Sunday 7 January 2018

The Diamond 13 Puzzle

L.E. Ott’s Diamond 13 - Upending the classic magic square.
Well time for the first blog post of the year and I am extremely grateful to Mike Desilets, the PuzzleMad foreign correspondent for jumping to my rescue. I have been trying to solve a few new and old toys but this week have barely managed anything. On top of that I have just begun to go back to the gym after 8 weeks off for my surgery and am absolutely shattered! I suspect that 45 minutes of cardio for my first time was not a good idea but then you know that I am not terribly bright. It is really useful every now and then to have Mike (or anyone else who wishes to write something interesting) take the pressure off. Over to you Mike.....

Aloha kākou puzzlers,

As you may have noticed, it’s been quite some time since I’ve filed a report with Puzzlemad. Work has gotten the best of me lately and I just haven’t been able to muster the energy to sit down and write anything (Kevin makes it look very easy (Ed - if only that were true!)). It got so bad that my dear editor even sent me a note asking if all was ok (Ed - I just like to keep in touch with my friends). That was really nice, and just the kind of thing you can expect from a puzzle friend. Things are indeed ok, and with the holidays upon us, I now even have time to catch up on my backlog. Although I haven’t made time to write, there are several interesting puzzles that I’ve been itching to talk about.

You’ll recall many many months ago we dove (Ed - as an American, I will let you away with that but all of us Brits are cringing and shouting the word "dived" at you now) into a small but happy corner of the vintage puzzle world. Thanks to my friend and puzzle compatriot Amanda, we were able to explore some very cool old sliders. Around that same time I managed to pick up a great puzzle from the redoubtable (and long defunct) Embossing Company, makers of the Time Puzzle and Line up the Quinties. I don’t think I went into any detail about the company at that time, not wanting to stray too far afield. However, the history of puzzle companies is a sideline interest of mine, so please indulge me for a bit. This is all stuff you can google for yourself, but I’ll save you the trouble. (Thanks!)

The Embossing Company was not a proper puzzle company, in truth, but rather more of a toy company that carried a wide range of wooden toys, games, and “novelties.” Their early puzzles were of the simple edge matching, construction, or pattern making type, all made using their trademark embossed blocks. “Toys that Teach” was their motto. The company was founded in 1870 in Albany, New York and stayed in business for 85 years, finally selling out to Chicago wooden (and plastic) toy giant Halsam Products Company in 1955. Halsam was a relative newcomer, having entered the business in 1917. The owners, however, invested heavily in automation and quickly dominated the market. After the acquisition, they continued to manufacture the Embossing Company’s successful and well-regarded ABC blocks, dominoes, and checkers at their Chicago factory. I find no indication that they continued to produce Embossing Company puzzles, however.

Very fine detail achieved by the Embossing Company.
Embossing Company products, including their puzzles, were made using an embossing process (did you see that one coming?). Embossing of wood involves the application of heat and pressure to imprint a pattern or design. You’d have to ask a woodworker to be sure, but it seems to me that the compression makes these wood game and puzzle pieces significantly more dense, and thus harder, as well. The examples I have are very robust and likely capable of withstanding much abuse, which I imagine the children’s toy block line received. The embossing process also supported very fine design detail. Upon close inspection you will find very small and highly detailed patterns pressed into the wood.  If you are in the market for a very nice set of dominoes or checkers, think about hunting down an old Embossing Company set. They are as good as the day they were made and are prized by top checkers players particularly.

Nice durable pieces, courtesy of the Embossing Company, but be careful with the cardboard!
Ok, enough context. Let’s get to the actual puzzle. Today we have The Diamond 13 Puzzle, listed as No. 911 in the Embossing Company line. The Diamond 13 was invented by a certain L.E. Ott, probably in the 1940s or 1950s. A quick patent search pulled up nothing, so we can’t be exactly sure. This puzzle falls into the “Pattern” class, but somewhat uncomfortably in my opinion. You’ll understand better as you read on. (Ed - it looks like being in the Pat-Numb category of the Dalgety-Hordern classification)

The Diamond 13 is a descendant of the Magic Square family, including as it does the vast menagerie of ‘magic’ shapes that have been explored more recently. These puzzles are a mainstay of the recreational math community and have a very long history. Because the literature is so vast and most of the readership are likely familiar with them, I won’t go on and on. Also, recreational math is far from my strong suite. If I attempt to hold forth, I will eventually put my foot in it, betray my ignorance, and bring shame on Puzzlemad (Ed - NEVER!!). That said, you can’t appreciate the Diamond 13 without knowing the basics. So for new initiates, here is a very quick overview:

