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

A puzzle will be referred to as two-dimensional (2-D) if its third dimension is irrelevant (e.g., thickness of paper or plywood or an operation involving a third dimension such as folding). Most standard jigsaws are 2-D, although jigsaws with sloping cuts in fact have a relevant third dimension, so they must be classed as 3-D. It will be noted that the deﬁnition of a puzzle excludes the infant’s “posting box,” which, while perhaps puzzling the infant, was contrived only to educate and amuse; it also excludes the archer attempting to get a bull’s-eye, the exercise of whose ingenuity is entirely incidental to the original warlike intent of the sport. Also excluded are puzzles that only require paper and pencil (e.g., crossword puzzles), unless they are on or part of some physical object.

It is understood that specialist collectors will further subdivide the subclasses to suit their own specialized needs. For example, Tanglement–Rigid & Semi-Rigid is awaiting a thorough study of the topology of wire puzzles.

The full abbreviations consist of three characters, hyphen, plus up to four characters, such as “INT-BOX.” These are the standard abbreviations for the classes that have been chosen for relative ease of memory and conformity with most computer databases.

**The fourteen main classes are as follows:**

¯ Opening Puzzles (OPN) are puzzles in which the principle object is to open it, close it, undo it, remove something from it, or otherwise get it to work. They usually comprise a single object or associated parts such as a box with its lid, a padlock and its hasp, or a nut and bolt. The mechanism of the puzzle is not usually apparent, nor do they involve general assembly/disassembly of parts that interlock in 3-D.

¯ Interlocking Puzzles (INT) interlock in three dimensions; i.e., one or more pieces hold the rest together, or the pieces are mutually selfsustaining. Many clip-together puzzles are “non-interlocking.” ¯ Assembly Puzzles (Non-Interlocking) (ASS) require the arrangement of separate pieces to make speciﬁc shapes without regard to the sequence of that placing. They may clip together but do not interlock in 3-D. Some have a container and are posed as packing problems.

¯ Jigsaw Puzzles (JIG) are made from cut or stamped-out pieces from a single complete object, and the principle objective is to restore them to their unique original form.

¯ Pattern Puzzles (PAT) require the placing or arrangement of separate pieces of a similar nature to complete patterns according to deﬁned rules. The pattern required may be the matching of edges of squares, faces of a cube, etc. The pattern may be color, texture, magnetic poles, shape, etc. Where the pattern is due to differences in shape, the differences must be sufﬁciently minor so as not to obscure the similarity of the pieces.

¯ Sequential Movement Puzzles (SEQ) can be solved only by moves that can be seen to be dependent on previously made moves.

¯ Folding and Hinged Puzzles (FOL) have parts that are joined together and usually do not come apart. They are solved by hinging, ﬂexing, or folding.

¯ Jugs and Vessels (JUG). Vessels having a mechanical puzzle or trick in their construction that affects the ﬁlling, pouring, or drinking therefrom.

¯ Other Types of Mechanical Puzzles and Objects (OTH). This group is for puzzle objects that do not easily fall into the above categories and cannot be categorized into sufﬁciently large groups to warrant their own major class. Included in this group are Balancing, Measuring, Cutting, Math, Logic, Trick, Mystery, and Theoretical Puzzles. Also, provision is made for puzzles pending classiﬁcation.

¯ Ambiguous Pictures and Puzzling Objects (AMB). Puzzles in which something appears impossible or ambiguous.

178 J. DALGETY AND E. HORDERN ¯Ephemera (EPH). This category has been included because most puzzle collections include related ephemera, which, while not strictly puzzles, need to be classiﬁed as part of the collection.

A detailed classiﬁcation with second-level classes is given in the separate table. Puzzle Class Abbreviations (PZCODE) are standardized to a maximum of eight characters: XXX-YYYY, where XXX is the main class and YYYY is the sub-class. Examples of puzzles in each class are given in the right-hand column.

Proposed Developments Prior to 1998 the subclasses attempted to incorporate the number of dimensions, the type of structure, and any group/non-group moves. This has resulted in an unwieldy list. We are now working on deﬁning what should be included automatically in the subclasses where they are relevant. We hope to reduce the number of sub-classes substantially in the near future.

Number of Dimensions. 2-D, 3-D, 2-D on 3-D, 2-D to 3-D, 2-D, 3-D, and 4-D. Required for INT JIG ASS PAT RTF SEQ FOL.

