Sunday 19 January 2020

A Great Tonic For When I'm Under the Weather - Dominola

Another vintage puzzle - Dominola
My good friend and the singular PuzzleMad foreign correspondent, Mike Desilets, always comes up trumps for me just when I really need it. I seem to have been poorly for quite a while (12 days and counting) with a spot of puzzler's lurgy that just won't go away! Mrs S is complaining about the 'drowning in mucus' noises at night and keeps asking why I feel the need to gargle with it when she is trying to get to sleep. I, of course, am completely unaware of my horrific sound effects and have to add being woken up multiple times a night by being kicked or shouted at to my woes. Needless to say, puzzling has been a bit tough and every time I settle down to work on something I lose consciousness before I get anywhere. Just as I was getting desperate to find something to blog about, Mike dropped me another wonderful email about an old and interesting puzzle that he has been working on. Over to you Mike...

Aloha Kākou Puzzlers,

This time of year I, like you, fantasise (Ed - we are NOT getting into my fantasies here!!!) about all the puzzling I am going to get done in my time off. That never really happens. Instead, new puzzles flow into the house at an accelerated rate and the backlog grows (OMG! Tell me about it!). I also continually get hung up on certain puzzles that catch my interest. Many of those end up as blog posts, and today I have one such for you. Once again, it isn't the puzzle, per se, that has me hung up, but rather the full family of puzzles it belongs to. We’ll get to all that shortly.

Today’s puzzle is called, rightly enough, Dominola. Not to be confused with my editor’s current Dominola, Mrs S. That is a completely different blog post for a completely different kind of site (We are NOT going there! She won't let me). And you’ll have to pay for it. I am of course referring to the domino-based puzzle designed by Eric Everett way back in 1935, or so. The extent to which it can be considered an original design is open to question, and we will get to that further down. My usual slapdash patent search turned up nothing on the puzzle itself, but the phrase “Dominola, unique puzzles with many solutions” was duly copyrighted on June 10, 1935. Mr Everett then applied for a trademark on the name DOMINOLA in a distinctive if unimaginative san-serif font on June 29, 1935. The name Dominola (in a different font) had been previously registered as a trademark by one John M Haddock on April 7, 1903, for a card game. It appears this earlier trademark had expired by the time Mr Everett applied. Unlike patents, trademarks are considered “abandoned” under US law after three years of non-use. Said use, must be continuous, ongoing, and demonstrable, otherwise the mark goes back into the stew pot. 

Dominola packaging.
Eric Everett’s Dominola puzzle consists of a complete set of double-six dominoes and a two-sided playing board. A double-six set consists of 28 tiles, 7 of each number. In keeping with the times (see Embossing Company post from way back), and likely the target price-point for the puzzle, the tiles are made of embossed wood. If this post makes you want to add a set of dominoes to your collection, I recommend an antique set of bone over ebony with brass spinners. They seem to be plentiful on the British and European Ebays and are reasonably priced considering many are over a hundred years old. Anyway, wooden tiles are quite fit for Dominola and the coloured pips add a nice splash of, well... colour.

Dominola instructions - no surprises here.
So you already have a bargain with this puzzle, in that it comes with a full set of dominoes. You can both play dominoes AND solve Eric’s two puzzles. That’s right, two, because the playing board is two-sided, with a different puzzle on each. Look at the pictures below if you don’t believe me. Now, the objective of the puzzle is to lay down the tiles, ends matching of course, in such a way as to complete a continuous circuit. In essence, this is an edge-matching puzzle. Although the dominoes have “numbers” of pips, there is no mathematical element to the solution. The pips could very well be replaced by symbols or just colours for that matter.

