# «Contents Ü Foreword Elwyn Berlekamp and Tom Rodgers ½ I Personal Magic ¿ Martin Gardner: A “Documentary” Dana Richards ½¿ Ambrose, Gardner, ...»

True planar burrs are rarely found. The ﬁrst design I saw is Jeffrey Carter’s, depicted in A. K. Dewdney’s Scientiﬁc American column (January 1986, p. 16). Carter’s puzzle has four pieces. The puzzle is not very difﬁcult to solve, with only three moves needed to remove the ﬁrst piece.

The idea of a two-dimensional burr immediately appealed to me, and in April 1986 I made some attempts to ﬁnd a design of my own. One result is the “Zigzag” planar burr, depicted in Figure 1.

This puzzle has two congruent large pieces and two congruent smaller ones. It takes ﬁve moves to separate the ﬁrst piece. To solve the puzzle, the two large pieces move into each other along a zigzag line, until the two smaller pieces are free. The same movements, in backward order, will separate large pieces.

166 M. O. VAN DEVENTER

** Figure 1. Zigzag.**

Some months ago, particularly inspired by Muroi’s “Four Sticks” (so closing the circle of mutual inspiration), I took up the challenge again and succeeded in ﬁnding a new design of a planar burr, which I have called “Nine and One-Half Moves.” This is is a true two-dimensional burr of only three pieces and it needs no less than nine and one-half moves to separate one piece from the other two.

The three pieces of the puzzle form a square with internal voids.

** Figure 2. Nine and One-Half Moves.PLANAR BURRS 167**

The pieces, and the moves required to separate the pieces are depicted in Figure 2. The ninth move is a slide-plus-rotate move, so I count it as a move and a half.

In order to prevent us three-dimensional people from cheating, we can glue one piece of the puzzle between two square plates as indicated in the top left corner of Figure 2. By using opaque plates the design is hidden as well. A round hole can be used to hide a coin.

Block-Packing Jambalaya Bill Cutler My primary interest over the years has been burr puzzles, but there is another small category of puzzles that is especially intriguing to me. It is 3-dimensional box-packing puzzles where the box and all the pieces are rectangular solids. The number of such puzzles that I am aware of is quite small, but the “tricks,” or unique features that the puzzles employ are many and varied. I know of no other small group of puzzles that encompasses such a rich diversity of ideas.

Presented here are 11 such “block-packing” puzzles. The tricks to most of the puzzles are discussed here, but complete solutions are not given.

The puzzles are grouped according to whether there are holes in the assembled puzzle and whether the pieces are all the same or different.

For each puzzle, the total number of pieces is in parentheses. If known, the inventor of the puzzle, date of design, and manufacturer are given.

7. No Holes, All Pieces the Same “Aren’t these puzzles trivial?,” you ask. Well, you are not far from being completely correct, but there are some interesting problems. David Klarner

**gives a thorough discussion of this case in [5]. The following are my favorites:**

8. No Holes, Limited Number of Piece Types The puzzles I know of in this category follow a common principal: There are basically two types of pieces — a large supply of one type and a limited supply of another. The pieces of the second type are smaller and easier to use, but must be used efﬁciently to solve the puzzle. The solver must determine exactly where the second set of pieces must be placed, and then the rest is easy.

**¿º unnamed (9) (Slothouber–Graatsma):**

Box: ¿ ¢ ¿ ¢ ¿ Pieces: (3) ½ ¢ ½ ¢ ½, (6) ½ ¢ ¾ ¢ ¾

**º unnamed (18) (John Conway):**

Box: ¢ ¢ Pieces: (3) ½ ¢ ½ ¢ ¿, (1) ½ ¢ ¾ ¢ ¾, (1) ¾ ¢ ¾ ¢ ¾, (13) ½ ¢ ¾ ¢ In the ﬁrst of these, the three individual cubes are obviously easy to place, but they must not be wasted. By analyzing “checkerboard” colorings of the layers in the box, it is easy to see that the cubes must be placed on a main diagonal. In the second design, the three ½ ¢ ½ ¢ ¿ pieces must be used sparingly. The rest of the pieces, although not exactly alike, function similarly to the ½ ¢ ¾ ¢ ¾ pieces in the ﬁrst puzzle. See [2] or [5] for more information.

9. No Holes, Pieces Mostly Different

Quadron also makes for a nice entrance into the realm of computer analysis of puzzles and the limitations of such programs. The programmer can use algorithms that are used for pentominoe problems, but there are more efﬁcient algorithms that can be used for block-packing puzzles. I wrote such a program on my ﬁrst computer, a Commodore 64. The program displayed the status of the box at any instant using color graphics. I painted pieces of an actual model to match the display. The result was a fascinating demonstration of how a computer can be used to solve such a puzzle.