Group Moves. Whether needed, not needed, or partly used in solution.

Required for SEQ.

Number of Pieces. Required for INT JIG ASS PAT.

Type of Piece Structure. Required for INT ASS PAT.

Examples of approved structure adjectives that may be used include ¯ Identical: All pieces identical ¯ Cartesian: Three mutually perpendicular axes ¯ Diagonal: Pieces rotated 45 degrees along their axis ¯ Skewed: Like a squashed puzzle ¯ Polyhedral ¯ Geometrical: Non-Cartesian but geometrical structures ¯ Organic: Amorphously shaped pieces ¯ Ball: Pieces made from joined spheres ¯ Rod: Square, hexagonal, triangular, etc.

¯ Linked: Pieces joined together by hinges, strings, ribbons, etc.

Thus a standard six-piece burr uses square rods in a Cartesian structure, and the standard Stellated Rhombic Dodecahedron has a six-piece Diagonal Cartesian structure.

CLASSIFICATION OF PUZZLES 179 Guide to Making a Catalogue or Database for Puzzle Collections Headings for cataloguing puzzle collections could include the following items. In practice, without paid curators, it is probably advisable to be selective and limit the amount of information recorded.

Generalized Information ¯ Generic name of puzzle or objective if not obvious. **required ¯ Class + Subclass **required Information Speciﬁc to This Object ¯ Theme or advertisement (subject) ¯ Materials ¯ Dimensions A, B, C; d = diameter. **A required for scale ¯ What the dimension refers to, i.e., box, envelope, biggest piece, assembled puzzle ¯ If powered, i.e., battery, clockwork, electric, solar ¯ Patent and markings ¯ Notes, references, designer ¯ Manufacturer or publisher’s name and country ¯ Type of manufacturing, i.e., mass produced (over 5000 pieces made), craft made (commercial but small volume), homemade or tribal ¯ Year of manufacture **required ¯ Country of manufacture if different from publisher’s country ¯ Manufacturer’s series name ¯ Manufacturer’s product name ¯ Number in complete set, if known ¯ Number of set in collection ¯ A picture or photograph Information Speciﬁc to This Collection ¯ Location/cabinet/bin ¯ Acquisition ¯ Number ¯ Condition (Excellent, Good, Fair, Poor) ¯ Condition qualifying note, i.e., puzzle may have a cracked glass top or be missing one piece, but otherwise be in excellent condition.

¯ Source ¯ Date ¯ Cost ¯ Insurance value 180 J. DALGETY AND E. HORDERN

The eminent Oxford don, Charles Lutwidge Dodgson, demonstrated the applicability of formal mathematical reasoning to real-life situations with

**such incontrovertible rigor as evidenced in his syllogism:**

All Scotsmen are canny.

All dragons are uncanny.

Therefore, no Scotsmen are dragons.

While his logic is impeccable, the conclusion (that no Scotsmen are dragons) is not particularly surprising, nor does it shed much light on situations that we are likely to encounter on a daily basis.

The purpose of this note is to exploit this powerful proof methodology, introduced by Dodgson, to a broader range of human experience, with special emphasis on obtaining conclusions having political or moral signiﬁcance.

** Theorem 1. Apathetic people are not human beings.**

Proof. All human beings are different.

All apathetic people are indifferent.

Therefore, no apathetic people are human beings.

** Theorem 2. All incomplete investigations are biased.**

Proof. Every incomplete investigation is a partial investigation.

Every unbiased investigation is an impartial investigation.

Therefore, no incomplete investigation is unbiased.

Numerous additional examples, closely following Dodgson’s original mod-el, could be adduced. However, our next objective is to broaden the approach to encompass other models of mathematical proof. For example, Reprinted from Mathematics Magazine, Vol. 67, No. 5, December 1994, p. 383. All rights reserved. Reprinted with permission.

192 S. W. GOLOMB it is a well-established principle that a property P is true for all members of a set S if it can be shown to be true for an arbitrary member of the set S. We exploit this to obtain the following important result.

** Theorem 3. All governments are unjust.**

Proof. To prove the assertion for all governments, it is sufﬁcient to prove it for an arbitrary government. If a government is arbitrary, it is obviously unjust. And since this is true for an arbitrary government, it is true for all governments.