Lack of maths does not mean the puzzle is easy. It actually requires a respectable amount of work. Why? Glad you asked Kevin! (Hahaha!) Picture a simple tile rectangle. It would be pretty easy to make this circuit with the dominoes. Plenty of solutions and only one constraint, continuity. Now picture a figure-eight; one crossing. All of a sudden you have a constraint, though a minor one, and thus slightly fewer solutions. Add more crossings and/or interconnections and you quickly introduce all sorts of complicated co-dependent constraints, and the number of solutions drops rapidly. These new constraints typically involve three, four, and even five-way linkages of same-numbered pips. It's an interesting and underappreciated fact that all of the tiles in a pip-number set are not equal. The double-sided tiles (1-1, 2-2, 3-3, etc.) are, in a sense, unique in the way they function. When you are working on your homework assignments (below), you’ll quickly discover that the placement of double-sided tiles requires especially careful consideration. (Homework? But but but...I'm poorly!)

The complete double-six set provided with Dominola
Mr Everett does offer the puzzler an olive branch of sorts, in that four tiles are pre-set on each board. This gives one a starting point, at least. In the case of the purple side, the fact that two of the pre-placed tiles have one pip is highly significant and immediately rules out certain arrangements. There is still a lot of work to do, but the pre-set tiles at least stimulate thought in an analytical direction, or they should anyway. That becomes the real fun of this puzzle.

I must confess, I have never been a huge fan of edge-matching (Me neither). All too often it is simply a seemingly endless trial and error search tree process; not much to think about other than keeping track of what you’ve already tried so you don’t spin around in circles. I had that mindset and expectation when starting out on Dominola. I quickly realized, however, that domino tiles are not anything like the conventional three, four, or six-sided geometric tilings we are used to. They have, as you can see, two ends, each of which can form three potential connections in as many cardinal directions. They are fundamentally “end-matching” if you will. While most edge-matching puzzles build into a solid shape, domino tile end-matching forms open patterns (and also shapes, if you want to take the maths approach). This makes them intrinsically interesting to a recreational aesthete such as myself. You have to admit, Mr Everett’s designs are quite eye-catching. But more importantly, the properties of domino tiles, and thus how they match, opens up opportunities for true problem-solving.

Purple side
Orange side
So solving Dominola involves more than just trial and error. But don’t worry, If you are a lover of trial and error, there is still plenty of it. Despite what I just said above, I could not find a method to fully “solve” the puzzle through algorithm or artifice. I still think it is out there, I just didn't have the patience or smarts to figure it out before resorting to brute force.

Orange in play.
There are certain rules that should be helpful, though. For example, given the layout of Everett’s purple board, every tileset (7 tiles of a given number/colour) must necessarily connect its tile ends in patterns of either 2-2-3, 5-2, or 3-4 (3-4, for example, means 3 interconnected tiles connected in one place and 4 in another). These are the only possibilities, and they apply to all seven tile sets. So if you find you have laid down, say, two connected tiles of a certain type, you cannot then lay down four connected tiles of that type, or else you will have one lone member leftover with no partner. You would need to go with a 5 or a 2 and a 3, assuming the circuit allows for these. Keeping this in mind, and constantly checking against the three fundamental set patterns, you can prevent yourself from creating some clear dead ends. It should allow you to select out a large number of losing arrangements as you work a certain approach. That said, there are still an infuriatingly large number of ways to get it wrong, even avoiding the obvious dead ends, and you won’t often find out until you get to the last few tiles. (My goodness that is a complex thing to think through!)

Consider, if you will, the perplexing lattice design below. I concocted this mostly for the cool interior-exterior optical illusion. I also thought that it would make a challenging puzzle. But is it actually solvable? Take a moment and analyze; see if you can figure it out in principle, without laying down any tiles.

Mike's Lattice Challenge.
The way to figure this one out is to first break the puzzle down into its components and analyze those first. You’ll notice if you stare at it long enough that the lattice is composed of two equivalent halves, mirror reflected. Looking at either instance, you will also eventually see that the layout necessitates a string of seven connected tiles (six is technically possible, but this doesn't help the situation at all since it leaves an orphan). You can arrange a seven-string once on a layout, but you cannot do it twice (which becomes increasingly necessary with two mirror lattices) unless the two strings are interconnected. There is one way to connect the seven-strings of each half, but it creates an insoluble dead-end. That explanation probably won’t make sense until you dig into the problem. The bottom line is that this lattice cannot be completely tiled, and more importantly, that you do not need to systematically attempt every possible tile combination to know this. (I'm pleased that you told me that! I am fairly certain I would have spent many hours trying it!)

