Colossal Gardner, ch. 34

The section on Combinatorics ends with Bulgarian Solitaire and Other Seemingly Endless Tasks. Gardner starts out with a simple example. Say you have a basket of 100 eggs and a supply of cartons. Your job is to put the eggs in the cartons with the following steps – you can put an egg in a carton, or take one egg from a carton and put it back in the basket. The procedure you follow is: After each two successive packings of an egg you move one egg from a carton to the basket. This is inefficient, but eventually all of the cartons will be packed. Assume, now, that the basket can hold any finite number of eggs. The problem is unbounded if you start with as many eggs as you like, but once you specify the number of eggs, a finite upper bound is set on the number of steps needed to finish the task.

If the rules allow you to transfer any number of eggs back to the basket whenever you like, there’s no longer an upper bound on the number of steps to complete the job, even if you start with only 2 eggs in the basket. “Depending on the rules, the task of packing a finite number of eggs can be one that must end, one that cannot end, or one that you can choose to make either finite or infinite in duration.” Martin continues on to introduce some mathematical tasks where it would seem intuitively true that you could keep delaying finishing the task forever, but that are impossible to avoid completing in a finite number of moves.

One of my favorite people was Raymond M. Smullyan, who died on Feb. 6th, 2017. He started out as a stage magician, and moved over to math and logic. I read several of his books of logic puzzles in the 80’s, and they were a lot of fun, although they did get repetitious after a while. I’m sad that I’ll never get a chance to meet him or see him in a live situation. He wrote a paper presenting the following problem. You have an infinite supply of pool balls, each marked with a positive integer, and for every integer there is an infinite number of pool balls. You also have a box containing an infinite quantity of numbered balls. The goal is to empty the box. The steps are to remove a ball and replace it with any finite number of balls of lower rank. The exception is the 1 ball, which has no replacements. While the rules allow you to take out a ball marked with 1,000 and replace it with one billion balls marked 999, it would seem that you’d never finish this. The problem as stated is unbound, so there’s an unknown number of steps, but because the numbers on the balls keep getting smaller, eventually you’ll only have 1s, and then the box empties in the same number of steps as there are 1 balls. Smullyan’s proof is straight-forward: Start by throwing out all the 1s. When you get to a 2, replace it with any number of 1s you want, then throw out the 1s again. Eventually, you’ll run out of both 1s and 2s. If there are any 3s in the box, throw one out, and then you’re back to the issue of dealing with 1s and 2s again, which has already been solved. Etc.

He also proved the game ends by using a tree model. “A “tree” is a set of line segments each of which joins two points, and in such a way that every point is connected by a unique path of segments leading to a point called the tree’s root.” For the ball game, each ball is represented by a point, numbered the same as the ball. To simulate filling the box with balls, join the points to the tree’s root. When a ball is replaced by other balls of a lower rank, erase the number of that one and all the intervening numbers are joined to the spot where the old one was. The tree will now grow upwards, it’s end points, points that are not the root and are attached to just one segment, indicate the balls still in the box. If the tree becomes infinite, at least one branch must extend upward forever, but that can’t happen because the numbers on each branch steadily decrease. Since the tree is finite, the game has to end at some time.

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

Smullyan’s theorem comes from Cantor’s work on transfinite ordinal numbers, and is related to Dershowitz’s and Manna’s work on the computer halting problem. The above figure shows a special case of Smullyan’s ball problem. We’re allowed to snip off any branch of the tree and replace it with as many other branches anywhere else we like as long as they are a lower rank. The tree may grow bushier, but it will eventually “get chopped closer to the ground until it eventually vanishes.”

(Fighting the hydra.)

The next illustration is for Hercules versus the Hydra. Every time Hercules cuts off a head, its segment goes with it but more heads grow from the next point down. The Hydra may become extremely wide, but at some point Hercules will slay it permanently.

(Line segmenting task.)

Then we get into the 18-point problem. Start with a line segment. Place a point anywhere along it you like. Place a second point so that each of the two points is within a different half of the line segment (not including the end points of the segment). Then add a third point so that all three points are in different thirds of the original segment. Keep doing this carefully so that all n points are in different 1/n parts of the original segment. How big will n get? The trick is that the points must be sequentially numbered, and it is impossible to put down 18 points without violating the rules of the game. The above illustration shows one solution.