Years ago I went on a collecting trip, visiting non-innumeretic friends asking for contributions of a very particular kind. Martin Gardner gave me some help, not as much as I expected; others scraped together a token from here and there. I browsed, I pilfered, but the bottom of my basket ended up barely covered and, except for the accumulated webs and dust, so it remains. At this point, before throwing it out, I pass it around once more for contributions.

When a cold engine starts, it often misﬁres a few times before running smoothly. There are mathematical tasks like that: Parameterized in n, each can be accomplished for some hit-and-miss pattern of integer values until, after some last “misﬁring” value, things run smoothly, meaning simply that the task can be performed for all higher integers. One such task is to partition a square into n subsquares (i.e., using all of its area, divide it into n non-overlapping subsquares). This can in fact be performed for n = 4, 6, 7, 8, where 5 is “missing” from the sequence.

A devilishly difﬁcult kind of mathematical problem, I thought, might be to pose such a question in reverse, e.g., What is a task parameterized in n that last misﬁres for n = 6?, or for n = 9?, etc. More precisely, a task that last misﬁres for N ¯ can be done for at least one earlier Ò (½ Ò Æ);

¯ cannot be done for Ò Æ;

¯ can be done for all Ò Æ;

and, to be sporting, ¯ the task must not be designed in an ad hoc way with the desired answer built in — must not, for example, involve an equation with poles or zeros in just the right places;

¯ it must be positively stated, not for example “prove the impossibility of ”;

¯ it must at least seem to have originated innocently, not from a generative construct such as “divide Ò things into groups of 17 and 5.” 194 K. KNOWLTON For a mathematician or logician, these are attrociously imprecise speciﬁcations. But here, for me at least, lies the intriguing question: Will we agree that one or another task statement lies within the spirit of the game? This is, of course, a sociological rather than mathematical question; my guess is that the matter lies somewhere on the non-crispness scale between agreement on “proof” and agreement on “elegance.” Meager as the current collection is, its members do exhibit a special charm. I have answers only for N = 5, 6, 7, 9, 47, and 77. Some of these

**are well known, others quite esoteric:**

´µ Stated above, well known, with a rather obvious proof: Partition a square into n subsquares.

´µ Divide a rectangle into n disjoint subrectangles without creating a composite rectangle except for the whole (Frank Sinden).

Design a polyhedron with Ò edges. (Generally known. The tetrahe´µ dron has 6 edges, the next have 8, 9, edges.) Cut a square into Ò acute triangles with clean topology: no triangle’s ´µ vertex may lie along the side of another triangle. Possible for 8, 10, 11, 12. (Charles Cassidy and Graham Lord, even after developing a complete proof, ask themselves “Why is 9 missing?”) ´µ Place n counters on an inﬁnite chessboard such that each pair exhibits different numbers of row-mates and/or different numbers of column-mates. (Impossible for n = 2, 5, or 9; conjectured by Ken Knowlton, proved by Ron Graham. One of Martin Gardner’s books contains essentially this problem as a wire-identiﬁcation task, but the stated answer implicitly but erroneously suggests that the task can be performed for any Ò).

´µ Cut a cube into n subcubes (called the “Hadwiger” problem, tracked for years by Martin Gardner, ﬁnally clinched independently by Doris Rychener and A. Zbinder who demonstrated a dissection into 54 subcubes).

´µ Partition n into distinct positive integers whose reciprocals sum to one, e.g., such a partition for 11 is 2, 3, 6 since 2 + 3 + 6 = 11 and ½ · ½ · ½ ½. (Ron Graham is the only person I know who would ¾¿ think of such a problem, and who did, and who then went on to prove that 77 is the largest integer that cannot be so partitioned.) What we have here is an inﬁnite number of problems of the form “Here’s the answer, what’s the question?” Usually such a setup is too wide open to be interesting. But a misﬁre problem, as I think I have deﬁned it, has such a severely constrained answer that it is almost impossibly difﬁcult. But “difﬁcult” may be the wrong word. The trouble is that a misﬁre problem is MISFIRING TASKS 195 a math problem with no implied search process whatever, except to review all the math and geometry that you already know. Instead of searching directly for a task last misﬁring, say, for Ò ½¿, it’s much more likely that in the course of your mathematical ramblings you will someday bump into one.

I invite further contributions to the collection. Or arguments as to why this is too mushily stated a challenge.