Puzzometry Solved Assignment

The Assembly or Put-Together class includes those puzzles which entail the arrangement of pieces to make specific shapes in either two or three dimensions, to mesh in a particular way (without necessarily interlocking) or to fill a container or tray. The pieces are free to be juxtaposed in many different configurations but only one or a few arrangements will be valid solutions. The best such puzzles often permit a seemingly valid construction of all but one of the pieces, where the last piece stubbornly just won't fit! For the most part, the order in which the pieces are put together does not matter - when order does matter, it is a sequential assembly puzzle. If the pieces truly interlock to form a free-standing construction that remains stable in various orientations, the puzzle belongs in the Interlocking class. If the pieces have designs on them and there are rules about how the designs must appear, check the Pattern class or the Jigsaws class.

Packing Puzzles

Simply stated, the challenge of a packing puzzle is to fit a given set of pieces into a container. The boundaries are either enforced by walls and a lid, or sometimes just walls, with the "lid" implied by the requirement that no piece extends beyond the level of the walls. The container might also be more of a tray, especially if the pieces don't stack in 3 dimensions. Now, if you consider this task in the abstract, the entire container could be construed as implied rather than physical, and then many assembly puzzles could be considered to be packing puzzles. For example, the SOMA cube could be re-cast as "fit the pieces into a cubic box." In addition, you can shoehorn dissections in here by thinking of the original form as the "container" - the objective is to re-construct the original form, which is tantamount to fitting the pieces back into this abstract container. For my purposes here, I will include a puzzle in the "packing" category if there is a physical container, and some pieces to cram into it. In rare instances the container is similar to the pieces themselves. Sometimes the puzzle is presented with a subset of all the pieces except for one of them packed into the container, with seemingly no room for the additional piece, and the objective being to rearrange the pieces to make the last piece fit, too. Take a look at Erich Friedman's Packing Center. Bill Cutler has written an interesting essay on box packing puzzles. In addition to his seminal designs of Interlocking puzzles, Stewart Coffin has designed many great packing puzzles. When Coffin's designs appear in the tables below, I have highlighted them like this.

Single-Layer Packing Puzzles with Identical or Similar Pieces

Hercules - B&P
Designed by Jean Claude Constantin
Nice quality and poses just the right amount of challenge.

Crazy L
A very nice little packing challenge, from the Puzzle and Craft Factory.

Four T's - Binary Arts/Thinkfun

Pack the Tray (8 triangles + 1 rectangle) - Saul Bobroff
I got this prototype from Saul at the 2009 NYPP.

Houses and Factories
Designed by Richard Hess - distributed by B & P

Houses and Factories 2 - Hess
Purchased at a get-together.

Foxes and Wolves
Designed by Richard Hess. Purchased at IPP 29 in SF.

Packing Quarters - B&P

Butterfly - Nature's Spaces
Fit nine identical penta-hexes into a triangular frame. Only one arrangement will work.

Frog Pond - Nature's Spaces
Fit nine identical tetra-hexes into a triangular frame.

3 Ls
Fit the 3 L-shaped pieces into the tray.

Lucky 7 - Melissa & Doug

Blockade - B&P
Blockade is like Lucky 7 - both use 3 small and 4 large L shaped pieces, but Blockade also has pins on the board and corresponding holes in the pieces. Lucky 7 is trivial to solve - Blockade adds a little (but not much) challenge.

Kinato is a very nicely packaged puzzle from Ravensburger. Sixteen triangles are threaded via clever swivel connections. Arrange them into a large triangle with the proper pattern. I found it at jigsawjungle.com.

Snake Pool
Eleven cubes are loosely strung along an elastic to form a cube snake. Fit the snake flat in the tray - the "pool." There are at least four different solutions. The cubes are 3/4", the tray opening is 3.25" square.
The snake configuration is: 3+2+2+2+1+1 (where a + denotes a right-angled bend that can swivel).
Erich Friedman shows various square in square packings on his Packing Center site, but I don't think the solution shown for 11 squares works with this particular cube snake configuration.

Ampelmann - Roman Götter and Peter Knoppers
Purchased from Roman at IPP31 in Berlin
A black case with a hidden complex interior and two circular openings.
Three red "Don't Walk" Ampelmann figures, and three green "Walk" figures (one mirror image or the other two), colored on only one side.
Two challenges: 1) place all six figures in one "compartment" with one red Ampelmann in the middle, and 2) place all six figures in the other "compartment" with one green Ampelmann in the middle.
The clear piece is a hint - it shows the shape of the cavity inside the case. Simple, eh?
These figures are the old East Berlin crosswalk signal symbols - one of the few vestiges of Communist rule that Berlin citizens want to keep. Read more about " Speciation and Competition in Berlin's Traffic Lights."

Mimi packing puzzles: A, F, H

Pack the four T-shaped pieces into the tray -
I obtained this from George and Roxanne Miller
but I don't know its name.

Modest Hexominoes by Dr. Richard Hess (IPP17)
Place all 20 pieces so that each hexomino shape contains five identical pieces. Includes a booklet with 100 additional problems to maximally cover polyomino shapes with congruent tiles.

The Massai packing puzzle from Siebenstein Spiele, 2011.
Pack the 5 identical L-shaped tetrominoes in the tray.
My copy might be defective, but I found one solution and my wife and kids found two more.

Quartet in F - Stewart Coffin (#253)

Octet in F, designed and made by Stewart Coffin, exchanged at IPP32 by Rosemary Howbrigg

FN Puzzle - pack the four pieces in the tray in three different ways
designed by Mitsuhiro Odawara
produced by Toshiyuki Kotani
Purchased at IPP32

Retrofit - designed by Goh Pit Khiam,
made by Eric Fuller,
from Rosewood, Ipe, Walnut, Bubinga,
Padauk, Maple, and Acrylic.
The following tray-packing puzzles were all designed by Edi Nagata.
Edi sells versions in 2-sided trays, made from MDF. A couple were offered by Bits and Pieces with wooden 2-sided trays and aluminum pieces, other single-sided versions in CD cases by Embrain via Torito.

Pencil Case

Cat Case
aka Cats in a Cradle - B&P

Shirt Case
Purchase the 2-sided MDF version from Edi, or the single-sided CD-case versions "Shikoku" and "Australia" from Torito. Philos offers a version, too.

