Hoffman's packing problem

Hoffman's packing problem, dismantled with the cubic wooden box. The 27 cuboids are made of different types of wood and labeled accordingly, which is irrelevant for the puzzle.

Hoffman's packing problem is a mathematical problem and a 27-part mechanical puzzle based on it , which was presented by the American mathematician Dean G. Hoffman at a conference at the University of Miami in 1978 and was later named after him. According to Dean G. Hoffman, mathematician David A. Klarner was the first to solve the problem. The time for the solution usually ranges from 20 minutes to several hours.

Problem description and game instructions

The game consists of 27 identical cuboids with three different edge lengths , and . It is usually sold with a cubic wooden box with the length of the inner edge . The problem description and game instructions are: ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle z}$ ${\ displaystyle l = x + y + z}$

Insert 27 cuboids with the length of the edge , and into a cubic box with the length of the edge ; , and must be different from each other and each greater than : ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle z}$ ${\ displaystyle l = x + y + z}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle z}$ ${\ displaystyle (x + y + z): 4}$

${\ displaystyle (x + y + z): 4 <x <y <z}$

Scope of play

The game can consist of blocks with any edge lengths , and can be constructed. The condition that the smallest edge length must be greater than a quarter of the sum of the edge lengths serves to exclude trivial solutions in which four cuboids are placed next to each other. If the edge lengths form an arithmetic sequence , the difficulty of the game increases because three joined average edge lengths fit into the cube, but only apparently lead to the solution of the problem. ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle z}$

mathematics

Each solution to the problem arranges all the cuboids similar to the 3 × 3 × 3 Rubik's Cube with their sides parallel to the sides of the receiving cube. Each edge of the resulting cube is formed by three edges of different lengths of the cuboids. If rotations and reflections are excluded, the problem has 21 different solutions; because of the position of the cuboid in the center of the cube, none of them have symmetries.

Inside the completed cube there are cavities that have no influence on the stability. The total volume of the 27 cuboids is smaller than the volume of the cube:

${\ displaystyle 27 \ times x \ times y \ times z <(x + y + z) ^ {3}}$

One third of the cube root of the volume of the 27 cuboids corresponds to the geometric mean of the values , and . In contrast, one third of the cube root of the cube volume corresponds to the arithmetic mean of the values , and . The different volumes result from the inequality of the arithmetic and geometric mean for different , and . ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle z}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle z}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle z}$

Solution example

The assembled cube can be shown in the top view of the three layers. Areas of the same size are shown in the same color, the interlocking of the levels resulting from the different heights of the cuboids is not taken into account:

Upper layer
	1	2	3	4th	5	6th	7th	8th	9	10	11	12	13	14th	15th
1
2
3
4th
5
6th
7th
8th
9
10
11
12
13
14th
15th

Middle location
	1	2	3	4th	5	6th	7th	8th	9	10	11	12	13	14th	15th
1
2
3
4th
5
6th
7th
8th
9
10
11
12
13
14th
15th

Lower layer
	1	2	3	4th	5	6th	7th	8th	9	10	11	12	13	14th	15th
1
2
3
4th
5
6th
7th
8th
9
10
11
12
13
14th
15th

Other dimensions

Solution of the two-dimensional problem, at the same time illustration of the inequality of the arithmetic and geometric mean

The two-dimensional modification of the packaging problem requires the accommodation of four rectangles with the different side lengths and in a square of the side length . The problem is always solvable and is often used to illustrate the inequality of the arithmetic and geometric mean. ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle l = x + y}$

Hoffman's packaging problem can be carried over to higher dimensions . In dimensions, it is required to pack the same -dimensional analogies to the rectangle ( ) and the cuboid ( ) in a -dimensional hypercube . The American logician and mathematician Raphael M. Robinson came to the conclusion that -dimensional problems are solvable if there is a product , the problem being -dimensional and -dimensional problems . From this follows the solvability for the dimensions 4, 6, 8, 9 and all other 3-smooth dimensions. As a result of the inequality of the arithmetic and geometric mean, the volume of the -dimensional hypercube is always larger than the summed up volumes of the -dimensional hyper-rectangles from which it is composed. It is not known whether there are solutions for five-dimensional and higher prime dimensions. ${\ displaystyle d}$ ${\ displaystyle d ^ {d}}$ ${\ displaystyle d}$ ${\ displaystyle n = 2}$ ${\ displaystyle n = 3}$ ${\ displaystyle d}$ ${\ displaystyle d}$ ${\ displaystyle d}$ ${\ displaystyle d_ {1} \ cdot d_ {2}}$ ${\ displaystyle d_ {1}}$ ${\ displaystyle d_ {2}}$ ${\ displaystyle d}$ ${\ displaystyle d}$

Individual evidence

^ ^A ^b John Rausch: Put-Together - Hoffman's Packing Puzzle. In: Puzzle World. Archived from the original on December 13, 2019 ; accessed on December 13, 2019 .
↑ ^a ^b ^c ^d ^e ^f ^g Dean G. Hoffman: Packing problems and inequalities . In: David A. Klarner (Ed.): The Mathematical Gardner . Springer, 1981, p. 212-225 , doi : 10.1007 / 978-1-4684-6686-7_19 .
^ ^A ^b Claudi Alsina, Roger B. Nelsen: A Mathematical Space Odyssey. Solid Geometry in the 21st Century (= Dolciani Mathematical Expositions . Volume 50 ). Mathematical Association of America, 2015, ISBN 978-0-88385-358-0 , pp. 63 , = problem 3.10 (English, limited preview in the Google book search).
^ Elwyn R. Berlekamp , John H. Conway , Richard K. Guy : Winning Ways for Your Mathematical Plays . Ed .: AK Peters. tape IV , 2004, p. 913-915 (English).
^ Nikolaj Ingemann von Holck: An Experimental Approach to Packing Problems . Bachelor thesis. University of Copenhagen, January 2018 (English, ku.dk [PDF; 1,2 MB ; accessed on December 13, 2019]).

[Rausch-1] A ^b John Rausch: Put-Together - Hoffman's Packing Puzzle. In: Puzzle World. Archived from the original on December 13, 2019 ; accessed on December 13, 2019 .

[Hoffman-2] ↑ ^a ^b ^c ^d ^e ^f ^g Dean G. Hoffman: Packing problems and inequalities . In: David A. Klarner (Ed.): The Mathematical Gardner . Springer, 1981, p. 212-225 , doi : 10.1007 / 978-1-4684-6686-7_19 .

[Alsina-3] A ^b Claudi Alsina, Roger B. Nelsen: A Mathematical Space Odyssey. Solid Geometry in the 21st Century (= Dolciani Mathematical Expositions . Volume 50 ). Mathematical Association of America, 2015, ISBN 978-0-88385-358-0 , pp. 63 , = problem 3.10 (English, limited preview in the Google book search).

[Berlekamp-4] Elwyn R. Berlekamp , John H. Conway , Richard K. Guy : Winning Ways for Your Mathematical Plays . Ed .: AK Peters. tape IV , 2004, p. 913-915 (English).

[Holck-5] Nikolaj Ingemann von Holck: An Experimental Approach to Packing Problems . Bachelor thesis. University of Copenhagen, January 2018 (English, ku.dk [PDF; 1,2 MB ; accessed on December 13, 2019]).