Metal Puzzle 2 – Clover


A couple days after getting G Clef, I started thinking. Two of my adult students like seeing some of the stuff I buy here, and they both like playing with the puzzles, although neither of them can solve them without help. Anyway, it might be interesting to get one more metal puzzle and see how they react. So, I went back to the capsule dispenser and put in another 200 yen ($1.80 USD). I turned the crank, hoping to get one of the most difficult ones in the machine, vaguely thinking of Clover as something I’d settle for. After the ball dropped out, I opened it and there it was – Clover. Fully extended, it’s about 6″ long, and made of light steel. The goal is to remove the 3 rings from the chain without breaking, bending or cutting anything.

This was undoubtedly the most frustrating puzzle I’ve successfully solved in a long time (some of the wood puzzles were annoying, and at least one I needed help on from youtube. But Clover is in a class by itself.) As I mentioned in the post on G Clef, I’m not good with these kinds of things, and I don’t like playing with them. But, after about 15 minutes I hit on the right idea for getting started. It’s just that the pieces are so slippery that I couldn’t get them to hold in place as I got closer to the solution. So, it took me over another hour to actually take it apart. Now, I’m down to about 3 minutes to remove the rings and then put them back on the chain. Woof. Not a bad challenge for the price.

 

Brain Works, comments


(All rights belong to their owners. Images used here for review purposes only.)

I got three books for Christmas. The first one up is Brain Works, the companion book to the National Geographic TV series. I haven’t seen the series, so I’m treating the book as stand-alone. It’s written by Michael S. Sweeney, who has authored other books for Nat. Geo., and is a professor at Ohio University’s E. W. Scripps School of Journalism. Overall, the writing is pretty clear and easy to follow, although there is an inordinate amount of medical jargon when it comes time to talk about various regions of the brain and the functions they perform, which kind of gets in the way of the message that Sweeney is trying to deliver (i.e. – how those functions mess you up in terms of optical illusions or cognitive misfires).

The book is about 220 pages long, and broken up into 3 major sections – Seeing, Thinking and Being – plus there’s a foreword by David Copperfield. The foreword is kind of irrelevant, in that Copperfield tries to frame this work in terms of how the brain tricks itself for illusions and misdirection as it relates to stage magic. But, actually, Sweeney uses things like optical illusions, psychological experiments and quotes from experts in various fields to explain what we currently know, and don’t know about how the brain is designed, and what it does. There’s very little that directly ties to direction and exclusive focusicity. That is, as a magician, I’m not going to use Brain Works as a guide book for how to trick audiences better. It may help understand WHY humans can be tricked, but it’s not going to tell me how to do that more efficiently.

Seeing talks about the parts of the eye, how it responds to light, how signals get sent to the brain along the optic nerve, and what happens when the signals get distributed in the brain for analysis, interpretation and response. There are 8 sub-chapters, each of which includes at least one illusion (with trick paintings from Giuseppe Arcimboldo and Bev Doolittle), talks about the part of the brain involved in the illusion, and supplies supporting example medical case studies.

Thinking has 7 sub-chapters, and it goes through the current theories of how our brains evolved, and how what we think we remember can change with time and repeated recollection of an event. There’s one case study on false memories (one psychologist had a nanny who faked an attempted kidnapping when the guy was very young and manufactured a story about the attack. The psychologist still has memories of the event, even though the nanny later confessed to making it all up.)

Being gets into the roles the brain plays in conscious, subconscious and unconscious thought. The current theories hold that subconscious reactions affect your decisions and emotions a lot more than had previously been believed, and that expectations can drastically impact how you approach those decisions. There’s also a fair amount of writing on artificial intelligence, and speculation on how machine AI will never match or surpass the human brain.

My opinion is that this book is great for anyone that has seen the TV series, and wants to explore some of the experiments in more detail. It’s also fine for the casual reader that is interested in brain function. But, it is superficial, meant more as an overview than as a doctorate-level textbook, and as such left me feeling disappointed. I would have liked a lot more examples of optical, auditory and sensory illusions, and more psychological experiments. In fact, a full book of “things you can try” would have been more fun, and the medical quotes and case studies should have been left for the appendix.