Arrow Case
aka Packing Arrows - B&P

Cup Case

Baby Ducks Case

Single-Layer Packing Puzzles using a Set of Related Pieces

This is a special group where the pieces aren't identical, but they are related by some rule or theme, which distinguishes them from those puzzles in the more generic group having an assortment of dissimilar pieces. Some of the puzzles in the latter group may languish there though they belong in this section because I am unaware of the rule relating the pieces...
One event at the International Puzzle Party (IPP) is called the Edward Hordern Puzzle Exchange. Qualifying attendees can sign up to participate - each must submit a new puzzle design, and if approved, bring enough copies of the puzzle to exchange one with each other participant (up to 100). IPP32 in Washington D.C. in 2012 was the first time I participated in the exchange. There were 79 puzzles in the exchange in 2012. For the exchange, I created a tray-packing puzzle I call Non-Convex Bi-Half-Hexes. Catchy and mellifluous, eh? I chose to use a subset of the hexiamonds as pieces. If one divides a regular hexagon in half along a line connecting opposite vertices, then re-joins the two halves along a side-length, there are only seven resulting shapes that are non-convex. Using the "standard" piece names, this set of seven includes { butterfly, chevron, crook, hook, snake, sphinx, yacht }. The other five hexiamonds are either convex { rhomboid, hexagon }, or not composed of two half-hexes { crown, pistol, lobster }. I used this set of seven (mathematically complete given the defining rule) and designed four different simple symmetric trays into which all seven pieces can be packed flat (allowing gaps), three of which have only one solution apiece. The puzzle was produced by Steve Kelsey. The case cover is made from a single piece of wood, with a clever laser-cut flexible "binding." We designed a nice dovetail closure. From what folks tell me, this is a difficult puzzle. If you are interested in purchasing a copy, please email me at the address on my home page.
Non-Convex Bi-Half-Hexes, designed by Robert Stegmann
produced by Steve Kelsey.

I designed an expansion set for my IPP32 exchange puzzle Non-Convex Bi-Half-Hexes. I have come up with a dozen new tray shapes into which all seven pieces will fit flat. Difficulty ranges from easy to hard, with several having only 1 or 2 solutions, but a few having 7, 9, and 12 solutions. All of the trays are "hollow" and require no internal islands.

Nine Squared - Tom Lensch
All nine pieces have identical thickness but each has a different combination of length and width selected from discrete increments within a narrow range. When arranged correctly into the tray they simply drop in and out with no binding. Several incorrect packings seem like they should fit, if only you press down a little... wrong!

Apothecary's Cabinet - Constantin
(purchased at GPP)
Each "drawer" has a combination of side tabs and portions of the row separators, and is equivalent to a rectangle with each side having either a tab or a notch. There are 2^4=16 possible arrangements including rotations and reflections. The knobs on the drawers require the reflections. The fact that the side tabs/notches are off-center requires the rotations. This puzzle is a nice realization of a 4x4 heads/tails edgematching puzzle, but includes a cabinet/tray/frame which constrains the solution, since it has all notches along the left and top, and all tabs along the right and bottom. If you assign a 4-bit binary ID to each drawer using 0 for a notch and 1 for a tab, the low bit for the top and high for the left side, then one solution is:
For issues 61 and 62 (Nov 2003) of the CFF newsletter, Dieter Gebhardt wrote articles analyzing this puzzle, and in issue 62 reports results derived by Jacques Haubrich.

Digits - Constantin
Fit the 10 digits into the tray.
The much-copied Digigrams, designed by Martin Watson.
Made by Eric Fuller, from Grandillo, Walnut, and laser-cut acrylic.

Num3er Cruncher - Mick Guy
Mick is Vice President of the British Origami Society, and he
kindly sent me a copy of his "Num3er Cruncher" puzzle. Thanks, Mick!
Packing digits in a tray (or box) has been done before, but Mick's design is a bit different.

Square Dance - designed by Derrick Schneider
Purchased from Pavel Curtis - I've been wanting one for a while and was pleased to find Pavel had resuscitated it!
Square Dance won an Honorable Mention in the 2002 IPP Design Competition.
There is only one way to join two 2x2 squares by a half edge,
and only four ways to join a third 2x2 square by a half edge to the first two.
These are the four pieces of Square Dance, and there is only one way to pack them into an 8x8 tray,
and only one way to pack them into a 7x9 rectangle. The included tray is two-sided.

Partridge Puzzle by Robert Wainwright
obtained from Robert at the 2007 NYPP
Kadon offers some of Erich Friedman's "Partridge" puzzles.
In an "anti-Partridge" puzzle, there is one largest piece, and the count goes up as the pieces shrink.

Windmill Key - Tyler Somer
I received this at the 2014 Rochester Puzzle Party
(RPP) that followed IPP34. Thanks, Tyler!

Lonpos Cosmic Creatures

Pentagon Tiles, designed and exchanged at IPP32 by Rene Dawir, made by Marcel Gillen

13 Triangles, designed and exchanged at IPP32 by Ed Pegg Jr., made by William Waite

Di-Half-Hexes, designed and exchanged at IPP32 by Peter Knoppers, made by Buttonius Puzzles & Plastics
I was really surprised to see this one, since it is so similar to my Non-Convex Bi-Half-Hexes IPP32 exchange puzzle. What are the odds? Peter and I must have been hit by a similar brain wave. Fortunately, his puzzle uses a different set of hexiamonds and different trays.

Triangle Edges - designed by William Waite in 2005
Pack the 12 pieces into the tray.
The puzzle is based on a triangular grid
and each piece is composed of five edges.

Square Dissection - N. Baxter
Received from Dr. R. Hess at a get-together - thanks, Dick!

Domino Peg
From PuzzleMist (William Waite)
Fit the 12 pieces in the tray to form
different patterns of holes -
ten goal patterns specified on the back of the tray.

Single-Layer Packing Puzzles using an Assortment of Dissimilar Pieces

Karin's Star Cluster
An entry in the IPP24 Design Comp..

Tessellating Galaxies - JVK

Sun Dance - JVK

Fantastic Island

The City
2001 Binary Arts (Thinkfun)
Pack six heptominoes (3 distinct pieces and their mirror images) in the 6x7 tray. Nice metal pieces with 3D abstract buildings on them which prevent the pieces from being flipped and exclude most of the otherwise possible 80 assemblies.