Magic squares present the puzzler with a square matrix of a certain size, the object of which is to fill the cells with numbers (the numbers representing the number of cells, specifically) that all sum to the same total—horizontal, vertical, and diagonal. The order 3 (3x3) magic square is the all-time classic, having only one Real (ℝ) solution (i.e. discounting rotation/reflection). Each magic square that is solvable has only one Magic Constant, being the number that must be totaled. Order 3 is 15, order 4 is 34, order 5 is 65, and so on. The number of Real (ℝ) solutions for magic squares increases extremely quickly (order 3=1, order 4=880, order 5=275,305,224 and order 6 is a staggering 1.8 × 1019, or thereabouts. No one knows the exact amount because no method has been discovered for its calculation). 

2 7 6 15

9 5 1 15

4 3 8 15

15 15 15 15
Order 3 Magic Square. Known and pondered since 650 BC.

(Ed - My goodness I've had to do some html editing here - getting the Real number sign and superscripts and that table with arrows in it was a bit of a challenge)

Magic squares are the original ‘magic’ shape, but you can apply the concept to just about any shape you like, and people have done so over the years. Probably the second most well-known magic shape is the magic hexagon, discovered by William Radcliffe in 1895. There is only one solvable magic hexagon: order 3 with a magic constant of 38. It’s an interesting shape because the totaled rows are of variable length, ranging from 3 to 5 cells. This structure hints at how to speed up your solution search.  The magic hexagon is one of the very few magic shapes to have transitioned from the mathematician’s page to an actual physical puzzle. Professor Puzzle produces a serviceable version in their Great Minds line. They are very widely available and cheap. I picked up my copy at Barnes and Noble.

Order 3 magic hexagon from Professor Puzzle.
Pieces are a loose fit, but they need to be in order to manipulate and work the puzzle.
There are all kinds of other possible magic shapes including stars, triangles, cubes, circles, pan-diagonal tori, and the mind-bending magic tesseracts (4 dimensional hypercubes). To learn about any and all of these objects, I highly recommend you go to Harvey Heinz’s website. You can spend hours there (Ed - I have!). I’ve only scratched the surface of what he offers. Harvey also has a page that will be of interest to mechanical puzzlers, in which he presents a number of magic mechanical puzzles supplied to him by Jerry Slocum.

Mr. Ott lays down the rules.
Totaling structure for the Diamond 13.
Now that we are all up to speed, lets return to the Diamond 13. Mr. Ott employs a magic diamond as his base shape. This is one of the least explored magic shapes. In fact, I can find almost nothing about them, even on Harvey’s site. No matter though, because the Diamond 13 is not intended to be a magic shape in the traditional sense. Rather, this puzzle requires one to create separate diagonal and horizontal/vertical totals, which are two different numbers. In the third challenge, for example, the four diagonals need to total 7 while the vertical and horizontal must total 21. Click the Show/Hide button to see what D-7 HV-21 this looks like solved.

The puzzle also diverges in its piece values. The diamond has 13 pieces ranging in value from 1 to 7. There are two 1s, three 2s, two 3s, two 4s, one 5, two 6s, and one 7. This is quite a mixture and certainly a departure from the sequential approach used for traditional magic shapes. Likely the major developmental work for the puzzle was in finding a mix of values that would generate the most ‘magic number’ problems. The image below shows 26 sample problems provided with the puzzle, and there are clearly more to be discovered. We take multi-challenge puzzles as a matter of course today, but they were not nearly as common in previous eras.

Sample problems for the Diamond 13 Puzzle.
D=diagonal, HV=horizontal/vertical.
With those radical departures from traditional ‘magic shape’ construction, one might reasonably ask if this is really even a ‘magic’ puzzle at all. The magic is usually to be found in the symmetry and deep patterning inherent in shaped arrangements of sequential number sets. And the magic constant of course underpins the whole edifice. The Diamond 13 takes great liberty with these rules, tossing most of them out the window. However, it still successfully exploits our primal fascination with the intersection of geometry and arithmetic, the quality responsible for the experiential “magic” felt when pondering magic squares or hexagons. From a historical perspective, at least, the relationship is crystal clear. The Diamond 13 was born conceptually from the magic square. 

