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The following identity is a bit isolated in the arithmetic of natural integers $$3^3+4^3+5^3=6^3.$$ Let $K_6$ be a cube whose side has length $6$. We view it as the union of $216$ elementary unit cubes. We wish to cut it into $N$ connected components, each one being a union of elementary unit cubes, such that these components can be assembled so as to form three cubes of sizes $3,4$ and $5$. Of course, the latter are made simultaneously: a component may not be used in two cubes. There is a solution with $9$ pieces.

What is the minimal number $N$ of pieces into which to cut $K_6$ ?

About connectedness: a piece is connected if it is a union of elementary cubes whose centers are the nodes of a connected graph with arrows of unit length parallel to the coordinate axes.

Edit. Several comments ask for a reference for the $8$-pieces puzzle, mentioned at first in the question. Actually, $8$ was a mistake, as the solution I know consists of $9$ pieces. The only one that I have is the photograph in François's answer below. Yet it is not very informative, so let me give you additional information (I manipulated the puzzle a couple weeks ago). There is a $2$-cube (middle) and a $3$-cube (right). At left, the $4$-cube is not complete, as two elementary cubes are missing at the end of an edge. Of course, one could not have both a $3$-cube and a $4$-cube in a $6$-cube. So you can imagine how the $3$-cube and the imperfect $4$-cube match (two possibilities). Other rather symmetric pieces are a $1\times1\times2$ (it fills the imperfect $4$-cube when you build the $3$-, $4$- and $5$-cubes) and a $1\times2\times3$. Two other pieces have only a planar symmetry, whereas the last one has no symmetry at all.

Here is a photograph of the cut mentioned above.

(source)

jeq
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Denis Serre
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    (+1) Regarding your definition of connectedness: Do you need the "arrows" to be of unit length? Otherwise, for example, two unit cubes along the same edge of $K_6$ but on opposite faces of it would be "connected" by an arrow of length 5. – John Bentin Jan 24 '11 at 12:31
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    A few years ago Raphaël Cerf played around similar problems for so-called polyominoes. This was in relation with the metastability of the 3D Ising model (no less), see http://www.combinatorics.org/Volume_3/Abstracts/v3i1r27.html. So he might know the answer. – Did Jan 24 '11 at 13:01
  • How about the more general dissection problem (into continuous connected pieces, mind you) for solids? Are there results for turning a cude into a prism of the same volume with few cuts? Gerhard "Ask Me About System Design" Paseman, 2011.01.24 – Gerhard Paseman Jan 24 '11 at 21:09
  • Also, suppose there is a proof that no two of the smaller cubes fit together inside the largest cube. That provides a proof that 5 is a lower bound for any such dissection. Gerhard "Ask Me About System Design" Paseman, 2011.01.24 – Gerhard Paseman Jan 24 '11 at 21:16
  • Can you point us to the 8-piece solution? – Gerry Myerson Jan 25 '11 at 00:11
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    @Gerhard The solution to a version of the dissection problem is known: one cannot turn a cube into the regular simplex of the same volume through a finite number of cuts by hyperplanes, since these have different so called Dehn invariants. This holds in every dimension at least 3. In dimension 2, the area is a complete invariant, meaning that every polygons of the same area are equivalent, and the only nontrivial step of the proof is that any rectangle $L\times\ell$ can be turned into the square with side $\sqrt{L\ell}$. – Did Jan 25 '11 at 06:50
  • This would make a nice physical assembly puzzle. In the usual sort of puzzle, the symmetric assembly is easily dismantled into irregular pieces which are hard to reassemble. This one would be a satisfying puzzle in both directions (though perhaps easier in one direction than in the other). A lot more challenging would be $11^3+12^3+13^3+14^3=20^3$. – John Bentin Jan 25 '11 at 11:11
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    @John Bentin : such a puzzle was actually constructed. We used it last week during a mathematical exposition aimed at high school students. I found it already quite challenging. – François Brunault Jan 25 '11 at 12:47
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    @Denis Where can one find the solution for N=8? Is it known if there are more then one solution for N=8 ? – fastforward Jan 29 '11 at 23:32
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    Here is an interesting link about the analogous problem for pythagorean triples : http://mathafou.free.fr/pbg/sol110d.html It turns out that for small pythagorean triples, a puzzle with only 4 pieces can be constructed. The smallest triples for which the minimal number of pieces doesn't appear to be known are (20,21,29) and (28,45,53). – François Brunault Mar 13 '11 at 09:31

3 Answers3

90

8 is the least.

You can't have a piece of length 6, thus no two corners of the 6x6x6 cube can be part of the same piece, the cube has 8 corners, we need then a minimum of 8 pieces.

Noam D. Elkies
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JHI
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JHI's elegant lower bound of $8$ on $N$ is achieved by an explicit dissection. I show my construction below; you might want to try to find a solution yourself before proceeding $-$ it makes for a neat puzzle. There may well be other ways to do it.

