Hoffman's packing puzzle is an assembly puzzle named after Dean G. Hoffman, who described it in 1978. The puzzle consists of 27 identical rectangular

cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cu ...

s, each of whose edges have three different lengths. Its goal is to assemble them all to fit within a cube whose edge length is the sum of the three lengths. writes that the first person to solve the puzzle was David A. Klarner, and that typical solution times can range from 20 minutes to multiple hours.

Construction

The puzzle itself consists only of 27 identical rectangular

-shaped blocks, although physical realizations of the puzzle also typically supply a cubical box to fit the blocks into. If the three lengths of the block edges are , , and , then the cube should have edge length . Although the puzzle can be constructed with any three different edge lengths, it is most difficult when the three edge lengths of the blocks are close enough together that , as this prevents alternative solutions in which four blocks of the minimum width are packed next to each other. Additionally, having the three lengths form an

arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...

can make it more confusing, because in this case placing three blocks of the middle width next to each other produces a row of the correct total width but one that cannot lead to a valid solution to the whole puzzle.

Mathematical analysis

Each valid solution to the puzzle arranges the blocks in an approximate grid of blocks, with the sides of the blocks all parallel to the sides of the outer cube, and with one block of each width along each axis-parallel line of three blocks. Counting reflections and rotations as being the same solution as each other, the puzzle has 21 combinatorially distinct solutions. The total volume of the pieces, , is less than the volume of the cube that they pack into. If one takes the cube root of both volumes, and divides by three, then the number obtained in this way from the total volume of the pieces is the

geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...

of , , and , while the number obtained in the same way from the volume of the cube is their

arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The coll ...

. The fact that the pieces have less total volume than the cube follows from the

inequality of arithmetic and geometric means In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and ...

Table of solutions

The 21 distinct solutions are tabulated here as described by the references cited above . All boxes below are entered in the format ''(east-west length) x (north-south length) x (up-down length)'', denoting the size of each box with the dimensions ''A'', ''B'', and ''C'', where ''A'' < ''B'' < ''C''. (In the above example, ''A'' = 4, ''B'' = 5, and ''C'' = 6). All 3x3 matrices describe a set of 9 boxes, with east-west neighbors along each row and north-south neighbors down each column, with the three stacked layers being listed in sequence for each solution.

Higher dimensions

A two-dimensional analogue of the puzzle asks to pack four identical rectangles of side lengths and into a square of side length ; as the figure shows, this is always possible. In dimensions the puzzle asks to pack identical blocks into a

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions ...

. By a result of

Raphael M. Robinson Raphael Mitchel Robinson (November 2, 1911 – January 27, 1995) was an United States of America, American mathematician. Born in National City, California, National City, California, Robinson was the youngest of four children of a lawyer and a t ...

this is again solvable whenever for two numbers and such that the - and -dimensional cases are themselves solvable. For instance, according to this result, it is solvable for dimensions 4, 6, 8, 9, and other 3-smooth numbers. In all dimensions, the inequality of arithmetic and geometric means shows that the volume of the pieces is less than the volume of the hypercube into which they should be packed. However, it is unknown whether the puzzle can be solved in five dimensions, or in higher

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

dimensions.

References

{{reflist, refs= {{citation , last = Hoffman , first = D. G. , editor-last = Klarner , editor-first = David A. , editor-link = David A. Klarner , contribution = Packing problems and inequalities , doi = 10.1007/978-1-4684-6686-7_19 , pages = 212–225 , publisher = Springer , title = The Mathematical Gardner , year = 1981 {{citation, url=https://johnrausch.com/PuzzleWorld/puz/hoffmans_packing_puzzle.htm, title=Put-Together – Hoffman's Packing Puzzle, work=Puzzle World, first=John, last=Rausch, accessdate=2019-11-16, archive-url=https://web.archive.org/web/20191117012519/https://johnrausch.com/PuzzleWorld/puz/hoffmans_packing_puzzle.htm, archive-date=2019-11-17, url-status=live {{citation, title=A Mathematical Space Odyssey: Solid Geometry in the 21st Century, volume=50, series=Dolciani Mathematical Expositions, first1=Claudi, last1=Alsina, first2=Roger B., last2=Nelsen, publisher=Mathematical Association of America, year=2015, isbn=9780883853580, contribution=Problem 3.10, page=63, contribution-url=https://books.google.com/books?id=FEl2CgAAQBAJ&pg=PA63 {{citation , last1 = Berlekamp , first1 = Elwyn R. , author1-link = Elwyn Berlekamp , last2 = Conway , first2 = John H. , author2-link = John Horton Conway , last3 = Guy , first3 = Richard K. , author3-link = Richard K. Guy , pages = 913–915 , publisher = A K Peters , title = Winning Ways for Your Mathematical Plays , volume = IV , year = 2004, title-link = Winning Ways for Your Mathematical Plays {{citation, url=https://www.math.ku.dk/english/research/tfa/ncg/paststudents/bs-theses/Holck_bsthesis_18.pdf, title=An Experimental Approach to Packing Problems, first=Nikolaj Ingemann, last=von Holck, series=Bachelor's thesis, supervised by Søren Eilers, publisher=University of Copenhagen, date=January 2018, access-date=2019-11-17, archive-url=https://web.archive.org/web/20191117012517/https://www.math.ku.dk/english/research/tfa/ncg/paststudents/bs-theses/Holck_bsthesis_18.pdf, archive-date=2019-11-17, url-status=live Mechanical puzzle cubes Packing problems