But there is really only one question that matters. Does it work as a puzzle? In my opinion, yes, it works very well. I had tremendous fun working out the various number sets. The sample problems are challenging but quite manageable. The Diamond 13 hits the sweet spot for my taste. I enjoy tough puzzles too, but for pure enjoyment, a puzzle like this can’t be beat. Part of the enjoyment is due to the fact that this isn’t just a hunt and peck puzzle, where you are simply exhausting possibilities. The structure of the puzzle can be used to your advantage, but the way it does so evolves with the changing target values of the magic numbers. Note that the diagonals are only three units long, with only one value not shared with another diagonal. The horizontal/verticals are five units long, sharing only one common piece. As you play through the first few challenges, the dynamics of the structure become apparent. Another pleasant surprise was the approach (not given in the instructions, I did it out of laziness) of transitioning from one solved state directly to the next solved state. Doing the challenges progressively, in other words, without scrambling the pieces between challenges. I recommend this style of play. Some of the transitions can be done in very few moves, if you can see them. Exploiting the structural limitations really pays off here. This form of progressive play is reminiscent of the Time puzzle.

The Diamond 13 Puzzle is yet another great, underappreciated example of mid-century puzzle design. Although it may seem a bit contrived compared to the mathematical elegance of a pure magic shape puzzle, I can assure you that the design functions as intended. It provide a series of moderate-level challenges that force you to exercise both trial-and-error and strategic thinking. The designer began with a magic square-like concept, then rewrote the rule book. This might easily have proved disastrous and it’s a tribute to the designer that the puzzle works so well. 

Well as it works, we certainly don’t see the Diamond 13 in stores today. I’m not always sure why certain puzzles catch on and others don’t. In this case, the Diamond 13’s fate was probably linked closely with that of the Embossing Company itself. Like most game and puzzle companies, the Embossing Company relied mainly on a line of anchor products — tried and true classics — for the bulk of their sales. But they also tried to liven things up occasionally with new innovative products. The Diamond 13, the Time Puzzle, and Quinties were beneficiaries of the Embossing Company’s drive to distinguish itself from all the other companies making similar products (toy blocks, checkers, and dominoes principally). When the Halsam Products Company took over in 1955 however, only the best selling products, those with over a half century of name recognition, were retained. The more innovative puzzles, probably never great sellers in their own right (and often included only within larger sets) were not retained. I’m reasonably sure Halsam’s acquisition was more about buying market share than buying a product line anyway. Halsam would go on to be purchased by Playskool in 1962, which would ultimately be bought by Milton Bradley, which itself would even more ultimately be bought by Hasbro. 

The Diamond 13 Puzzle. Unlikely to be reissued by Hasbro.
Although there are a diminishing number of Diamond 13 puzzles out there for puzzlers and collectors, they are not especially uncommon. Both Amanda and I have copies, which should tell you something right away. But you don’t need to buy the actual Embossing Company puzzle, great as it is. All you need is paper and pencil. This hold for all the magic shapes, bar the terrifying magic tesseract. If you close your eyes and concentrate, you might even be able to do the order 3 magic square in your head. Try it. 

One final thing. The question of whether L.E. Ott’s diamond shape has a magic constant in the traditional sense may have crossed your mind. Looking at the layout, and thinking about the numbers involved (1–13), I would have to say I think not. But don’t take my word for it. I really don’t know and, embarrassingly, haven’t lifted a finger to try (Ed - shocking!!). But I will give it a go and report back in the next guest post. If anyone out there in the Puzzlemad Army wants to take up the challenge, please do. Even better, can you prove mathematically what the constant must be, or alternatively why there cannot be one? It’s an interesting problem which I am entirely unequipped to solve. Some enterprising rec math enthusiast should have a go at it. I’m sure you could get a paper out of it. Minimally, my editor will splash it all over Puzzlemad. (Ed - I definitely would even if I couldn't understand it)

That’s about enough for this post. Usually I try to squeeze in a few different puzzles for variety, but this seems like enough for one Sunday. Thanks for tuning in. Back to you Kevin . . . 

Thank you so much for that Mike.....really GREAT article! I certainly do need to try and find some of these vintage puzzles for my own collection. 

Tomorrow is my first day back at work and I am seriously not looking forward to it but I daren't be at home much longer or there will be a murder! "She" has already told me that I am NEVER allowed to have another operation that needs time off and I am never going to be allowed to retire! It would appear that I really got in the way AND pissed her off! Whack! Ouch! Ooops! Caught!


  1. I promise to use this from now on!

    1. Hahahaha! Interestingly it allowed the word “dove”!!!