The Commodore 64 is such a wonderously slow machine — when running the program in interpreter BASIC, about once a second a piece is added or removed from the box! Using compiled BASIC, the rate increases to 40 pieces/second.

When running these programs on more powerful computers, the difference between the three boxes is stunning: The ﬁrst box can be completely analyzed in a small fraction of a second. The second box was analyzed in about a minute of mainframe computer time. In early 1996, I did a complete analysis of the third box. There are 3,450,480 solutions, not counting rotations and reﬂections. The analysis was done on about 20 powerful IBM workstations. The total CPU time used was about 8500 hours, or the equivalent of one year on one machine. By the end of the runs, the machines had constructed 2 1/2 trillion different partially ﬁlled boxes.

º Parcel Post Puzzle (18) (designer unknown; copied from a model in

**the collection of Abel Garcia):**

Box: ¢ ½ ¢ ¾ Pieces: all pieces are of thickness 2 units; the widths and lengths are ¢, ¢ ½, ¢ ¾½, ¢, ¢ ½¼, ¢ ½¿, ¢ ½, ¢ ½, ¢ ½½, ¢ ½¿, ½¼ ¢ ½½, ½½ ¢ ½½ and two each of ¢, ¢, and ¢ ½¿.

Since all the pieces are of the same thickness and the box depth equals three thicknesses, it is tempting to solve the puzzle by constructing three layers of pieces. One or two individual layers can be constructed, but the process cannot be completed. The solution involves use of the following obvious trick (is that an oxymoron?): Some piece(s) are placed sideways in the box. Of the 18 pieces, 10 are too wide to ﬁt into the box sideways and 4 are of width 5, which is no good for this purpose. This leaves 4 pieces that might be placed sideways. There are four solutions to the puzzle, all very similar, and they all have three of these four pieces placed sideways.

**º Boxed Box (23) (Cutler, 1978, Bill Cutler Puzzles):**

Box: ½ ¢ ½ ¢ ½ Pieces: ½¿ ¢ ½½¾ ¢ ½ ½, ½ ¢ ¼ ¢, ½ ¢ ¢ ¼, ½ ¢ ¢ ½ ¼,

## ½ ¢ ¾ ¢, ½ ¢ ¾ ¢ ¾, ½ ¢ ¿ ¢, ¾¼ ¢ ¼ ¢ ¾, ¾½ ¢ ¾ ¢,

172 B. CUTLER## ¾¾ ¢ ½¼ ¢ ½¿½, ¾¿ ¢ ½ ¢ ¿, ¾ ¢ ¢, ¾ ¢ ¿ ¢, ¾ ¢ ¢ ½¾¿,

## ¿¼ ¢ ¢½¿, ¿½ ¢ ¢, ¿¿ ¢ ¢ ¼, ¿ ¢½½¼¢½¿, ¿ ¢ ¾ ¢½¾,

## ¿ ¢ ¿ ¢ ½¾½, ¿ ¢ ¾ ¢ ¼, ¢ ¢,¢¢

The dimensions of the pieces are all different numbers. The pieces ﬁt into the box with no extra space. The smallest number for which this can be done is 23. There are many other 23-piece solutions that are combinatorially different from the above design. Almost 15 years later, this puzzle still fascinates me. See [1] or [3] for more information.**10. Holes, Pieces the Same or Similar**

º Hoffman’s Blocks (27) (Dean Hoffman, 1976) Box: ½ ¢ ½ ¢ ½ Pieces: (27) ¢ ¢ This sounds like a simple puzzle, but it is not. The extra space makes available a whole new realm of possibilities. There are 21 solutions, none having any symmetry or pattern. The dimensions of the pieces can be modiﬁed. They can be any three different numbers, where the smallest is greater than one-quarter of the sum. The box is a cube with side equal to the sum.

I like the dimensions above because it tempts the solver to stack the pieces three deep in the middle dimension. See [4].

º Hoffman Junior (8) (NOB Yoshigahara, 1986, Hikimi Puzzland) Box: ½ ¢ ½ ¢ ½ Pieces: Two each of ¢ ¢ ½¼, ¢ ¢ ½½, ¢ ½¼ ¢ ½½, ¢ ½¼ ¢ ½½

11. Holes, Pieces Different ½¼º Cutler’s Dilemma, Simpliﬁed (15) (Cutler, 1981, Bill Cutler Puzzles) Box: ¼ ¢ ¾ ¢ ¾ Pieces: ¢½ ¢¾, ¢¾¼¢¾¼, ½¼¢½½¢ ¾, ½¼¢½¾¢¾, ½¼¢½ ¢¿½,

