Is This Curve Infinite?

Hilbert from Abhishek Ruikar
This version in Oak

Out of the blue, a couple of months ago, I was contacted by Abhishek informing me of his latest creation and asking whether I wanted one. I had just mortgaged my soul to the devils (aka Dee Dixon and Tye Stahly) and asked whether he would wait a month for me to replenish my PayPal and also soothe over the disgruntled first wife (Whack! Ouch!)

He was very happy to wait for a bit and duly reminded me after a suitable period. Luckily, I had a little spare cash and it flew over the wires to India. This puzzle is available in Oak, Ash, Teak and Mahogany - I decided on the Oak version but they all look nice.

It drives in a nice green box (the corners of mine had taken a slight beating from the postal service) and inside is a little folded leaflet with the instructions (and if you need it, the solution).

Abhishek obviously has a penchant for knots and topology as his previous puzzle also involved wooden pieces with magnets that needed to be assembled into an intertwined shape. I needed to resort to Google to understand the name of the puzzle. I was aware of David Hilbert as a famous mathematician but did not know why this puzzle was named after him. Within a minute the reason sang out to me - the Hilbert curve is a fascinating concept describing fractal curves that can be both 2D and 3D space filling:

Hilbert curve courtesy of Tim Sauder

3D filling by Robert Dickau

When I removed the pieces, I realised that there were 8 of them and they all have similar but not identical shapes and now the reason for the name really became clear. This is a 3D filling puzzle:

8 very similar shapes with magnets!

I set to work making chains of pieces to try and form a cube. It didn't look too tough and was helped by the fact that the polarities of the magnets were the same on the equivalent ends of all the pieces. After about a ½ hour, I realised this was not quite as simple as expected. The pieces fit together making interesting shapes but after 5 or 6 of them the curve interested itself or blocked the insertion of the next piece. I made several dozen interesting shapes that didn't go anywhere:

Not getting anywhere!

Finally, I got fed up of random assembling of pieces and actually looked properly at the shapes that I had - there are 4 pairs of identical pieces which need to be arranged into the cube shape. It could not possibly be a random assembly - they had to work in a logical sequence which needed me to to some think©ing.

4 pairs of pieces

Once I understood this, then there was much less random trial and error. Thinking in terms of 6 faces and dividing up 8 pieces as 4 pairs was not helping me at all. It needed a few attempts at looking how the pairs could be arranged and thinking of edges (there are 8 in a cube) and I had a lovely little Aha! moment and managed to arrange my magnets in such a way that they all met end to end and formed a cube. Simply delightful - combining mathematics and mechanical puzzling cannot be beaten!

A Hilbert cube

Thank you, Abhishek, this was a delight. I have just realised that putting the pieces back in the box will be another challenge!

 I am sure that he would be delighted to sell you a copy if you contact him.

Two Half Voxels Make All The Difference

Sukiyaki by Frederic Boucher

I am catching up today with a puzzle from the Pelikan workshop that I did not manage to solve within the time frame that I had been given before they went on sale. Unbelievably, there are still 7 of these left available for sale. Stop reading now and go buy one - it is an amazing puzzle. 

This puzzle is one of three designed by Frederic Boucher named after Japanese foods (Frederic lives in Japan). Frederic wrote this about it:

“SUKIYAKI is the first in a series of 3 packing puzzles I created some years ago, named after famous Japanese foods (SUKIYAKI, TAKOYAKI and TAIYAKI). The box size is 2x3x4 units, with 3 openings on the sides plus the entrance on the top. At first glance the pieces seem identical, but there are actually 3 kinds of pieces. No rotation is required to solve this puzzle, but that doesn’t mean it’s easy. To solve this puzzle you’ll have to think “outside the box” to find the right configuration and then how to make the pieces interact with each other. I hope you will have as much fun solving it as I had creating it!”