Two comments, though. Sweeney discusses computers as tools, and states that tools are designed to perform certain tasks, and as such computers are good at number crunching and sorting through vast amounts of data quickly, things that humans are poor at, but that computers can’t become intelligent. That computers are bad at facial recognition and understanding poetry. Duh. Digital computers based on binary instruction sets were designed for number crunching, so yeah, that’s what they’re going to be good at. If you want facial recognition, or cognitive functions, start over and build a new machine specifically for those tasks. Don’t design a hammer for pounding nails into concrete, repurpose it as a paint brush, and then claim that hammers will never replace paint brushes in the house painting industry. Can humans make machines smarter than us? I think so. If that’s what the original design specs call for when you create an all new machine. If you want machine AI, design the tools to be machine AI’s upfront; don’t pull a number cruncher from off the shelf and then complain that all it’s good for is number crunching.

Second comment: In the section on Being, there’s a sub-chapter regarding pain. The “take away” is that you can trick the brain into thinking that pain is not that bad through a combination of expectation, and ending on a high note. That is, if you tell a patient that they can control how and for how long a certain procedure will last, they’re more likely to tolerate it than if you make all the decisions for them and they’re just in the room as a passive subject. And, in one example, male patients being give colonoscopies were divided into two groups. In one, the scope was removed right after the work was completed. In the other, the scope was left in place for another 20 seconds before being removed. The second group reported that they were more likely to get exams in the future because the procedure wasn’t all that bad. The reason? The scope had stopped moving for a while and it didn’t hurt as much after it was pulled out. Meaning that prolonging the time the scope was in place actually made the overall experience feel better because it ended on a less painful note, very clearly tying pain and pleasure together. What’s strange, though, is the discussion is accompanied by two pictures of an attractive, undressed woman. In the first picture she’s screaming in pain, and in the second she’s apparently enjoying something fairly intimate. But, there’s no text box saying why these pictures were chosen. The reader is left to guess if this is a suggestion about how to conclude a rough date.

Anyway, more optical illusions. That’s all I’m saying.

 

Prime Eval, Part 18 – Topology


Topology as a branch of mathematics goes back to the 1700’s, arguably to when Euler wrote his 1736 paper on the Seven Bridges of Konigsberg. It’s an outgrowth of geometry and set theory, and is a relatively old branch that wasn’t taught very well when I was in school. In fact, the only time I’d really heard of it was when I was in my 30’s and some science writer posed the question of how donuts are related to coffee cups.

In reading Shing-Tung Yau’s The Shape of Inner Space, I’ve gotten a somewhat better understanding of just what topology is trying to do. One thing that’s interesting to me is Yau’s position that gravity is all about geometry, and that space has a shape. The goal of String theory, and by extension M-Theory, is to determine what that shape is from a geometrical viewpoint. And that’s where topology comes in.

When you have two objects, there’s the question of how to convert between them. We can see this in examples of anamorphosis, in which an object has been painted such that you have to be standing in a specific location or at a specific angle to see it properly. Actually, making a 2D painting of a 3D object is a form of conversion. So, when you have two objects that are both three dimensional, having a conversion formula for mapping the surface of one object onto the surface of the other revolves largely around whether the objects are of related types. Topologically, the coffee cup and the donut each have only one hole, and, using silly putty, you can massage one into the other without tearing the surface to make a new hole, or plugging up an existing one. Therefore, taking a picture painted on the donut, and then mapping it onto the cup will be both smooth, and easy. That’s not the case if the first object is a canvas painting of the Mona Lisa, and you want to map that onto the donut. The result will have a wrapping error, and you’re not going to get the same image if you map it from the donut to the cup and then back to the canvas.

What this means is that topology is about multidimensional shapes, and what the most basic shapes are for a given set of dimensions. And that depends on how many holes you have in the shape. A golf ball, a silver dollar and an egg are all examples of the sphere object (no holes); while the donut is a torus, and glasses frames, certain types of pretzels and a pair of scissors with the blades fused together are 2-holed torii.

I haven’t finished reading Inner Space yet, so I’ll put off talking about that for a while. But, I was thinking about the issue of relating a sphere to a torus. The point is that to get a sphere from a torus, you either have to tear the torus apart, or you have to plug the hole. Now, you can make the hole look like it’s disappeared by making it shrink, and then moving it really close to the surface of the torus. In fact, that’s kind of what you get when you pierce someone’s ear really, really close to the edge of the lobe. The result would be a sphere with a very tiny loop that goes below the surface and then immediately comes back up again. The donut would be almost indistinguishable from a sphere, but the donut hole would still be there if you knew where to look. So, no happiness that way.