Fit To A Tee - Thinkfun
A nice metal tray-packing puzzle from Thinkfun. Pack the 9 pieces representing golf holes complete with tees, sand traps, and pins, into the base. The base presents a challenge on each side (the front and back nines), with different arrangements of fixed water hazards to work around. Oh, and just as on a real course, abut each flag with the tee of the next hole!

Geometrex Set - Ormazd, Nabucho, and Quirinus
In each case the pieces can be rearranged within the tray to fit in an extra square.

The "845 Combinations" puzzle is almost like pentominos... here is a solution to the 845 puzzle.

On 6/22-25/13, made a trip to Niagara Falls. At Niagara Falls, I stopped in at Turtle Pond Toys. They carry several nice puzzles of various types. I picked up the IQ Puzzle from Toyland Company. It has 9 curved pieces that must be fit into the channels within a 4x4 grid of circles. It is an easier version of the 845 Combinations puzzle, which has 10 pieces to be fit into the same grid.

Adam's Cube

One Way

Boxed In - Milton Bradley

The IQ Link puzzle from Smart Games designed by Raf Peeters

Circle Challenge - Melissa & Doug
A good one for kids - work on it from the inside out. The pictures on the pieces are merely decorative.

Double Cross - Mag Nif
There are four pink plastic pieces and the tray. The objective is to form a cross (plus sign) in the tray.

Magic Block (MCS promo)

Figa Block

IQ Block

Quartet Quandary
From PuzzleMist (William Waite)
Fit the four pieces in the tray. Two solutions.

Build the Block - a metal square dissection puzzle branded "Arco."

Sleazier - Pavel Curtis
based on Stewart Coffin's Four Sleazy Pieces (#169A)
Fit the 4 pieces into the tray. IPP25

Stewart Coffin's Sunrise / Sunset (#181)
Fit the 5 polyominoes into each side of the tray, making a symmetric pattern in each case. Gift from Bernhard Schweitzer (thanks!). IPP22

Stewart Coffin's Drop In (#153B) aka The Trap
Fit the four pieces into the box through a small slot. They must be arranged so all fit within the inside perimeter of the box walls. Saul Bobroff IPP23

Stewart Coffin's Few Tile (#133)
Made by John Devost
A beautiful Padauk frame about 5.75" squared, with corner splines, and Birch plywood pieces.
A gift - Thanks, John!

Eccentric's Dream - designed and made by Pavel Curtis

Stewart Coffin House Party (#250)
Fit the four Marblewood pieces into each side of the tray, which is made from Poplar on Baltic Birch.
Made by Tom Lensch

Stewart Coffin's Four Fit (#217)
Made by Tom Lensch. Purchased from Tom at the Dartmouth College Mechanical Puzzle Day in Feb. 2008.

Stewart Coffin's Five Fit
From Dave Janelle at Creative Crafthouse.
Fit the five pentomino pieces into the square tray.
The tray has a handy storage space for one of the pieces
should you be unable to solve it.

A Stewart Coffin Tray Puzzle Set (#181), in Poplar and Lyptus woods, made by Tom Lensch. Purchased at Puzzle Paradise .ca. This set includes six of Coffin's tray-packing puzzles - a single-sided rectangular tray (#181, 1 solution), a two-sided pentagonal tray (#181-C, The Housing Project, 1 solution each side), and another two-sided pentagonal tray having a movable wall segment on one side (#181-A, The Castle Puzzle, 3 solutions; #181-B, The Tree Puzzle, 2 solutions, other side #181-B, The Vanishing Trunk Puzzle, 1 solution).

Stewart Coffin's Cruiser (#167)
Made by Walter Hoppe.

Lean 2, designed by Stewart Coffin, made by Tom Lensch, exchanged at IPP32 by Dave Rossetti

Buridan, designed, made, and exchanged at IPP32 by Vladimir Krasnoukhov

Heart and Bud, designed, made, and exchanged at IPP32 by Yoshiyuki Kotani

Think Square - Pressman
There are 4 small right triangles, 4 large right triangles, 4 stair-case shaped pieces, and 5 small squares. The pieces can be fit snugly into the tray with and without one of the five small squares.

Triaden spass - Logika

Pack It In - Great American Puzzle Factory 1996
Pack a set of 16 items into a suitcase frame. Flat cardboard pieces.

The Trapped Man - Tom Jolly
Laser cut by Walter Hoppe. Five unusually convoluted pieces, including the little "man." The first challenge is to fit them into the tray so that none can slide or rotate. Next, try it with only four of the five pieces, then with only three! Several other puzzle goals accompany the Trapped Man puzzle.

The Suitcase Puzzle - made in China for Bear, Bear, & Bear Ltd. England 1996

Pac-Man - Milton Bradley
First create 4 Pac-men with open mouths. Then use the same pieces to create 3 Pac-men with closed mouths. There are eye stickers on some pieces, which must be positioned correctly. The pieces can be flipped.

The Jayne Fishing Puzzle - A 1916 advertisement of Jayne's Tonic Vermifuge (yuck!). Discussed in Slocum and Botermans' "The Book of Ingenious and Diabolical Puzzles" on page 15. You were to cut out the fish and the ring and then pack the fish inside the ring. The fish names are (left to right, top down): Codfish, Shad, Red Grouper, Cowtrunk Fish, Flying Fish, Bluefish, Mackerel, Tarpon, Sheepshead, Moonfish, Striped Bass, and Weakfish.
Also see No Fishing by Bepuzzled.

No Fishing - Bepuzzled 1998
Remove the water then fit all twelve fish into the bowl. This is a nice wooden laser-cut, colorful, and faithful copy of the Jayne Fishing Puzzle of 1916.

In the Raging Rapids puzzle from Thinkfun (Binary Arts), you have to fit all the men into the raft, facing the right way. The figures' bases have various patterns of tabs and indents.

In the Mayan Calendar puzzle from William Waite, you have to fit all the glyphs into the tray, facing the right way. The glyphs have various patterns of tabs and indents. (Similar to Raging Rapids.)

Alex Randolph's Moebies - Springbok 1973
There are 8 sockets at various positions in the orange board. Six pieces and six pegs are included - the object is to find a way to peg the six pieces to the board so that all fit within the edges.

