# Pentomino solution maximizing straight lines length in rectangle - wood cutter problem

Recently in my free time I cut from wood with my scroll saw two pentomino sets. One set made from 10x6 pattern, and then the other set 20x3 pattern. Think of wood cutter difficulties. I would like to post a problem of finding the rectangle solution with the most straight lines.

The solution 20x3 has straight lines of 46 units. The straight line length comes entirely from the rectangle perimeter.

The solution 10x6 has straight lines of 44 units. Here 32 units come from the whole rectangle perimeter and there are 12 bonus units from a straight line cutting the rectangle in two halves (red line in picture). So alternatively we may count perimeters of two smaller rectangles which emerged after red line cut.

I suspect that there might exist a 12x5 solution which may consist of three smaller rectangles. Such solution, if exists, might give us 54 unit of straight lines (34 from perimeter and 10x2 from two internal straight lines).

Is there a solution in which a large rectangle can be composed of three smaller rectangles? Is there any solution which might beat the trivial 20x3 solution? If not, do you have any ideas for solutions maximizing strait line sections.

Being forced to choose one of two excellent answers my choice falls on Daniel Mathias's answer. The answer of Daniel Mathias replies to my original question "is it possible to make rectangle from three smaller rectangles?" and the answer provides useful link to all possibilities.

After it became apparent that the answer is NO, Dr Xorile sank deeper and stated another question "how few straight cuts of any length can be done". I do appreciate it. It was not an easy choice for me and I really wish I could upvote it triple times. Thank you, Dr Xorile.

• That's a really pretty set of pentominoes - I love the animal designs!
– Deusovi
Feb 11, 2019 at 16:10

There are no solutions in three rectangles.

For an index of all pentomino solutions:

• Do you mean that there are no solutions in three rectangles whatever the design or is your answer limited to 12x5 set only? Feb 11, 2019 at 16:32
• @PrzemyslawRemin I mean there are no solutions in three rectangles that can be arranged in a single rectangle. Feb 11, 2019 at 20:15
• @DanielMathias can you help me understand why #747 is interesting? Feb 12, 2019 at 16:08
• @DrXorile #747 has a straight cut of length $8$. As such, I suggested that it may be of interest, not that it necessarily is. Feb 12, 2019 at 16:12
• I must cut in to this talk:-) Strait lines are interesting because they might be done with different tool - table saw. Table saw is good for strait lines and it is very fast for that task. Perpendicular or curved cutting is done with scroll saw. Cutting true strait lines with scroll saw is difficult because the blade is very thin 0.12 inch (0.3 milliliter) and it tends to wander. Cutting strait line with a scroll saw is like riding on a bike. You have to constantly correct the course. Feb 13, 2019 at 8:46

@Daniel_Mathias gave a very helpful link which has all the 12x5 solutions in a text file. So some simple code allows us to see that of the 1010 12x5 solutions, there are 264 with 1 straight cut. But, sadly, none with 2 or more cuts. A few examples of the former are:

12
FFPPP IIIIINN
LFFPP ZZWNNNT
LFXUU VZWWTTT
LXXXU VZZWWYT
LLXUU VVVYYYY
81
FYYYYWZZUUU I
FFFYWWZTUXU I
PFVWWZZTXXX I
PPVNNNTTTXL I
PPVVVNNLLLL I
464
I LLLLUUXYYYY
I VVVLUXXXYPP
I VZZFUUXTWPP
I VZFFFNNTWWP
I ZZFNNNTTTWW

Edit: Now that I've understood @DanielMathias's point more thoroughly (see discussion under his answer), the question becomes "how few straight cuts of any length can be done". For 12x5, there are 18 with only 31 internal cuts (which is the fewest cuts for any 12x5). Examples include:

12
FFPPPIIIIINN
LFFPPZZWNNNT
LFXUUVZWWTTT
LXXXUVZZWWYT
LLXUUVVVYYYY
60
FLLLLIIIIINN
FFFXLZZWNNNT
PFXXXVZWWTTT
PPUXUVZZWWYT
PPUUUVVVYYYY
570
IPPPNNNWWVVV
IPPNNZZXWWTV
IUUFFZXXXWTV
IUFFZZYXLTTT
IUUFYYYYLLLL
623
IVVVWWNNNPPP
IVZZXWWTNNPP
IVZXXXWTFFUU
IZZYXLTTTFFU
IYYYYLLLLFUU
747
LLLLNNNFFPPP
LIIIIINNFFPP
VZYYYYWTFXUU
VZZZYWWTXXXU
VVVZWWTTTXUU

Further edit:

If we are just looking to minimize the number of cuts, then we can consider other lengths. I used the 3x20, 4x15, 5x12, and 6x10 solutions from the pentomino website and found that there are 9 solutions that have only 30 cuts. They are:

308
IIIIINLLLL
PPPFFNTTTL
PPFFWNNTZZ
UUXFWWNTZV
UXXXYWWZZV
UUXYYYYVVV
609
IIIIIWWVVV
LLLLWWXUUV
LNNNWXXXUV
PZZNNFXUUT
PPZFFFYTTT
PPZZFYYYYT
1089
IPPYYYYVVV
IPPXYFFUUV
IPXXXWFFUV
IZZXWWFUUT
IZLWWNNTTT
ZZLLLLNNNT
1324
IVVVLLLLPP
IVWWTTTLPP
IVFWWTZZXP
IFFFWTZXXX
IFNNYZZUXU
NNNYYYYUUU
1419
IWWTTTLLLL
IYWWTFNNNL
IYYWTFFFNN
IYVPZZFXUU
IYVPPZXXXU
VVVPPZZXUU
1653
LLLLIIIIIT
LFZZPPPTTT
FFFZPPXUUT
FWWZZXXXUV
WWNNNYXUUV
WNNYYYYVVV
1654
LLLLIIIIIT
LFZZPPPTTT
FFFZPPXUUT
FWWZZXXXUV
WWYNNNXUUV
WYYYYNNVVV
1963
LPPYYYYVVV
LPPFFYXUUV
LPFFWXXXUV
LLZFWWXUUT
ZZZNNWWTTT
ZNNNIIIIIT
1964
LPPYYYYVVV
LPPXYFFUUV
LPXXXWFFUV
LLZXWWFUUT
ZZZWWNNTTT
ZIIIIINNNT

Yet another edit:

Getting the fewest internal cuts (i.e. straight-line cuts that do not reach the side) favors solutions with lots of external cuts (i.e. straight-line cuts that do reach at least one side), which means bigger aspect ratios. In this way, you can find two 3x20 solutions that have 17 internal cuts. They have 34 total cuts:

1 (17, 34)
UUXIIIIINNNFTWYYYYZV
UXXXPPLNNFFFTWWYZZZV
UUXPPPLLLLFTTTWWZVVV
2 (17, 34)
UUXIIIIIZWWTTTFLLLLV
UXXXPPZZZYWWTFFFNNLV
UUXPPPZYYYYWTFNNNVVV
• It is interesting finding you have done. Feb 13, 2019 at 8:58
• Thank you. I appreciate your effort. Please see updated question. Feb 27, 2019 at 8:10
• Hi Dr Xorile, do you have any code for groping through the solution? I wonder which solution has the highest number of internal cuts - and in this sense which one is the most complicated. Mar 12, 2021 at 15:49
• @PrzemyslawRemin I still have my code, so can easily adjust it for whatever. What do you need? By "internal cut", you mean a straight cut of any length that does not intersect with the outside? And you are looking for the most complicated or the least complicated? Mar 12, 2021 at 20:10