How about this, then? Treat the sphere like a big ball of bread dough, and roll it out into a long, thin cylinder. Split both ends in half length-wise so they look like two fingers giving a peace sign. This is bifurcation. Split the end of the fingers in half so both ends now have 4 small fingers each. And keep repeating the bifurcation as if you’re making a tree root system. This is a very simple process, it doesn’t change the topology of the sphere, and you can still reverse it to go back to the original sphere. Continue to infinity.

Here’s the trick – before you start the bifurcation, curve the bread dough so the opposite ends of the cylinder face each other and are almost touching (ignore the stickiness of the dough for a second, because at this point the object is still a deformed sphere with identifiable surfaces). Now, initiate the bifurcation process, and extend the fingers from both ends so that they interweave. As the fingers get smaller and smaller, they fill the gaps between the two ends and get entwined like the world’s worst case of velcro locking. Continue to infinity. Writing the formula for the bifurcation and applying it to the sphere still leaves you with a sphere, topologically. But, if you curve the cylinder and aim the ends at each other, you’re going to create an object that will be virtually indistinguishable from a torus (now, make the fingers sticky and imagine what you’ll get). At a minimum, taking such an object and mapping the bifurcations back to a sphere may well be close to impossible. Is it possible then to say that a curved, velcroed sphere is topologically equivalent to a torus? Would untangling the knots be the same as cutting through the torus? Would there be a workable reverse-conversion formula?

———–

This entry makes the last of my rambling on math and other stuff I don’t know very much about. At least, for now. You may commence rejoicing.

Metal Puzzle 1 – G-Clef


I’d kind of cleaned out the capsule ball dispenser in trying to get Plain Cross a few weeks ago. After that, I wanted to hold off on spending any more money on capsules, so I didn’t bother visiting the shop right away. Finally, I figured that I’d at least check whether the dispenser was still there, and if it was refilled. Instead, the machine was used to hold an all-new series. This time, simple metal puzzles. I’m not that good at these, and as a result I don’t have much interest in them. But, I did want to get just one for the blog. (There’s a total of ten in the series.)

This one is “G Clef”, and is probably the simplest of the set. About 3″ long, 200 yen ($1.80 USD). Took me less than a minute to take it apart and put it back together again. Virtually no challenge at all.

Switched-On Bach 2000 comments


(All rights belong to their owners. Image used here from Amazon for review purposes only.)

I want to like Wendy Carlos’ work, I really do. But, she doesn’t make it easy.
When I started messing around with synthesizers a few years ago, one of the things I wanted to do was get my hands on a copy of the original Switched-On Bach, to see what she (at the time, Walter) had done using the Moog synths. Unfortunately, S-OB was long out of print, and it wasn’t until this Christmas that I received a used copy of the 2000 25th anniversary re-master.

I can’t remember much at all about the original S-OB, except that it had received a lot of airplay on the rock FM radio stations, and that I liked the one song that the stations kept replaying. At the time, I didn’t know anything about electronic music (I was 11 when the album came out in 1968), but I thought it sounded cool. (Hot Butter released their version of Gershon Kingsley’s “Popcorn” in 1972, and THAT I REALLY liked. I became very interested in synth music at that point, but all the music teachers and majors I talked to at the time were convinced the “pock” sounds in the song were the result of African wooden blocks, and weren’t done on the Moog.)

Anyway, the reason I wanted the S-OB CD was to hear the original tracks again now that I have a better understanding of synthesizers. But… anyone that has heard S-OB 2000 already knows that Wendy re-wrote all the songs to be more in keeping with Bach’s original spirit of the music. She wrote 20+ pages of liner notes for the re-release detailing why she hated working with the early synths, and how she disliked the “turgid messy” sounds they made. Part of the notes describe the new tunings she used for each song, and how this makes the songs closer to what they would have been like in Bach’s period. Really, Wendy’s focus was on the use of electronic instruments to make an “authentic-sounding” reproduction of Bach’s music, rather than re-interpreting it to take advantage of things synths can do that acoustic instruments can’t. It’s another example of using a $5,000 synth to imitate a piano, when you could have just as easily just bought the piano outright.