Finally, Bulgarian Solitaire. First, we have to go back to “triangular numbers.” Partial sums of the series 1+2+3+4… are called triangular numbers because they correspond to triangular arrays, like with the 10 bowling pins or 15 pool balls. If you are going to work with any triangular number of playing cards, the highest you can get with one deck is 45 (from summing 1-9). “Form a pile of 45 cards, then divide it into as many piles as you like, with an arbitrary number of cards in each pile. You may leave it as a single pile, […] or cut it into 45 piles of one card each.” Next, keep repeating the following procedure. Take one card from each pile and place all the removed cards on the table to make a new pile. Stop when you have 9 piles, with 1 card in one pile, 2 cards in the second, 3 in the third, etc. Turns out that Bulgarian solitaire is a way of modeling problems in partition theory. It can be modeled as below for the 6-card case. As shown in the diagram, you’re never more than 6 steps from finishing the game (piles of 1, 1 , 2 and 2 cards are considered “worst case”), assuming you don’t get stuck in an endless loop where the states don’t change from step to step.

(Bulgarian solitaire tree.)

The below diagram is the tree for 10 cards.

The rest of the chapter goes into more theory and a little history (involving G. H. Hardy and Srinivasa Ramanujan), and a comment that partition theory, in the form of the Young tableau is used in particle physics.

Game: Taking two decks of cards, try playing Bulgarian solitaire with 91 cards (1+2+3…+13). See how long it takes to reach the end state, based on how you divide the piles at the beginning.


Proof, The Science of Booze, comments

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

I received three books for Christmas this time, all science and math-based, that I’d requested. (I also received vol. 8 of The Norm newsletter/comic strip collection, but that was for a different reason.) The first book I sat down with (after finishing The Norm #8) was Adam Rogers’ Proof: The Science of Booze. I wanted this one because I’d read Tom Standage’s A History of the World in 6 Glasses when it came out in 2005, and I was wondering how much overlap there’d be. Standage is currently the deputy editor of The Economist, and Rogers was the articles editor at Wired magazine when this book came out. So, you’d expect some variations in writing style at a minimum. One thing that struck me right away about Proof is that Standage and 6 Glasses aren’t mentioned at all in the index or bibliography. I don’t know if this is a deliberate slight, or an intentional decision. I liked 6 Glasses, and I felt that the history element in the creation of beer, and the fact that Mesopotamian beer was a sweet watery millet mixture, would have a place in Proof in showing how beer evolved in some way.

Actually though, Rogers doesn’t come right out and say that he wasn’t going to get into the history of alcohol until the second to last page of the afterword. There’s history throughout the book, which according to Rogers was the simplest way of explaining the science, just not really complete history, and not the level of what’s in 6 Glasses. Which to me strikes at the core of what left me disappointed in the end.

Proof came out of an article Rogers wrote for Wired about black mold that was growing outside the Canadian Club distillery, and the search for what caused it and how to get rid of it. The book is a very detailed description of the human-based processes of making ethanol under controlled conditions, and is divided into the chapters on Yeast, Sugar, Fermentation, Distillation, Aging, Smell and Taste, Body and Brain, and Hangover. In order to say what yeast is, and what it does to make different kinds of alcohols (beer, wine, rum, sake) within a scientific context, Rogers does recognize that talking about how no one even knew that yeast existed until relatively recently does make it easier to come out and say that there’s still a lot we don’t know about it. Same as with the other topics. My problem, though, is that the book reads like one long Wired article – lots of breathless writing, the tossing around of terms that aren’t explained with the expectation that the reader already knows what they mean or that they’ll look them up on wikipedia (such as with the difference between pot and columnar stills, or what all the varied chemistry terms are). He does explain some science words a bit at the outset, then veers off to describe his visit to some lab, or distillery or bar or other, before jumping to a completely different topic, and then (maybe) returning to an earlier interview with someone, followed with yet another diversion elsewhere. Yeah, there’s science all over the place, but it’s like it’s being presented by someone on speed, not booze. I kept getting this “too cool for you” impression seeping between the lines. I never felt lost at any point, but I’m also not one of Wired’s targeted readers. In the end, I don’t feel any smarter than when I started, but now it’s like I have a whole bunch of tidbits and trivia I could “amaze” people with at a party. (Such as the one about the Harvard medical student that participated in a study on the effects of alcohol which was administered to him rectally. Want a name? He was only identified in the study as “Ius”.)