Springbok Fitting & Proper
Here is a nice set of small, pocketsized tray packings designed by William Waite, purchased from his PuzzleMist website: From left to right, they are: Triangle Quorn, Square Quorn, Hex Quorn, Diamond Teaser, and Mix Teaser 2.

The Kitchen Ceiling Puzzle - designed by William Waite in 2006
Pack the 12 pieces into the tray so that the holes make symmetric patterns.

Optimal Tumble - designed by William Waite in 2010
Pack the 12 pieces into the tray so that the holes make symmetric shapes.

Vintage 1969 packing puzzles from Lakeside. So far they include:
  • 16 Trains and Planes
  • 18 Horses and Riders
  • 19 Animals
  • 20 Cars and Trucks
  • 21 Fish and Birds

JVK Tessellating Hexagons

Galaxies & Stars - JVK

"Tripple 7" - 3-piece packing (prototype) - JvK

Easy Eight / Hard Eight - Bob Hearn

Wetten Dass...
Also known as FACT
Purchased in Berlin.
The tray has a moving bar, pivoted at one corner. When the bar is aligned along the top edge, the five pieces are easy to pack into the tray. When the bar is aligned along the side edge, it's more difficult.

Toysmith 11 pc. wood puzzle

Mind the Gap - Chris Morgan
Some tray packing puzzles designed by Naoyuki Iwase (Osho) - Mouse, Tulip, and Seals: See Osho's website, Puzzle-In.

eLeL4 - designed by Hiroshi Yamamoto
presented at IPP30 by Hiroshi Uchinaka
Fit the four pieces into the 8x8 tray. Each piece is composed of 2 'L' shapes.

Unique U - designed by Hiroshi Yamamoto
Fit the six U-shaped pieces into the 9x9 tray.

The Nifty Fifty from Jean Claude Constantin requires you to pack the four pieces into the tray.

The Quartet Puzzle - the Quartet's tray has a movable end wall, and you must pack specific subsets of pieces into the tray depending on where the wall is positioned.

Four in a Frame - a two-sided four-piece tray packing puzzle based on a triangular grid, designed by Markus Götz

Pack-Man - Chris Enright

Hexagon 10

Game Ball Puzzle

Animals of Australia
Dump out the ten nicely cut
animal pieces and try to fit them back in the tray.
No peeking at the solution!

Forever Wild - Animals of the Adirondacks
Pack the ten nicely laser-cut animals into the tray. The animals all go in with a specific side upwards.
From Creative Crafthouse.

Forever Wild - Animals of the Appalachians - a tray packing puzzle, one of a series - each comprising a high-quality tray and nice laser-cut pieces in various colors.

The Cook's Cupboard Puzzle
Pack the eleven kitchen items into the tray.
From Creative Crafthouse.

Hexus, a packing puzzle from Brainwright. Seven pieces and a movable "challenge block" to be placed on a hexagonal grid according to a series of 44 challenges. Purchased at Necker's.

Quadrillion - designed by Raf Peeters, produced by SmartGames
Thanks, Raf!
Arrange the four base plates per a challenge, then pack the pieces on.

Artefacts - designed by Frederic Boucher
made by Eric Fuller, from Walnut, Baltic Birch, Chechen, Maple, and Steel.
The five wooden pieces pack flat in the tray with the steel dowel three ways,
but with the dowel in the hole the pieces pack only one way.

The Hive
Pack four honeycomb pieces in the tray,
and capture the bees. Many ways to capture 3, only one way
to capture all four.
From Puzzleguy (Dan Diehl)

Four piece tray packing.
From Puzzleguy (Dan Diehl)

Jurassic Pack (V2) - by Jerry Loo
I purchased four new acrylic laser-cut tray-packing puzzles from Jerry Loo -
clockwise from top left: Tick-Pack-Toe, Fish Tank (designed by Pit Khiam Goh), Fly in My Pumpkin Soup, Hash Pack

Penguins Pool Party - SmartGames
Tray packing on a hexagonal grid - graduated challenges using
four "ice floe" pieces and up to four single-hex penguins. Charming!

IQ XOXO - SmartGames
120 graduated challenges. Pack the 10 2-sided pentomino pieces in the tray. Where an 'O' appears on one side of a piece, an 'X' is on the opposite side, and vice versa. Bumps in the tray demand 'O' sections over them; every 'O' must be adjacent to 'X' segments, and vice versa.

Parking Puzzler - SmartGames
A tray-packing puzzle with graduated challenges.

Puzzometry Jr. - Jim Fox at Puzzometry

The 12th Tile - designed by Masaki Watanabe

Packing Squares

This section describes several types of puzzle in which assortments of square pieces or tiles must be packed in various ways. Much study and analysis has been done in this area, and there are some great resources on the web. Topics include:

Mrs. Perkins' Quilt

The problem of Mrs. Perkins' Quilt (or Mrs. Perkins's Quilt) appeared as no. 173 in Henry Ernest Dudeney's 1917 book Amusements in Mathematics. You can find the book and the problem online in a few places, including at www.gutenberg.org, and at www.scribd.com. The problem: given a large square quilt made of 13x13 small squares (169 small squares total), find the smallest possible number of square portions of which the quilt could be composed - i.e. a dissection of the large square into a number of smaller squares that don't all have to be different. However, only prime dissections are allowed - the side lengths of the component squares cannot all have a common factor - they must be relatively prime. There can be no sub-square which is itself divided - such a solution is termed "primitive" - primitive quilts are quilts without sub-quilts. Martin Gardner devotes chapter 11 in his 1977 book Mathematical Carnival to Mrs Perkins' Quilt and Other Square-Packing Problems. Ed Pegg discussed the problem on his Math Games site. The problem is also discussed at mathworld.wolfram.com. The solution comprises 11 squares and is shown at gutenberg.org. It contains the following number of squares of given sizes: 1x72, 2x62, 1x42, 2x32, 3x22, and 2x12. The smallest numbers of squares needed to create relatively prime dissections of an n�n quilt for n=1, 2, ... are 1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, ... (Sloane's A005670). Karl Scherer discusses additional variations at his website. Karl defines a nowhere neat tiling - in which no two tiles have a full side in common, and a no touch tiling - where tiles of same size cannot touch, noting that no-touch are always nowhere-neat.

