So the below are not rectangular because they have holes.

This is the complete set (up to symmetry) of optimal arrangements constrained to a 9x7 bounding box determined by exhaustive brute force computer search.

There also are multiple 33s in an 8x8 bounding box, for example:

• Yes your non rectangle solution it is ok Commented Sep 1, 2023 at 0:06
• Nice find! When I said rectangular arrangements, I of course meant only those without holes. I subsequently checked all 5x11 solutions without the 1x5 pentomino to see if any of those would give a nice solution when that pentomino was added on the side, but none of those improved on the record. Commented Sep 3, 2023 at 15:59
• @JaapScherphuis Of course. I've updated the post to make this clearer. Commented Sep 3, 2023 at 17:07
• Have you tried 8x8? The website I referenced has limited solutions for that with none better than 34, but other placements of the holes may provide further improvement. Commented Sep 3, 2023 at 17:49
• @DanielMathias Not yet. Unfortunately, every additional hole is quite expensive. Commented Sep 3, 2023 at 18:01

All best solution I know needs 34 lines