S-OB 2000 sounds so generic, like something anyone could have played. I’m not a connoisseur of classic music, and I can’t tell the difference between a “well-tempered” tuning and a “medium-rare” one. To me, all the songs seem the same – pretty much an acoustic interpretation of standard, well-known classical music. I guess if you want to know what the music would have been like if Bach himself had played it on a period instrument, S-OB 2000 would give you that experience. Instead, I wanted something a little more along the lines of what Tomita had done. There are some places where I can pick out little glides, or buzzy squarewave notes, but they don’t stand out at all. All the instruments sound too much like regular acoustic pianos and such. Where are all the synthesizers?

Summary: Switched-On Bach 2000 is Wendy Carlos’ vision of what Bach’s music would be like if you were listening to him live. It’s nice and all, but if what you want is the sound from the original 1968 album, you’ll be disappointed. Recommended only to classic music buffs.

Prime Eval, Part 17 – Heavy


Gravity is a weird thing. As Walter Lewis says in For the Love of Physics, it’s not something that most people think about. If we do think about it, we limit ourselves to the pull of the Earth on us, and possibly the Moon’s pull on the tides and the Sun’s pull on earth. Very rarely, someone will talk about gravity wells, as if there’s a “wall” around the planet that a rocket has to climb out of to reach outer space. But, as Einstein put it, gravity warps space-time, so it’s not so much that we have an attractive force that must be surmounted, as it is that space is more concentrated around larger clumps of matter, and we have to work harder to get past that space to escape that pull.

But, that pull goes on to infinity. Think about it – entire solar systems spiral around a galactic center. The gravity of the Milky Way as a collection of masses is so great that it’s pulling on our sun, and it’s not even like the center of the Milky Way is all that close to us. And then we have clusters of galaxies that rotate around each other, or in which the pull of masses from one galaxy is acting on the masses of another one. Einstein predicted gravitational lensing, where light from one galaxy is bent by the gravity of another galaxy it passes by, and the lensing can act like a gigantic focuser that we can use to view things that otherwise may be too far away to observe conventionally. And it is true, we’ve discovered examples of this phenomenon happening. That gravitational warping of space is all-pervasive.

The thing is, we can’t sense smaller variations in gravity normally, and it’s not like any changes are going to occur quickly. Gravity as we perceive it is a smooth collection of additive forces. And, we are going to be influenced more by the Earth, Moon and Sun than by Jupiter and Mercury, even though all of the planets are pulling and tugging on each other and the Sun to produce chaotic orbits. We know that Jupiter, Uranus and Saturn are making the Sun wobble as they go around, but we don’t see that as it’s happening.

Which brings me to my point – Have you ever wondered if you’re heavier in the middle of the night than you are during the middle of the day?

Gravitational forces are vectors, and we can do vector math with them. The combined vectors for gravity from the Sun and the Earth as they work on us as humans will be at a maximum when the Sun and the Earth are in a straight line “under” us, and a minimum when the Sun is directly over head (the Sun’s gravitational pull will be fighting against the Earth’s from our viewpoint). So, if sunset is at 8 PM and sunrise is at 6 AM, then “mid-night” would be at 1 AM, and that’s when you would weigh the most (the Earth and Sun are both pulling on you in the same direction). “Mid-day” would be 1 PM and you would weigh the least (the Sun is pulling “up” on you as the Earth is pulling “down”). Couple this with the Moon, and the biggest impacts would be during the lunar and solar eclipses, when everything is in more or less straight lines. If you’re in the U.S., you’d weigh the most when Japan experiences a solar eclipse, and the least when you see one.

And then, when you get a cosmic alignment of all the planets…!

Anyway, the next time you complain of having a heavy gravity day, check to see where they put the moon.

PlainCross 3D Puzzle


This is the same puzzle as demonstrated in the American Woodworker video. I figured that I would like to complete the capsule ball set by trying to get this one as well, in the hopes that it would be as clean and elegant as shown in the video. I’d also gotten paid for the month and was feeling a bit flush. There seemed to be 10 puzzles remaining in the dispenser, which would work out to 2,000 yen ($18) if I had to clean it out, and thus proving if PlainCross was even in the machine or not. At this stage, it wasn’t about the money, but about verifying the math. On the plus side, I’ve been showing these puzzles to my students, some of whom are old enough to have children of their own, and they seem to be intrigued by them. So, I gave away the extras as Christmas presents.