Having failed to solve it for the sale, I have carried it with me for several weeks and have been playing with it intermittently for 3 weeks. It looks like a fairly straightforward packing puzzle with 6 identical S-shaped tetrominoes (made from Maple) to place in the 2x3x4 unit box (made from a rather lovely American Walnut). The opening seems quite large and there are 3 holes in the box for manoeuvring pieces out of the way to allow more to be placed inside. I genuinely didn't think it would be too much of a challenge (one reason, I had saved it to last in that batch) but, oh boy, I was very wrong!

Looking closely, you can see that whilst there are 6 S-shaped pieces, 2 of them have a half voxel removed from it and, for each of those, it is the opposite side of the piece. The obvious question to ask is, why? My initial thought was that these two pieces would interlock inside the box to form a whole voxel. As advised by Frederic, I set to "thinking outside the box" and created some shapes with the 2 odd pieces interlocked. Much to the amusement of several orthopaedic colleagues, I began swearing at myself under my breath and had to deny their urge to do the orthopaedic thing and hit it with a hammer. Let's just say that particular surgical specialty is better known for their brawn and not their brains, unlike my Urologist friend who is better at solving things down a long tube and surrounded by rather smelly liquid - hence his preference for copious amounts of alcohol!

Needless to say, I resisted the urge to use force and decided to use thought©. I worked out quite quickly that the 2 half voxels are not interlocking with each other and they must be present to make space for internal moves to happen in the box. I struggle to think© in terms of whole voxels and so half ones are really going to confuse me! This is why it took me 3 weeks to solve this fabulous puzzle. It was useful to know ahead of time that there were no rotations involved as there was a huge temptation to start turning pieces about inside or in the entry hole. Despite the relatively large entry hole in the top of the box, the pieces are quite restricted in how they can be inserted if rotations are not allowed. Having thunk outside of the box, and using the position of those single holes, I had found what I thought was the likely assembly and none of the gaps formed by the half voxels would be visible through the holes in the box. Of course the 2 odd pieces could be placed in several possible positions and I just needed to decide where so they would be useful. 

Interestingly, this ended up being solved as an assembly puzzle outside the box followed by a disassembly puzzle before actually packing the pieces inside. It really took a whole lot of logic and thought before I had my sequence and packed the pieces away for this photo:

Sukiyaki solved at last

The assembly sequence for this is absolutely stunning! It really cannot be solved by random movements unless you are willing to be completely exhaustive. The proper way to solve it is by genuinely applying multiple steps of logic. There was a huge grin on my face when the last piece went inside and a complete look of disbelief on my orthopaedic friend's face when it was done. In retrospect, I actually think this might be the pick of the bunch from Jakub and Jaroslav's last release.

There are 7 of these left in stock as I type this. Go buy it now! You really will not be disappointed!

I have been in discussion with Frederic about the Minima puzzle series and he has written a wonderful article on the whole series including additions by other designers. It is a wonderful, large and comprehensive document which is downloadable here. If you have any comments on this series which you would like me to pass on to Frederic then please get in touch and I will pass on your thoughts.

Minima(lly) Effective - So Others Try It

Minima Fibonacci series

I apologise for last week...I appear to have had Covid for the second time and it's just as nasty as ever! The only difference for me this time is that I had to obey the government's instructions and continue working with it if I was physically able. So having been poleaxed on Sat/Sun, I continued to go to my usual Monday meetings via Teams and then back to the operating theatre after that. I was told on a few occasions by colleagues that I looked shocking (and even once by a patient) but I soldiered on wearing my mask to protect others - not really sure how effective that will have been but that's the instructions. I'm currently on day 8 and still feeling pretty hellish but am at least more functional. Mrs S, on the other hand is convinced that I am trying to murder her with the virus. Only another week to go before we feel a bit better. Stay safe out there guys, it's still not a trivial thing.

On to the puzzling...

It has reached the point now that my friend Tye Stahly, knows what sort of puzzling I like and contacts me with news of things he is making that I will want to buy when they are released. He is a savage businessman as my PayPal keeps emptying into his bank account! Having fallen for the Minima challenge set by Allard and Frederic Boucher a few weeks ago, I couldn't resist it when Tye showed me that another renowned puzzle designer had jumped on the bandwagon with the Minima series and found some other wonderful challenges. One of the best designers in the world, Laszlo Molnar had decided that this was a good format. I have raved about many of Laszlo's puzzles over the years and been encouraged about them by my sadly missed friend, Felix, who saw the incredible talent and encouraged me to try them and even made me a whole load of them himself. 