## ½¼ ¢ ½ ¢ ¾, ½¼ ¢ ½ ¢ ¾, ½½ ¢ ½½ ¢ ¾, ½½ ¢ ½¾ ¢ ¾, ½½ ¢ ½ ¢ ½,

## ½½ ¢ ½ ¢ ¿¼, ½½ ¢ ½ ¢ ¾, ½¾ ¢ ½ ¢ ½, ½ ¢ ½ ¢ ¾½, ½ ¢ ½ ¢ ¾½

The original design of Cutler’s Dilemma had 23 pieces and was constructed from the above, basic, version by cutting some of the pieces into two or three smaller pieces. The net result was a puzzle that is extremely difﬁcult. I will not say anything more about this design except that the trick involved is different from any of those used by the other designs in this paper.BLOCK-PACKING JAMBALAYA 173

12. Miscellaneous

The Melting Block is more of a paradox than a puzzle. The eight pieces ¢ ¢ ½¿¾. This ﬁts into ﬁt together easily to form a rectangular block the box with a little room all around, but seems to the casual observer to ﬁll up the box completely. When the ninth piece is added to the group, the ¢ ¢ ½¿¿ rectangular solid. (This pieces can be rearranged to make a second construction is a little more difﬁcult.) This is a great puzzle to show to “non-puzzle people” and is one of my favorites.

By the way, one of the puzzles listed above is impossible. I won’t say which one (it should be easy to ﬁgure out). It is a valuable weapon in every puzzle collector’s arsenal. Pack all the pieces, except one, into the box, being sure that the unﬁlled space is concealed at the bottom and is stable.

Place the box on your puzzle shelf with the remaining piece hidden behind the box. You are now prepared for your next encounter with a boring puzzle-nut. (No, readers, this is not another oxymoron, but rather a tautology to the 99% of the world that would never even have started to read this article.) Pick up the box and last piece with both hands, being careful to keep the renegade piece hidden from view. Show off the solved box to your victim, and then dump the pieces onto the ﬂoor, including the one in your hand. This should keep him busy for quite some time!

Written on the occasion of the Puzzle Exhibition at the Atlanta International Museum of Art and Design and dedicated to Martin Gardner.

Classiﬁcation of Mechanical Puzzles and Physical Objects Related to Puzzles James Dalgety and Edward Hordern Background. “Mechanical Puzzles” is the descriptive term used for what are also known as “Chinese Puzzles.” Several attempts have been made to classify mechanical puzzles, but most attempts so far have either been far too specialized in application or too general to provide the basis for a deﬁnitive classiﬁcation. Many people have provided a great deal of help, but particular thanks are due to Stanley Isaacs, David Singmaster, and Jerry Slocum.

Objective. To provide a logical and easy-to-use classiﬁcation to enable nonexperts to ﬁnd single and related puzzles in a large collection of objects, and patents, books, etc., related to such objects. (As presented here, while examples are given for most groups, some knowledge of the subject is required.) Deﬁnitions. A puzzle is a problem having one or more speciﬁc objectives, contrived for the principle purpose of exercising one’s ingenuity and/or patience. A mechanical puzzle is a physical object comprising one or more parts that fall within the above deﬁnition.

Method. A puzzle should be classiﬁed by the problem that its designer intended the solver to encounter while attempting to solve it. Consider a three-dimensional (3-D) interlocking assembly in the form of a cage with a ball in the center. The fact that the instructions request the would-be solver to “remove the ball” does not change the 3-D assembly into an opening puzzle. The disassembly and/or reassembly of the cage remains the primary function of the puzzle. An interlocking puzzle should be classiﬁed according to its interior construction, rather than its outward appearance (e.g., a wooden cube, sphere, barrel, or teddy bear may all have similar Cartesian internal construction and so should all be classiﬁed as Interlocking– Cartesian). In cases where it seems possible to place a puzzle in more than For updated information and illustrations, go to http://puzzlemuseum.com.

176 J. DALGETY AND E. HORDERN one category, it must be classiﬁed in whichever is the most signiﬁcant category. A few puzzles may have to be cross-referenced if it is absolutely necessary; usually, however, one category will be dominant.

A good example of multiple-class puzzles is the “Mazy Ball Game” made in Taiwan in the 1990s. It is based on a ¿ ¢ ¿ sliding block puzzle under a clear plastic top. The pieces have L-shaped grooves, and a ball must be rolled up a ramp in the lower right onto one of the blocks — the ball must be moved from block to block, and the blocks themselves must be slid around so that the ball can exit at the top left. Thus the puzzle requires Dexterity, Sequential movement, and Route-ﬁnding. It would be classiﬁed as Route-ﬁnding because, if the route has been found, then the dexterity and sequential movement must also have been achieved.