Squared Rectangles and Squares

The problem of Mrs. Perkins' Quilt leads to other questions. In general, how might it be possible to dissect various rectangles or squares into smaller squares? Such puzzles are known as Squared Rectangles and Squared Squares. If a dissection results in pieces all of different sizes, the dissection is called perfect, otherwise it is imperfect. If the dissection does not contain any smaller square or rectangle that is itself further divided, it is called simple (or primitive), otherwise it is compound. The order is the number of tiles used. When describing solutions, it is convenient to use a notation called Bouwkamp code. One lists the side lengths of the tiles as they appear in the solution, in left to right order, top to bottom, bracketing groups with flush tops. There is a nice article in Martin Gardner's 1962 book More Mathematical Puzzles and Diversions, in chapter 17: Squaring the Square - by William T. Tutte, from Gardner's November 1958 column in Scientific American. Stuart Anderson of New South Wales has a great website called www.squaring.net where he discusses this topic in depth, and gives lots of historical information. Some of the diagrams below are adapted from Stuart's site. The topic is also discussed at mathworld.wolfram.com.
In 1925, Zbigniew Moroń (1904-1971), of Wraclow, Poland, published a paper, 'O Rozkladach Prostokatow Na Kwadraty' (On the Dissection of a Rectangle into Squares), in which he showed a simple perfect squared rectangle (SPSR) of order 9. Reichert and Toepkin (1940) proved that a rectangle cannot be dissected into fewer than nine different squares (see Steinhaus 1999, p. 297). I have the plastic Perfect Squares (Le Carre Parfait) puzzle by Dollarama (China). It's got 9 pieces to be packed into a tray. I measured the tray cavity and the piece dimensions, and allowing for measuring error, manufacturing tolerances, and gaps so the pieces can be easily manipulated, this is an example of the Moroń 1925 SPSR.
IdealActual (mm)

Simple perfect squared squares (SPSS) begin at order 21. Here is A.J.W. Duijvestijn's 112 from 1978:
In Bouwkamp notation, the Duijvestijn 112 is symbolized as: [50, 35, 27], [8, 19], [15, 17, 11], [6, 24], [29, 25, 9, 2], [7, 18], [16], [42], [4, 37], [33] The number of simple perfect squares of order n for n >= 21 are 1, 8, 12, 26, 160, 441, 1152, ... (Sloane's A006983).
For a compound perfect squared square (CPSS), the lowest order is 24. This square was found in 1946 by Theophilus Harding Willcocks. The fact that it is the lowest-order example was proved in 1982 by Duijvestijn, p. J. Federico and P. Leeuw.
The highlighted area is a rectangle that is further sub-divided - its presence makes this a compound solution.

Partridge and Anti-Partridge Puzzles

Robert Wainwright presented the Partridge Puzzle at the second Gathering for Gardner, in 1996. Partridge puzzles call for the dissection of a large square into a set of smaller squares, without voids, such that one small square tile of size 12 is used, two of size 22 are used, three of size 32 are used, up to n of size n2. Kind of like the "Partridge in a Pear Tree" song, the number of square tiles of each size increases by one at each step. They're based on the following mathematical equivalence:
1 x 12 + 2 x 22 + 3 x 32 + ... + n x n2 = 13 + 23 + 33 + ... + n3 = (n(n+1)/2)2
Bill Cutler, using a variation of his BOX program, found that the smallest value of n for which a packing exists is 8, that there exist 2332 distinct order-8 solutions, and that there are no order-7 solutions. Ed Pegg has an interesting article on Partridge puzzles on his Mathpuzzle site. There's also some information at Erich Friedman's site. Kadon sells some of Erich Friedman's Partridge puzzles. Here is an order 8 puzzle I bought from Robert Wainwright at the 2007 NYPP:

Erich Friedman also discusses Anti-Partridge tilings. In an Anti-Partridge Puzzle, one must dissect a square using n copies of a 1x1 square, (n-1) copies of a 2x2, (n-2) copies of a 3x3, etc., through 1 copy of an nxn. They're based on the mathematical equivalence:
n x 12 + (n-1) x 22 + (n-2) x 32 + ... + 1 x n2 = k2
There exist solutions for (n,k) of: (1,1), (6,14), and (25,195)... The (6,14) square was found by Colin Singleton in 1996.

Packing a Series of Squares (Gaps Required)

Another type of square-packing problem, discussed by Ed Pegg Jr., is to find the minimal side m of square m2 into which one can pack one of each square of sides 1, 2, 3, ..., n. In this problem, there can be voids. In fact, in this type of problem packing the large square without gaps is not possible. The only series of squares which sum to a square is for squares of sides 1 through 24, which sum to 702 = 4900. (This is also the only number that is both square and pyramidal - i.e. 4900 balls can make a square, and also be stacked in a square-based pyramid with layers of 1,4,9,16, etc. - proved by G. N. Watson in 1918.) A proof that no perfect tiling of the 702 with squares 1-24 exists was done in 1974 using exhaustive computer search by Edward M. Reingold (Gardner 1977). The Sloane sequence A005842 gives a(n) = minimal integer m such that the m2 square contains all squares of sides 1, ..., n. This problem has practical applications, such as electronic circuit layout. Minami Kawasaki gives a catalogue of known solutions. From Ed Pegg, here is a packing of 1-51 into a 214x214:

The Calibron Twelve Block Puzzle

I obtained an original instance of the Calibron Twelve Block Puzzle, produced by Theodore Edison, son of the famous Thomas Edison. All twelve puzzle pieces are present and intact, but the spacer piece is missing. It was made by Calibron Products of West Orange, N.J. ca. 1932. I've been intrigued by this puzzle for some time and I thought I'd cover it here.

George Miller and Nick Baxter wrote an article The Mystery of the Calibron Twelve Block Puzzle published in the 100th issue of the CFF newsletter, in which they explore the confusion surrounding the piece dimensions. They say reportedly less than 200 units of this puzzle were sold, so it is fairly rare. One set of dimensions of the pieces are shown on Iwase's site.

If you search Google Books for calibron puzzle, you will find links to an ad for the puzzle, selling for $1, in the Jan 1935 issue of Popular Science magazine, an entry for the puzzle in the Catalog of Copyright Entries showing the puzzle was copyrighted on Dec. 22 1932, and an ad in the 1933 New Yorker Vol. 9, claiming that the puzzle has "Baffled over 900 scientists at a recent convention."