The Fibonacci name could only mean one thing:

Fibonacci 1 pieces

Fibonacci 2 pieces

Yep! The Fibonacci series - polyomino sizes 1, 1, 2, 3, 5 all to be fitted into the 2x2x3 box to ensure that all the holes are covered (in fact the box is packed tight). Having said that the name makes sense, I am a little mystified about number 3:

Fibonacci 3 pieces
Still has 12 voxels but it just doesn't add up!

Tye has made them using his usual multicoloured wooden blocks and beautifully cut acrylic sheets - I am informed that it is actually Tye's own Mrs S who makes these and for that he is a very lucky man. I couldn't resist playing with them as soon as I was physically fit enough to concentrate on anything other than my chest. They are just as confusing as ever with the requirement to assemble shapes outside the box first being important and then gradually working out how to get the bigger pieces inside and what constraints they would have on fitting in the small ones. Just because there were single and dual voxel pieces does not necessarily make these easier - the 5 voxel shapes are very restricting and take up a lot of space inside. The requirement for rotations makes for a lovely challenge and interestingly, one of these is solved using linear movements only. I won't spoil it for you by saying which one.

Not to be outdone, the original Minima designer, Frederic Boucher, has extended his design prowess into flat (2D) versions - except the solution is most definitely not restricted to 2 dimensions. Not content with making me rotate blocks around in a box, Frederic has designed a tray packing puzzle - Flat Minima which is still available:

Flat Minima

In this wonderful little challenge the aim is to fit all the pieces inside the tray. Again, there is a limited entry way (in fact there are 2) and rotations are required. The end result is to achieve a 2D image of a 2x2x3 block inside the "box". 

How hard can it be? Well, I know that I am awful at tray packing puzzles and this has the added challenge of restricted entry and rotations as well as a viral infection - I figured I was buggered!

The first thing to do as usual is make your shape and that wasn't actually too hard. Next, work out a possible order of piece insertion. Finally, insert the bloody things. Erm! It's not quite that simple. The hole in the front and the side is not big enough to actually insert the pieces. This is why it's not really a 2D puzzle. The insertion of at least 2 requires rotational moves in the third dimension and can be quite confusing for am an of my limitations. Having achieved that, I worked on the remaining pieces and noticed that there is not a lot of room left after the first couple are in for the remainder to move about. The Aha! moment with this one is lovely. It probably only took me about a ½ hour but it was a delightful voyage. There's not much of a clue in the solved picture but look at your own risk.

Solved it. Looks like a 2x2x3

Finally, another of Laszlo's amazing designs was thrown in as a gift for me from Tye. The Hardcore puzzle is a sphere packing puzzle:

Hardcore
Pack these 3 shapes in the sphere with no force to close and a snug fit

This is one that Tye had asked me whether I had heard of it and might have the exact dimensions of the pieces. I searched my database and pictures for Hardcore and pronounced that I had no knowledge of it. When this duly arrived, I had a sneaky suspicion that I had seen it before but another search of my database did not reveal that name. I worked on this for a day and solved it with another lovely Aha! moment before realising that I had definitely seen it. In fact, Laszlo had used me as a guinea pig for it before he had come up with the name (hence I had come up blank) - I showed it off back in 2017.

Laszlo's prototype (he called it the Spherical packing puzzle)

I have to apologise to Tye for not realising and helping more - I had seen it before it was named. 

It is currently for sale here and is well worth your attention. It is not impossibly hard...it is just lovely and nice to show to non-puzzling friends as a gentle challenge.

No post this week

Sorry guys, I’m too sick to write anything today. Caught a nasty viral thing at work and am spending as much time in bed feeling sorry for myself as I can get away with.

Sniff, sniff, cough, sneeze!