As my indefatigable editor and I have said to you many times, Puzzlemad is a full-service blog. No, we won’t check your car’s oil level, but we do enthusiastically support our fellow travellers. Thus, we present you with a free digital copy of Dominola for your use and enjoyment. The purple side is downloadable here and the orange side is here. I drafted these on an 11x17 board, so use your best judgment when printing them out at work. I used the Dominola tiles as a gauge, and they are a bit smaller than modern standard tiles. Fiddle with the enlargement/delargement (ed - is that a word?) settings and it should work out. You get what you pay for, so please don’t bother contacting the Puzzlemad customer service department. It's Mrs S’s mobile. (Whack! Ouch!

True metagrobologists are by this point wringing their hands and scowling mightily.  Of course, they are correct, Eric Everett did not invent this type of puzzle, and it is less than certain that he even originated the Dominola designs. I haven't seen them anywhere else, but of course, I have a meagre puzzle library. Those of the readership that have shelves and shelves of puzzle books, old ones especially, please feel free to research this issue (Ed - I actually have almost no books of puzzles at all). But regardless of Dominola design origins,  the idea of making symmetrical circuits using dominoes has been around a very long time, probably nearly as long as dominoes themselves.

One puzzle book that I do own is Slocum and Botermans’ The Book of Ingenious & Diabolical Puzzles. It contains a selection of eight historical domino circuit puzzles (see page 59). I haven't tried all of these yet, but they look very interesting. The important thing to remember is that each one has its' own dynamic. They do not all behave the same way, and rules you might have developed to help you solve one will not necessarily be useful on another. You really have to think about each one on its own terms.

Since my decades-old copy of Adobe Illustrator CS3 was all warmed up making the Dominola boards, I went ahead and drew the Slocum and Botermans set, just for you. FULL service, my friends. Now you have no excuse at all. Tramp out to the corner market and order up a double-six set and some chips get puzzling. These eight designs will unquestionably keep you busy. Download them all in a single zip file from here.

Although none of the puzzles in this post requires arithmetic, there exists a whole array of domino puzzles that do. If you have that Slocum and Botermans book I’ve been referring to, you can study several pages of them. I would also like to make special mention of a third sort of domino puzzle. This is the use of dominoes to perform simple computation. Here’s a quick link to get you started - a half-adder from 2007. I think others have improved on it since. This will be VERY interesting to a certain narrow segment of the puzzle community.

Even the great Sam Lloyd had something to add to the Domino puzzle sphere - he published an article which you can read below:

I hope you have good eyesight!
To wrap up, I found Dominola, and the whole family of domino tile-matching puzzles, to be both fun and challenging. They go a step beyond any edge matching puzzle I’ve done previously, in a good way. Eric Everett did what many puzzle producers do, take an existing idea, tweak it a little, and package it in a way that is attractive to consumers. It is a great puzzle to own for the collector, although I suspect there are not many copies floating around, even fewer in good condition. More importantly, Dominola led me into the larger, deeply historical world of domino-based puzzles. This has not only caused joy (and frustration) but has saved me a significant amount of cash. If Mrs S finds out about this, I think Kevin will also save a lot of money, and also curse me to the grave. (Luckily for me, she doesn't read any of my drivel at all - otherwise, I am a dead man!)

Ok boss, bring us home...

My goodness! You have completely opened up a new sphere of puzzling to me! I had never even contemplated dominoes as a source of puzzles! I will need to get myself a nice set and have a play once my post Xmas finances have settled down a bit. The nice vintage domino sets I have found on eBay are quite pricey!

Thank you so much, Mike, as always, for helping me out in my hour of need! I will hopefully be back to full strength puzzling very soon or Mrs S may well euthanase me with her bare hands and teeth!

I hope that this has been of interest to all you puzzlers out there? I certainly found it quite fascinating and will certainly be trying some of these challenges myself.

Follow up - the reference given above to Slocum's book was an error - it should have been to Van Delft and Botermans' Creative Puzzles of the World. The page numbers are correct. Sorry if we confused anybody.

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