I just started throwing money in the machine. When a capsule came out, I’d pop it open, drop it on the floor, and try again. In short, I got 4 CrossNeos (which is good, because that’s the one everyone likes), 1 Galaxy, 1 Diamond and 1 GetaCross. I got PlainCross on my eighth try, and at that point I really couldn’t tell how many capsules were still in the dispenser (maybe 3 or 4).

PlainCross is disappointing in that it wasn’t machined very well. The locking bar of the pin piece is too thin, so the puzzle sags when you prop it on its corners, giving away the secret to disassembling it. Beginners may still have trouble putting it together again, but it’s sort of self-explanatory if you’ve worked with 3D puzzles very much at all. There are three pieces, about 5 cm long. Two are identical, and the third is the pin piece. Take one of the identical pieces, and put it on the table with the big notch pointing up, and the half-notch facing to the right.

Take the pin piece and put it into the notch of the first piece to form a cross, and rotated to have the crossbar at the upper left. Slide the pin piece to the right to leave a long half-notch gap as shown in the photo.

Pick up the remaining piece and hold it upright, at 90-degree angles to the other two, with the large notch facing to the right. Slide it into place. Then rotate the pin piece to lock the puzzle.

Anyway, I now have the complete 6-puzzle set, with 7 duplicates I gave away to my students. Hopefully this will tide me over for a few years, until I get a larger apartment.

Big Beat Boppers comments


(All rights belong to their owners, Images used here for review purposes only.)

Full disclosure time again (not sure if I want to make a habit of this or not). I received this CD as a gift from Bon. He produced it and has 2 songs on it from his bands Bon DX and ARTS. It’s a sampler of Japanese reggae and Ska bands from around Kyushu. I’m not familiar with any of the other bands, and in fact I still haven’t heard ARTS performing yet, either. But, this is a good introduction to the various groups if you’re interested in this kind of music. A couple songs are in Japanese, the rest are in English with attempts at getting the accents right. I like Ska, but I’m not as big a fan of reggae, and I don’t care for the religious elements that permeate some reggae songs. So, I do like all the Ska pieces on this CD, especially the two by Bon. The reggae songs are ok, but I don’t listen to them as much as I do the others.

If you’re interested in hearing more, either get the CD, or check out the bands on youtube and then get the CD.

Wave Islands (from Miyazaki)
Rude Vibes (from Kagoshima)
The Rhinocerosmiles (from Kumamoto)
The Convictions (from Fukuoka)
ARTS (from Kagoshima)
Ram Jam Vendors (from Fukuoka)
Club Sandinista (from Fukuoka)
The Little Bitch (from Kagoshima)
DUB creation (aka Dub Rockers) (from Fukuoka)
SKA’SH Onions (from Fukuoka)
The Holidays (from Fukuoka)
The Ska-Phonics (from Oita)
Bon DX (from Kagoshima)
V.S. Honour (from Fukuoka)
No Kidding (from Kumamoto)
Vic Bongo (from Fukuoka)
The Explosions (from Fukuoka)
The Dicksweets (from Miyazaki)

Prime Eval, Part 16 – Spin


I once posted a question on a science board regarding whether the universe rotates, which would explain expansion without needing to resort to repulsive forces like dark energy. I got one reply back saying that I need to learn science more better. That expansion of the universe is more like inflating a 3-dimensional balloon. The universe isn’t expanding to fill outside space, the “balloon” is just getting larger. Which means that there’s nothing “outside” the universe to spin relative to.

I find this argument unsatisfying. Everything spins or has rotation, from electrons and subparticles, up to planets, stars, galaxies and galactic clusters. Why would this feature stop when you get to the universe? As to what it would spin relative to – the outer envelop would rotate relative to the center.

Think of it this way. If the Big Bang theory holds true, then we have this explosive release of energy that expands until it cools down to allow the formation of matter and the transmission of light. This expansion continues until it stops, if the universe is a closed system, and then reverses to recompact and repeat the cycle all over again. The conventional view of this cycle is straight line. Our galaxy goes straight out and then straight back. But, why would it do that?

