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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.

enter image description here

Update after answers.
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.

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

2 Answers 2

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There are no solutions in three rectangles.

For an index of all pentomino solutions:

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  • $\begingroup$ Do you mean that there are no solutions in three rectangles whatever the design or is your answer limited to 12x5 set only? $\endgroup$ Commented Feb 11, 2019 at 16:32
  • $\begingroup$ @PrzemyslawRemin I mean there are no solutions in three rectangles that can be arranged in a single rectangle. $\endgroup$ Commented Feb 11, 2019 at 20:15
  • $\begingroup$ @DanielMathias can you help me understand why #747 is interesting? $\endgroup$
    – Dr Xorile
    Commented Feb 12, 2019 at 16:08
  • $\begingroup$ @DrXorile #747 has a straight cut of length $8$. As such, I suggested that it may be of interest, not that it necessarily is. $\endgroup$ Commented Feb 12, 2019 at 16:12
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    $\begingroup$ 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. $\endgroup$ Commented Feb 13, 2019 at 8:46
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@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
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    $\begingroup$ It is interesting finding you have done. $\endgroup$ Commented Feb 13, 2019 at 8:58
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    $\begingroup$ Thank you. I appreciate your effort. Please see updated question. $\endgroup$ Commented Feb 27, 2019 at 8:10
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    $\begingroup$ 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. $\endgroup$ Commented Mar 12, 2021 at 15:49
  • $\begingroup$ @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? $\endgroup$
    – Dr Xorile
    Commented Mar 12, 2021 at 20:10

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