Sunday, 12 December 2021

Mike Saves My Bacon...Again!

Happy Sunday everyone! I have been rather strapped for time yet again - work has been very busy both with clinical stuff and quite a lot of admin too. I had to work yesterday and have not really had any time for puzzling. Despite this, thanks to my good friend, Mike Desilets, the official PuzzleMad foreign correspondent, I still have a fascinating post for you. I don't know how he does it, Mike always seems to sense when I am struggling for time and comes up with the goods for me/you. This time he did it in the guise of asking for some help with a puzzle he had been struggling with but I know that he is too good a puzzler to need help and it is just an excuse to get in touch with yet another article. I hope to get back to normal puzzling this week (although I am working next weekend as well) - if anyone else has a nice article for the PuzzleMad readers then please get in touch. Over to Mike...

Aloha kākou puzzlers,

I know we have yet to close out the story of the Great 13, but let’s please forget about that for a while longer. Instead, the PuzzleMad Foreign Office, Hawaii Branch, offers up this quick and dirty post, designed specifically to provide Kevin with a free weekend (Ed - yay!). Free from blog posting, that is to say, not free from his actual day job, which does not respect weekends. He deserves so much more, but this is all I can muster at the moment. Today I share with you some recent output from the PuzzleMad Workshop, which is conveniently located in a sub-basement 200 feet below the current Mrs S’s house. (Ed - how come I don't have access to this area? It would definitely solve my storage issues)

John Horton Conway (1937-2020)
Years ago, George Bell turned me on to a very interesting problem known as Conway’s Soldiers. This problem was first proposed in 1961 by the eminent mathematician John Horton Conway. John has, tragically, left the room, having been one of the very early covid victims. But he left behind a massive legacy in a variety of fields. The Conway’s Soldiers problem, or puzzle, is surely among the mildest of Conway’s accomplishments. If you’re not familiar with John Conway’s varied contributions to mathematics, go check the wiki right now. They’re quite remarkable. And, unlike the great majority of mathematics (and mathematicians), also interesting, relevant, and accessible to the average fellow, at least in broad strokes. Most puzzlers probably know Conway from his combinatorial game theory work, of which Conway’s Soldiers is a small piece.

 Disentanglers might be familiar with his knot theory work. Everyone is surely aware of his Game of Life (Ed - I programmed this into my school Tandy TRS80 in 1979 and even learned Assembly language because the Basic was too slow). If you hadn’t kept pace with his recent work, Kevin, then I highly recommend checking out his (and Simon Kochen’s) Free Will theorem and its implications for the ever dubious, but somehow inextinguishable, deterministic worldview. I actually thought that whole thing was put to bed decades ago, but I suppose people are still free to choose determinism (or are they?). Anyway, the point is that you will benefit greatly, and in unexpected ways, from engaging with Conway’s vast and varied body of work. 

Conway’s Soldiers, the mechanical puzzle.
For my part, being a simple puzzler, I was drawn especially to Conway’s Soldiers and its peg solitaire aspects. A couple years ago, in a flash of brilliance, I realized that creating a physical version of this abstract puzzle was within my ability. It is truly the simplest of designs, consisting of gridded peg holes and a line dividing the plane. A full year and a half after the initial inspiration, I made the puzzle. The lag was partly due to the location of the PuzzleMad Workshop, which is not convenient for me (or Mrs. S, for that matter).  (Ed - I NEED to know how to get there - it will allow me to make puzzles and even get a 3D printer!)

Conway’s Soldiers poses a simple question: Using conventional (but non-diagonal) peg jumping rules, how far above the “line” can you get your pegs? No amount of trial will ever prove an answer, of course, so in terms of a mechanical puzzle, it is not really a “solvable” problem. The question can only be answered (proved) mathematically, and Conway did this, showing that the fourth row is the maximum possible. His proof utilizes the golden ratio, by the way, which is fascinating in itself. It was later shown that allowing diagonal moves will get you as far as the eighth row. There are numerous web pages about the puzzle where you can study previous work and variations. You need to read the 2007 article by George Bell and his colleagues Daniel Hirschberg and Pablo Guerrero-Garcia, The Minimum Size Required of a Solitaire Army. Bell et al. solve a problem that actually does make a good puzzling objective: what is the minimum number of pegs needed to achieve a given row?