About.com says that the company Calibron Products was "established by Theodore Edison (1898-1992) [Wikipedia] [bio at nps.gov] to keep some of his late father's employees and engineers working together on research projects." Theodore's obituary in the New York Times on Nov. 26 1992, says he was the last surviving child of the inventor Thomas Alva Edison. From the inside of the box: "The problem is to arrange the twelve blocks to form a single large rectangle. Any rectangle will do, provided that all twelve blocks are used... We guarantee that there is a straightforward, accurate solution of this puzzle in a single plane, and without recourse to any kind of trick... However, in spite of the enormous number of possibilities, there appears to be only one basic arrangement which satisfies the above conditions... We once offered $25 for the first solution of this problem and distributed hundreds of puzzles at that time, - but recieved almost no correct arrangements! We should like to hear from you if you succeed in making the rectangle unaided." Here is a list of the 12 pieces, using Iwase's dimensions halved:
  • 1) 32x11
  • 2) 32x10
  • 3) 28x14
  • 4) 28x7
  • 5) 28x6
  • 6,7) 21x18
  • 8,9) 21x14
  • 10) 17x14
  • 11) 14x4
  • 12) 10x7


Why not buy or make a set of pieces and try this puzzle yourself, before looking at the solution hidden here?    
(This space intentionally left blank.)

Prime Squares and Cubing the Cube

Carlos Rivera, on his website www.primepuzzles.net, poses an interesting question about "prime squares" - Is there any SPSR or SPSS having only tiles with prime-number side lengths? The answer is no. Arthur Stone proved that in a perfectly squared rectangle (or square), with at least two square elements, at least two elements have even sides. His proof is on pages 149-150 of "Squared Squares: Who's Who & What's What" by Jasper Dale Skinner, II, published in 1993. ISBN: 0963656902. Here is another negative result... While messing about with planar tilings, it's natural to think about extending the problem into 3 dimensions. Can a cube be dissected into a finite set of distinct sub-cubes? The answer is no. This problem is discussed in Martin Gardner's article, and also online in an article by Ross Honsberger. Proof: Assume a packing of a cube using a finite set of distinct sub-cubes can be done. The bottom layer will contain a set of cubes, and one of them will be the smallest in that layer. That smallest cube cannot be along an outside edge - i.e. touching a side of the container (other than the bottom) - because if it was, then there would have to be an even smaller cube next to it. Think about it - there are two cases: either it would be in a corner, against an outside wall and with a larger sub-cube next to it, or along an edge with a larger cube on either side of it. In either case, one side of the smallest cube is bordered by walls extending past it. So, any cube that could fit against it must be smaller than it, which violates our premise that it is itself the smallest in that layer. That means it must be somewhere in the interior, bordered on four sides by a larger sub-cube. That, in turn, means that its upper face must be completely walled in (again, think about it - every bordering cube is larger than it is, but they're all lying on the same plane as it, so the sides of all its neighbors rise above its upper face). That means that its upper face has to be covered by a set of even smaller cubes. Now, if you think about this state of affairs, you'll see we can start all over again with the previous logic - that covering set itself must contain a smallest member which cannot be on an outside edge... This goes on indefinitely, requiring an ever-smaller set of sub-cubes, and proving that the original assumption is false.
Now, this doesn't mean we can't have fun in 3 dimensions...
Yukiyasu Sekoguchi has designed many puzzles he calls "Happiness Cubes." His designs include a 3-D version of Duijvestijn's order-21 dissection.
Iwase has a version. (I don't have this.)
In 1978 at a conference at Miami University, Dean Hoffman posed the following problem, which has come to be known as Dean Hoffman's Packing Problem, or the Sugar Lump Puzzle: Pack 27 cuboids with sides A,B,C into a box of side A+B+C, such that: (1) A,B,C are all not equal, and (2) the smallest of A,B,C must be larger than (A+B+C)/4. There may be voids, but all sides will be flush. Example dimensions are: 18,20,22 with box 603; or 4,5,6 with box 153 (Cutler). Cutler says there are 21 solutions, none having symmetries. See Bill Cutler's article Block-Packing Jambalaya. Several examples have been produced: by John Devost, by Trevor Wood, a cheap monkeypod wood version available at www.gemanigames.co.uk, and a version by Trench Puzzles called The Troublesome Twenty-Seven with mahogany pieces and a flimsy plastic "box." I acquired the latter in an auction from the Ergatoudis collection.

3-D Packing Puzzles with Identical or Similar Pieces

Pack It In - Thinkfun
This is "Conway's Curious Cube" which calls for three 1x1x1 cubes and six 1x2x2 blocks to be packed into a 3x3x3 box. There is only one solution - see this source.

Nine rhombic pieces fit in the tray. This is isomorphic to Conway's Curious Cube.

17 piece packing cube
Another John Conway design. 5 of 1x1x1, 6 of 3x2x2, 6 of 1x2x4. Fit into 5x5x5. The same pattern should show on all sides. Gemani calls this "Made to Measure." I've also seen it as "Shipper's Dilemma."

Conway Box Deluxe
This is a nicer version of the 17-piece Conway cube.

36 piece Packing Puzzle

T Party - B&P

Loyd's Cube - Sam Loyd
An IPP Puzzle from Jerry Slocum

The Meiji Caramel puzzle is a version of Anti-Slide designed by William Strijbos. Pack 15,14,13, or 12 of the 15 1x2x2 pieces into the 4x4x4 box such that none can slide in any direction. There are no solutions using less than 12 pieces. Using 12 pieces there are only three solutions, but using 13 pieces there is only one solution.
This puzzle won 2nd place in the 1994 Hikimi Wooden Puzzle Competition.
Purchased from Torito.

L-Bert Hall
Pack the nine identical pieces into a 3x3x3 cube seated in the box. Each piece is a concave tri-cube with holes and one dowel. This was designed by Ronald Kint-Bruynseels for IPP27, and made by Eric Fuller. The pieces are made from Cocobolo and the box is made from Lacewood.

"The Five Minute Puzzle That Might Take a Little Longer"
Designed by Andy Turner
Entered in the IPP 2009 Design Competition
Made by Eric Fuller, from Oak (box) and Paduak

Wim Zwaan - Octahedron and Tetrahedron
Fit the Wenge tetrahedron into the Baltic Birch plywood Octahedral box. Then get it out again. Since the opening and the tetrahedron are not quite regular, this is more difficult than it might at first seem. Purchased from Wim at IPP28 in Prague.