If somebody can make a "$3$-dimensional" graphic or picture of the $8$-piece dissection, you're welcome to add it by editing my answer. My diagrams are two-dimensional, labeling each piece with its height. Fortunately the dissection is simple enough for this to be possible; in particular, the eight pieces comprise four boxes and four L-shaped prisms. This also made it possible to find the solution using just pencil and paper on an otherwise uneventful international flight.

Begin by cutting the $6 \times 6 \times 6$ cube top to bottom into three pieces, as shown in top view in the first square diagram. Then cut each piece horizontally in two, dividing AB into $3+3$,$\phantom.$ C into $4+2$, and D into $5+1$. Each AB piece is then further subdivided into a box B and an L-shaped prism A. The second diagram shows (say) the bottom layer of four pieces, and the third diagram shows the top. Note that the AB subdivisions are not quite the same.

(source)

Pieces with the same color will come together to form a smaller cube. The larger C piece is a $4$-cube, and the two A pieces form a $3$-cube as shown. It remains to construct the $5$-cube from the remaining five pieces. The last two diagrams show the bottom and top of the $5$-cube.

(source)

The two $5$'s are the larger B piece, rotated to span the entire height of the cube, and the thick D piece. The thin D piece completes the bottom, with width $1$. The top is filled by the thinner C piece and the smaller B, both rotated to height 4. QEF

I guess that a physical model won't make for a hard puzzle to reconstitute into either one or three cubes (e.g. the AB, C, and D parts of the $6$-cube are independent) but would still make a nice physical model of the identity $3^3 + 4^3 + 5^3 = 6^3$.

This dissection is specific to the solution $(a,b,c;d)=(3,4,5;6)$ of the Diophantine equation $a^3+b^3+c^3=d^3$; I don't know whether an $8$-piece dissection is possible for any other solution. JHI's analysis shows that one can never get below $8$, and in some cases even that's not possible: if $a<b<c$ and $a+c<d$ then there's at least one corner of the $d$-cube, say $(1,1,1)$, that contributes to the $a$-cube, but then any cell $(x,y,z)$ with $\max(x,y,z) = a+1$ cannot connect to any corner. This first happens for $(a,b,c;d) = (6,32,33;41)$.

What's the minimal dissection for the "taxicab" identity $1^3 + 12^3 = 9^3 + 10^3$? JHI's corner-cutting argument shows that at least nine pieces are needed.

jeq
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Noam D. Elkies
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    Beautiful! I'm especially impressed that you found a way to explain it in a convincing way by using 2-D figures. – Denis Serre May 05 '13 at 20:06
  • Actually, I should like to accept your answer. Unfortunately, I accepted already JHI's. – Denis Serre May 05 '13 at 20:07
  • @DenisSerre, I think you can change your accepted answer. Just click the check mark for Noam's answer. – JRN May 06 '13 at 04:21
  • @Joel. Yes, I can! – Denis Serre May 06 '13 at 05:25
  • Thanks! The 2-D illustration was possible because it's how I was able to find the dissection in the first place $-$ fortunately no coordinate permutation was needed except for boxes. It would have been much harder both to find and to illustrate the dissection if some of the pieces were as complicated as in the 9-piece dissection that you mentioned and F.Brunault posted. – Noam D. Elkies May 06 '13 at 20:30
  • Thanks too for accept ing my new answer. This curiously gives GHI the rare, perhaps unique, distinction of earning a gold Populist badge for his first and only MO answer! Populist or not, GHI's answer is necessary to show that the 8-piece dissection is minimal, and greatly helped focus my search using the key condition that each piece much include a corner of the $6 \times 6 \times 6$ cube. It also has the virtue of applying to all $(a,b,c;d)$, not just $(3,4,5;6)$. $$ $$ While I'm at it: who is "GHI"? When citing his contribution I'd like to credit a real person rather than a pseudonym. – Noam D. Elkies May 06 '13 at 23:44
  • Not that this helps much, but it appears to be JHI, not GHI. – Gerry Myerson May 07 '13 at 06:02
  • Yes, JHI, as I wrote correctly in my actual answer; here I must have been distracted by the alphabetical progression $-$ sorry. Unlike answers, comments cannot be edited here except by deleting the original comment and entering the corrected text for a new comment... – Noam D. Elkies May 07 '13 at 06:13
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The solution mentioned in the question consists in fact of $9$ pieces, as shown in the following picture I took yesterday. I don't know whether a solution with $8$ pieces exists.

Solution

Glorfindel
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François Brunault
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  • It seems to be quite a puzzle to fit the pieces together properly, both into the 3,4,5 cubes and into the 6 cube. – Thomas Sep 25 '13 at 04:31