Look at those rings Kevin!
(Ed - drool! That is some beautiful wood)
Although not rigorous, you can kind of work this out yourself on the board. The trivial row 1 and 2 arrangements are shown below, mostly as an excuse to insert more pictures of my pretty game board. But I won’t give the solution to rows 3 or 4; you’ll have to read Bell et. al. or do the experimental work to figure those out. You can, however, assume that the number of pegs needed to achieve row four is equal to or less than the number peg holes on either side of my board (Ed - OMG - that looks quite a tough challenge). 
Solution for row 1, requiring two soldiers.
Solution for row 2, requiring four soldiers.
As you can probably guess, the PuzzleMad board is made from American Black Walnut and the pegs were scavenged from an old Setko puzzle. It is fairly simple to construct, just drill a grid of evenly spaced holes and insert a dividing line. Drilling the grid was tedious but straightforward. The line, in this case, is an inlaid half-rod of brass. Inlaying it was actually very tricky and stressful, but it did turn out well in the end. I learned in the process that you can use a drill press as a rudimentary router. I was afraid this might overly stress the bit, but it was actually a really smooth cut. Overall, a nice little DIY project for the amateur puzzle-maker.  (Ed - sob! One day I will make some puzzles...probably after I have retired)

In perspective.
John Conway has left us with a very enjoyable little diversion in the Conway’s Soldiers puzzle. Even if you cannot follow his fourth-row proof, you can surely figure out, eventually, how to arrange and sequence your army to achieve it. You can also, if you like, dwell on the more philosophical “diminishing returns” aspect of this problem. I think most people would intuitively believe that any level above the line is achievable, given enough soldiers, but the truth is that there is a hard theoretical limit, and that limit is surprisingly low indeed. 

That's about all I can muster for this mini-post Kevin, please take us home my friend...

My goodness! Thank you so much Mike. That was wonderful! You have given me (us) a nice bit of enjoyable reading to do and anything involving JHC and also George bell is always fun! I absolutely love the beauty of your home made constructions (even if they were apparently made under my home - I was totally unaware that I even had a basement).

Mrs S has had her booster yesterday and is feeling quite poorly - I had better go and offer her something sweet and sustaining as well as maybe some paracetamol or ibuprofen. Be careful out there everyone. Go and get your booster shots. The evidence seems to be that Omicron is significantly more transmissible and even if it tends to cause slightly less severe illness overall (we don't actually know that for certain yet) then by sheer numbers, a lot of unvaccinated people are going to end up in hospital or even dead. In our hospitals, the ONLY Covid admissions are the unvaccinated or the immunocompromised. The current vaccines with booster DO protect against severe illness. A Covid death is a very unpleasant way to die - go and get yourself immunised and help protect your health services as well as the vulnerable and yourselves.

New PuzzleMaster kickstarter campaign
Whilst you are here, there is a Kickstarter campaign being run by the PuzzleMaster guys: They are crowdfunding the production of three fabulous looking new metal puzzles. The skull is a design by Jerry Loo and consists of 67 interlocking pieces - it looks incredible! The other two puzzles are level 9 and 10 on the PuzzleMaster scale and also look fun.  This will run until New Years Day and certainly looks worth your attention - I have backed it (backer number 25).

Mind the gap
Also, my friend Andrew Coles, has created another puzzle lock, the "Mind the gap" puzzle. It will be available to order from his site quite soon. I have received my copy and hope to play soon. 

I have had no luck yet solving Boaz Feldman's latest creation - Loki, and suspect that locks are too difficult for me!


  1. Bravo, Mike! That is a beautiful board! Now maybe you can make one on a hexagonal grid? That is more difficult, and you'd need a larger board, and more pegs!


  3. I'm on it George, thanks! (might take a while...)

    1. Wow! I’m now looking forward to the next article in the series on peg solitaire variants! Thank you so much guys.



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