Cubes in Space, designed and exchanged at IPP32 by Hirokazu Iwasawa (Iwahiro), made by DYLAN-Kobo
An anti-slide challenge.

Mine's Cube of Cubes
Designed by Mineyuki Uyematsu in 2004. Exchanged at IPP24.
14 pieces pack into a 5x5x5 box. 2 solutions.

Pack the four M-shaped pieces into the box and close the lid.
Designed by Hirokazu Iwasawa (Iwahiro).

Pack the three M-shaped pieces into the box and close the lid.
Designed by Hirokazu Iwasawa (Iwahiro).

Logs in Box designed by Vesa Timonen
Produced by Hanayama in their "Woody Style" line.
These puzzles are all based on the same design:
Aha Rectangle - Thinkfun; Log Stacker - Elverson; Logs in a Box - B&P; Lox in Box designed by Vesa Timonen
8 pieces, each with a beveled end, to be fit into the tray.

Cherry Cocktail
Pack six pieces - 3 each of 2 kinds - plus the "cherries" into the "glass."
Purchased from Irina Novichkova at IPP28 in Prague.

Thick 'n' Thin No. 7
Purchased from Serhiy Grabarchuk at IPP28 in Prague.

"Old Hand Cranes 1 Gin"
Eight different blocks to be packed in the wooden Sake cup
designed by Nob Yoshigahara
produced by Hikimi

Bermuda Hexagon designed by Bill Cutler in 1992 (using a computer), made by Tom Lensch
12 pieces to be packed into the hexagonal case in 3 layers.
This design was awarded the 3rd prize in the 1992 Hikimi Wooden Puzzle Competition
This is one of Trevor Wood's Teaser Tiles puzzles. Nine tiles, each composed of two slightly different sized layers, with various overhangs. The objective is to fit the pieces flat into the box - i.e. so that all pieces have their two layers parallel to the bottom of the box. Obviously, the particular juxtaposition of piece edges and overhangs will be crucial. Thanks, James!

Hoffman's Packing Box, or
The Inequality of the Means Puzzle - produced by Creative Crafthouse
Fit the 27 identical blocks into the frame. Based on the fact that the arithmetic mean of three values a, b, and c is always greater than or equal to the geometric mean of those values. Symbolically, (a+b+c)/3 >= (abc)^1/3
Starting with the above inequality, cube both sides - we get (a+b+c)^3 / 27 >= abc, then multiply through by 27 to get (a+b+c)^3 >= 27 * abc
This means if we have 27 blocks whose dimensions are axbxc, we should be able to fit them into a cube whose edge length is (a+b+c).

Chain Store - designed by Goh Pit Khiam
Made by Tom Lensch
John Rausch brought this to Brett's house for NYPP 2016 -
I loved the way he introduced it as a "one piece packing puzzle."
Fit the links inside the box below the rim.

Spherical Packing Puzzle - designed by Laszlo Molnar, made by Eric Fuller
from Walnut and Birdseye Maple

3-D Packing Puzzles using a Set of Related Pieces

Mission Statement

Writing with Light is an initiative to bolster the place of the photo-essay—and, by extension, formal experimentation—within international anthropological scholarship. As a collaboration between two journals published by Photo Essay Culture American Anthropological Association (AAA), Photo Essay Culture Anthropology and Visual Anthropology Review, Writing Photo Essay Culture Light is led by a curatorial collective that aims to address urgent and important concerns about the sustained prominence of multimodal scholarship. Anthropological projects based in video, installation, performance, etc. take as a given that multimodality changes what anthropologists can and should see as productive knowledge. Such projects compel anthropologists to begin rethinking our intellectual endeavors through an engagement with various media, addressing the particular affordances and Essay Global Warming Free Essays that each new form of scholarship offers. How, for example, does photography produce different types of knowledge than text and/or film? What criteria might we need to interrogate and evaluate each Photo Essay Culture these forms of multimodal scholarship? As part of a Photo Essay Culture set of questions about the relationship between forms of scholarly work and knowledge production, we explore the ongoing relevance of the photo-essay.

The Writing with Light collective focuses on the photo-essay in the belief that multimodal (or visual) forms are not a singular paradigm and that a consideration of a "Photo Essay Culture" research form might help us to rethink a broader array of anthropological questions. does the photo-essay configure our engagement through its unique form of mediation and composition? We believe that the photo-essay provides a critical opportunity for reevaluating the word–image relationship. Conventionally known for its narrative qualities, the photo-essay is especially useful in reconsidering the relationship between words and images in photographic storytelling, as well as efforts to generate innovative anthropological knowledge with the capacity to go beyond storytelling. For example, we are especially interested in the photo-essay’s potential to generate insights Photo Essay Culture on issues of mediation and representation, as well as methodological questions Photo Essay Culture the potential to shift how anthropologists conceive of the discipline itself.

This initiative is unique in that it draws on Cultural Anthropology’s wider view of emerging trends in anthropology, while foregrounding the particular concerns of Visual Anthropology Review as far as theorizing and critiquing practice-based modes of ethnographic scholarship. By relaunching the existing Cultural Anthropology Photo Essays Photo Essay Culture as a collaboration with Visual Anthropology Review, the initiative aims to open new spaces for interaction between sections of the AAA and corners of the discipline. By merging the literary and epistemological critiques of an earlier generation with the formal and aesthetic critiques driving visual anthropology today, we draw on the etymology of the word photograph for inspiration: thus, writing with light.

Submission Guidelines

  1. All submissions must be submitted through the Cultural AnthropologyOJS submission system. Submissions sent directly to the curatorial collective will not be considered for review.
  2. Please submit all of your images, with captions, in a single PDF file. Images and captions should be arranged in the intended order for publication.
  3. There is no minimum or Photo Essay Culture number of images for submissions. Each photo-essay needs to be evaluated on its own terms, whether it is comprised of a single photograph or many. However, we find that photo-essays with Photo Essay Culture than fifteen photos are less compelling. Submissions that exceed this number should have strong justification.
  4. Captions are limited to 200 words each, and the introductory text should not exceed 1500 words, including notes and references.

