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A wooden beam (which is a rectangular parallelepiped) with edges of 8, 8, and 27 inches is to be sawed into 4 parts out of which a cube can be made. Naturally, one should draw first and saw later.

8 x 8 x 27 wooden beam

How should the beam be cut?

Clarification: Assume that the sawing does not reduce the overall volume of wood (no sawdust will be produced).

Attribution: The Moscow Puzzles by BORIS A. KORDEMSKY.

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    $\begingroup$ The answer you accepted is not the one from the book and does not respect the requirement of “sawability” – you wouldn’t be able to saw pieces like that. Armin Rigo’s answer is the one described in the book (found the pdf online). $\endgroup$
    – Didier L
    Commented Oct 26 at 0:19
  • $\begingroup$ @DidierL I agree that Albert.Lang's solution would be challenging to saw. Perhaps a master carpenter using a Jig saw could do it. $\endgroup$
    – Will.Octagon.Gibson
    Commented Oct 26 at 4:01

6 Answers 6

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The figure shows cross sections of the beam at depths given in the first row. A,B,C,D indicate 4x4 squares of the 4 pieces.

    0-3   3-6   6-9   9-12  12-15 15-18 18-21 21-24 24-27
    AA    AB    AB    AB    CB    CD    CD    DD    DD
    AA    AA    AB    AB    CB    CD    CD    CD    DD

Similarly, cross sections of the 4 pieces rearranged into a 12x12x12 cube. The B and C pieces are rotated by 90 degrees.

    0-3   3-6   6-9   9-12
    AAB   ABB   ABB   ABB
    AAD   AAD   ADD   ADD
    CCD   CCD   CCD   CDD

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  • $\begingroup$ That's great (+1) The illustration from @Xavier helped to visualise it, but how on earth did you find that solution?! $\endgroup$
    – John Hunter
    Commented Oct 22 at 9:00
  • $\begingroup$ @JohnHunter Thanks! Thinking in terms of 3x4x4 blocks is an obvious thing to try. And then you have that looking down the 4-blocks-of-3 axis of the target cube no piece can cover more than one corner. Therefore each piece must cover one fixed corner over the entire depth. With these constraints it doesn't take that much trying and erring. $\endgroup$
    – Albert.Lang
    Commented Oct 22 at 10:09
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Illustration of answer by @Albert.Lang

enter image description here

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  • $\begingroup$ It's very helpful $\endgroup$
    – John Hunter
    Commented Oct 22 at 9:02
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    $\begingroup$ That can’t be done with a saw, right? $\endgroup$
    – Didier L
    Commented Oct 22 at 15:23
  • $\begingroup$ @DidierL - each colored piece can just be assembled from smaller rectangular prisms, so why not use a saw? $\endgroup$
    – The Chaz 2.0
    Commented Oct 22 at 16:31
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    $\begingroup$ @TheChaz2.0 because they would not be 4 parts. The question says "sawed into 4 parts", $\endgroup$
    – Weather Vane
    Commented Oct 22 at 21:15
  • $\begingroup$ Ok then use an oscillating multi-tool! $\endgroup$
    – The Chaz 2.0
    Commented Oct 23 at 16:25
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You can do it in two steps,

each with a simple 2D visualization. First look at the beam from the top, and saw the block in two halves to turn a 27x8 rectangle into a 18x12 rectangle (with depth 8):

                                                    9       9
              9        9        9              +--------+--------+
         +--------+--------+--------+          |                 | 4
       4 |        |                 |          +--------.        +
         +        '--------.        +    ==>   |        |        | 4
       4 |                 |        |          +        '--------+
         +--------+--------+--------+          |                 | 4
                                               +--------+--------+

Then, look from the side and repeat: saw in two independent halves and reassemble, this time turning a 18x8 rectangle into a 12x12 rectangle (with depth 12).

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    $\begingroup$ Very clever! At first I thought you'd end upwith more than four pieces so I made one. And the beauty is, you can cut it with a regular wire saw and turn the corners. There are no hidden cuts. $\endgroup$
    – Weather Vane
    Commented Oct 23 at 10:39
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    $\begingroup$ The puzzle only requires that the beam be "sawed into 4 parts", so your reduction of the number of cuts from 4 to 2 can only be seen as a great triumph. $\endgroup$
    – Mark Dominus
    Commented Oct 23 at 21:39
  • $\begingroup$ Sorry you didn't get the green check mark. It wasn't the first answer, but IMO is better, with the 27 dimension able to be split and reorganised this way. $\endgroup$
    – Weather Vane
    Commented Oct 24 at 18:01
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    $\begingroup$ This is the actual answer given in the book. $\endgroup$
    – Didier L
    Commented Oct 26 at 0:24
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Here are pictures for Armin Rigo's fine answer.
The pieces can be cut with a wire saw – no magic saw needed!

enter image description here
1 – starting with 8 x 8 x 27 inches
2 – first cut
3 – rearrange as 8 x 12 x 18 inches
4 – tip forewards
5 – second cut
6 – rearrange as 12 x 12 x 12 inches
enter image description here

I made the shapes with some 1 cm cubes that I have, and resized the pictures in the horizontal dimension so they represent 4 x 4 x 3 inch cuboids.

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    $\begingroup$ +1 for the great pictures $\endgroup$
    – MaxD
    Commented Oct 24 at 0:49
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Not a full answer yet, will think about this more later. However, here are some initial observations:

1. The cube will have each side 12" long.
2. The long side of the beam will consequently have to have cuts run go through all its sides at least twice.
3. A trivial solution is possible with 7 parts like this: Cut the beam into 2 12" lengths. Cut along the middle of one of the newly acquired 12" pieces, so that you have 2x 12x4x8". Cut the remaining 3" part into three quarters of 3" length.

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  • $\begingroup$ When you divide something into quarters, there are four of them. $\endgroup$
    – phoog
    Commented Oct 23 at 20:50
  • $\begingroup$ @phoog You are correct, and that is the way I was using the word - which is why I was referring to 7 total parts. These are as follows: 1 12x8x8", 2 12x8x4" and 4 4x4x3" (that can then be stacked to one 12x4x4-pillar). This is not a solution since it is more than four parts total, though. $\endgroup$
    – Brain404
    Commented Oct 24 at 9:23
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Here is a solution with 4 cuts (although it does lead to 12 pieces, but if you really worked in a sawmill and needed to do it efficiently...)

Cut longways in half and half again leading to 4, 4x4x27 pieces. when back together in original cuboid shape, cut them shorter, 12 inches, 12 inches and 3 inches.

Now 12 pieces, 8 are 4x4x12 and 4 are 4x4x3. since a 12 inch square can be divided into nine 4inch squares, these pieces would make the 12 inch cube.

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