In other words, what would cause the universe to slow down and contract again? The conventional view had been that the total mass of the universe would apply an attractive force on all matter (i.e. – gravity) that would overcome the expansion velocity. It’s not that the universe is getting bigger, but that there’s more space between galaxies; or, the space between galaxies is increasing. If gravity within a closed system is enough for it to cause contraction, then the space between galaxies would decrease and the galaxies closest to the “core” would probably collapse into an ultra-massive black hole that would then speed up the collapse. And what does matter do when it’s approaching a greater mass? It spirals. And galaxies spiraling around the core would influence their neighbors. Given enough time, the spirals will all go in the same direction. Hence, the universe would rotate relative to its center. Just reverse this process and ask what happens when something with an outward trajectory slows down under the pull of gravity and begins to reverse direction. Attractions to nearby galaxies will cause each to pull towards the others sideways, developing an angular velocity relative to the center – they’re not going in straight out-and-back lines anymore. Angular velocity implies rotation, even if it’s not uniform and all in the same direction. This still allows rotation of the universe, and that rotation would be relative to the center, not to something “outside”.

Which brings me to the idea of “the center of the universe”. According to the conventional theory, because expansion is like with that 3D balloon, and it’s straight-in-straight-out, there’s no technical “center”, and no “outside”. So, let’s do this. Let’s do a search of all observable galaxies, and assign directional vectors to them. We may not be able to tell exactly which direction any given galaxy is going because we may have to take measurements over millions of years, but when certain galaxies “collide”, the dust trails resulting from the collision will give us clues as to where they came from and where they’re going. And if we have enough vectors, we should be able to draw oppositional lines to the general direction of the “core”. And if the core seems off-center from all the galaxies we can see, then that might position us relative to the rest of the universe. In other words, there’s got to be an “outer edge” and we’re either closer to, or farther from it than our neighbors are. Then, start galaxy hopping. Go from one galaxy to the next in the direction opposite from the core. If the universe is closed, and will eventually start contracting, then it’s not infinite. If it’s not infinite, then at some point, as you’re galaxy hopping away from the core, you’re going to run out of galaxies to hop to. What’s that going to look like if you’re facing outward and there’s nothing left to jump to?

I’ve been reading Walter Lewis’ For the Love of Physics. Interesting book; it’s a combination semi-autobiography/informal physics lecture. Lewis was an astrophysicist before becoming a professor of undergraduate physics at MIT. He mentions that the space between galaxies is expanding faster than the speed of light. Think about that – there are galaxies that are being formed right now, at the opposite side of the “observable universe” that are so far away, and moving so fast because of expansion, that the light from them will NEVER reach us.

Which brings me to a question: Can God create a genie so powerful that He can’t put it back into the bottle?

If God created the universe, and the universe isn’t closed, then how is He planning on rebottling the genie and putting the stopper back? Or, was the initial concept that there’s not SUPPOSED to be a reset button? This really does make the question of “what existed before the Big Bang” a lot more intriguing. All this energy and matter had to come from somewhere… And, if the universe isn’t closed, will there ever really be a “heat death”? Will the universe’s temperature go to 0, or will the space between stars become so great that it just looks that way to whatever civilizations are still out there?

Pyramid 3D Puzzle


Ok, the day after I wrote the entry on CrossNeo, I went back to the capsule dispenser shop and convinced myself that I would keep spending money (200 yen, or $1.80 USD, each) until I got one of the two remaining puzzles. Largely this was because I’d failed to watch the rocket launch from Tanegashima island and I was feeling frustrated. Up to this point, I had 4 unique puzzles out of the available 6 – Galaxy, Diamond, GetaCross and CrossNeo (in that order); leaving PlainCross and Pyramid. I put in my money and got Galaxy again. I put in more money and got Diamond again. I started banging on the dispenser in the hopes that that would have some kind of positive effect. On my third try, I got Pyramid.

While it looks good in the photos, this is the easiest puzzle of the group, and is kind of disappointing. It doesn’t help that the surfaces are so smooth that it falls apart when set on smooth tables, or when you try to pick it up. If you want to display this one, you may want to glue it together first.

It consists of 6 pieces: 4 pieces that have three lathed spheres, and 2 pieces with 4 lathed spheres. To reassemble it, put one 3-sphere down on the table, and prop two others against it as shown in the photo.

Lay a 4-sphere on the two standing 3-spheres to make the leading rising edge.

Lay the other 4-sphere piece down flat on the table alongside the 3-sphere piece that you started with to complete the three horizontal edges of the pyramid. Finish by putting the last 3-sphere piece in the gap facing you. It took me less than a minute to figure this puzzle out…