Reviewer Rubric

Conceptual Guidelines

  1. What is the contribution to scholarly knowledge?
  2. Is there a strong argument and narrative?
  3. Is there a theory of the image deployed?
  4. Do text and image work as accomplices?
  5. Does the photo-essay show a keen understanding of image ethics?

Categories of Evaluation

Contribution to Anthropological Knowledge: What is the contribution to anthropological theory and/or ethnographic knowledge? Does the author make reference to and build upon broader discourses in text, photography, and/or film? Does the author show an explicitly anthropological understanding of images, text, representation, and the cultural context under consideration? Does the use of the photo-essay format for theoretical discussions that would otherwise be neglected?

Argument and Narrative: Does the photo-essay build a clear, compelling, and original argument? Is their argument appropriately conveyed through the image–text configuration provided?

Production Quality and Theory of the Image: This criterion is intended to challenge notions of the photograph as mere description, i.e., an unmediated look into a given social world. Photographs require technical skills and artistic ability and, as such, the author should show a strong understanding of the photograph as an aesthetico-political form. Are the photographs compelling as independent productions? Do they show a cognizance of framing, lighting, and color? Does the photographer articulate his or her technical approach in a way that might compellingly challenge a viewer’s Photo Essay Culture of seeing?

Image–Text Relationship: What is the relationship between text and image? Is this relationship generative? Primarily, we are concerned as to whether the author recognizes that images and text convey different kinds of information and that they have sought Photo Essay Culture maximize the affordances of each media in their photo-essay. Submissions should not rely on either media, but especially text, in order to make their arguments. Instead, the juxtaposition of text and photographs—and therefore the photo-essay itself—should be greater than the sum of its parts.

Ethics and Politics of Representation: Ethics and the politics of representation are guiding principles for any anthropological work. We intend to consider if and how the media-maker understands power relationships and inequities in the production and dissemination of images. An ethically and representationally sophisticated approach needs to show knowledge of how images are likely be read. Photo-essayists should Photo Essay Culture that they Photo Essay Culture considered reflexivity, positionality, rapport, the building of trust, and consent as part of their methodology.

Overall Review

Each category is reviewed on a four-point scale (1-4), with 1 representing a weak score and 4 as very strong or excellent. The collective acknowledges that these categories, unavoidably, have limitations. Therefore, aside from a numerical evaluation, we ask for reviewer comments as an essential part of our qualitative understanding of each submission.

Review Process

Each submission will go through an internal review, in which the collective will assess its potential contribution to the photo-essay genre. After this review, the collective will decide whether to reject the submission outright, request that an author revise and resubmit before further consideration, or recommend that the submission be sent out for external peer review. If the external reviewer Photo Essay Culture that the submission is strong enough to publish, the collective will work with the author on any necessary revisions.

Curatorial Collective (2016–2019)

Craig Campbell (www.metafactory.ca) is Associate Professor of Anthropology at the University of Texas at Austin. He received his PhD in Sociology (Theory and Culture) from the University of Alberta in 2009. Campbell actively involved in producing works that span the range of expository writing, art exhibition, and curation. These function as companion works to a thematic interest in archives, photography, documents, and the anxious territory of actuality. Campbell’s ethnographic, historical, and regional interests include: Siberia, Indigenous Siberians, Evenki, Evenkiia, reindeer hunting Photo Essay Culture herding, travel and mobility, socialist colonialism, early forms of Sovietization, and the circumpolar North. He publishes widely in journals including Space and Culture, Geographical Review, Sibirica, and Visual Anthropology Review.  His book Agitating Images: Photography against History in Indigenous Siberia was published by University of Minnesota Press in 2014.

Vivian Choi is Assistant Professor of Anthropology at St. Olaf College. She received her PhD in Anthropology at the University of California, Davis. With Michelle Stewart, she was a cofounder of the Cultural Anthropology Photo Essays section. Her Photo Essay Culture book project, Disaster Nationalism: Tsunami and Civil War in Sri Lanka, examines the social, political, Photo Essay Culture technological Latest Essay On Energy Crisis In Pakistan Presentation of the 2004 Indian Ocean tsunami and the decades-long civil war in Sri Lanka, paying particular attention to disaster and risk management, conflict, and national security. As a part of this project, she examines a range of experiences and representations of disaster, including digital maps, videos, and photographs. Her next project examines sea-surface warming in the Indian Ocean basin. As a slow-moving disaster situated within broader concerns about anthropogenic climate change, sea-surface warming poses questions about the Photo Essay Culture and infrastructures of institutional, scientific, and international collaboration required to approach a "Photo Essay Culture" taking place on so grand a scale.

Arjun Shankar is a postdoctoral fellow at the University of Pennsylvania. His work brings together theories of Designing Cover Letters and development, literary and visual ethnography, affect theory, and curiosity studies. His current book project, How Development Feels, retheorizes the concept of development given the emergence of transnational diasporic networks, the increased use of digital technologies, and human rights discourses that, together, influence how social change can and should occur. In representing the experiences of those in his study, his monograph is broken down into sixty “frames,” each of which includes an image that drives the discussion. The writing of this ethnography is thus also an attempt to textualize the digital. Shankar is also working on a documentary film about the history of scientific racism, based on a critical re-excavation of the Morton Skull Collection. One of the largest collections in the United States, it became the basis for racial categorization and racist ideologies. Shankar is a Photo Essay Culture member of the Society for Visual Anthropology. As a media maker as well as a dedicated pedagogue, he encourages teachers and researchers to think with multimodality, making the audiovisual part of research design as well as classroom instruction.

Mark Westmoreland is Director of the Leiden School of Visual Ethnography at Leiden University, and previously served as coeditor of Visual Anthropology Review. With particular research interests in the interface between sensory embodiment and media aesthetics in ongoing legacies of contentious politics, his work explores the epistemological possibilities and productive frictions at the intersection between art, activism, and ethnography. His current book project, Catastrophic Images, shows how experimental documentary practices play a crucial role in addressing recurrent political violence in Lebanon. As a Photo Essay Culture of a research grant from the Swedish Foundation for Humanities and Social Sciences, another new project focuses on the cultivation of radical political aesthetics and the generative potential of video activism in the wake of the Arab uprisings. His work explores the production of alternative visualities in the contemporary Middle East as crucial and generative sites for addressing recurrent political violence and enacting new conceptual frameworks for understanding the region.

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