I did finish Proof in 2-3 days, and there was a lot of stuff in it that I did like reading about (the study of the black mold was good, as was the background on Jokichi Takamine). In fact, I would have liked to have read even more on Takamine. He was born in Japan in 1854, just before Japan was opened up to trade with the rest of the world. He attended the University of Tokyo as a chemistry major, and did his postgraduate studies at the University of Glasgow. He was the one that first figured out how koji (mold) works in converting starch in rice (for sake) and sweet potatoes (for shochu) to sugar for the yeast to eat, and came this close to setting up a koji factory in the U.S. to replace malt in the manufacture of whiskeys (which would have cut the time in making whiskeys significantly) except that a jealous malter burned his lab to the ground. Takamine went on to produce the first epinephrine extract, and he filed a patent on adrenaline in the U.S., and he was the one that donated many of the Japanese cherry trees now in Washington, D.C. So, yeah, I didn’t know that stuff before, and I live in an area where shochu is BIG. (One of my students works at a shochu distillery, another is trying to get a shochu certification, and I just recently found out that an American English teacher living nearby is in the process of becoming a bartender at a shochu bar, or something like that). Yes, that part was good.

After I finished the book, I was kind of intent on harping on that “we’re too cool” Wired vibe that I feel got in the way of my enjoyment of the writing, and the lack of a more solid historical perspective to alcohol. However, as I was looking for links to the book and to Rogers on the net to use in this commentary, I stumbled across an interview between Rogers and one of the guys from Tested. And Rogers comes across as a really nice guy that really knows what he’s talking about. If you have any interest in reading Proof, I suggest that you watch the below video first (but, ignore the comments, which are largely negative and largely from people that are anti-drinkers). And I do recommend Proof if you want to know what’s going on in the world of alcohol production and the science behind it.

Direct youtube link

Ramsey answer

Erdos and Andras Hajnal asked “What’s the smallest graph of any kind, not containing K6, that forces a monochromatic triangle when it is two-colored. Ronald Graham (mathematician at Bell Labs, who started out as a trampolinist, and is a former president of the International Juggler’s Association) proposed the below solution for an 8-point graph. His proof is reductio ad absurdum: assume that a two-coloring that avoids monochromatic triangles is possible, and then force such a triangle. At least two lines from the top must be one color, say, gray. The graph’s symmetry allows the two outside lines to be gray with no loss of generality. The end points of the two lines must be colored something else to prevent the formation of a gray triangle.
This week’s puzzle: Complete the argument.

According to Gardner, the above figure is supposed to be gray and black (it’s also 7 points). The gray lines are blue-empty, and the 4 (minimum) red triangles are drawn in black lines. I assume that the correct drawing is for the outside lines to be gray and the inner ones to be black, but I may be wrong.

Colossal Gardner, ch. 33

Martin originally wrote the chapter on Ramsey Theory for the first appearance of The Journal of Graph Theory, in 1977. “Graph theory studies the sets of points joined by lines.” There had been a few papers in the 1930’s on Ramsey Theory, by Paul Erdos and a couple others, but no one really focused on Ramsey Numbers until the late 50’s. One stimulus for this growth spurt was the puzzle, “Prove that at a gathering of any six people, some three of them are either mutual acquaintances or complete strangers to each other,” (problem E 1321, the American Mathematical Monthly, June-July, 1958).

To turn this puzzle into a graph problem, draw 6 points to represent the 6 people. Join every pair of points with a line; you can use a red pencil for people that know each other, and a blue pencil for two strangers. Next, prove that regardless of how the lines are colored, you can’t avoid having either a red triangle (joining 3 mutual acquaintances) or a blue one (for 3 strangers). Ramsey Theory is named for Cambridge University mathematician Frank Plumpton Ramsey, who died at age 26 following abdominal surgery for jaundice in 1930. He also contributed to economic theory, simplified Bertrand Russell’s ramified theory of types in logic theory, and divided logic paradoxes into logical and semantical classes.

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

Ramsey read a paper in 1928 to the London Mathematical Society, “On a Problem of Formal Logic,” that included what’s now known as Ramsey’s theorem. It applies to graph-coloring theory. If you connect all pairs of points with lines, you have a complete graph on n points, and it’s given the symbol Kn. If we only talk about the topology of the graph, the placement of the points and how the lines are drawn are unimportant. The above figure shows the complete graphs for points 2 through 6, and each graph includes all the subsets of n that have exactly 2 members. Next, color the lines either red or blue. They can be all red, all blue, or a mix. If we wanted to divide the lines into three sets, we could use a third color. “In general, an r-coloring divides pairs of points into r mutually exclusive classes.”

Now we get into confusing terminology, with the introduction of “subgraphs.” Any graph that is contained in the complete graph, in that all of the points and lines in that graph are contained in the larger graph, is called a subgraph of that complete graph. Additionally, any complete graph is a subgraph of any other complete graph that has more points. The bottom figure of the above illustration shows the names associated with the common simple figures. Note that the Wheels are the complete graphs of subsets with Kn points (the tetrahedron is the complete graph for K4).

So, say we have six pencils of different colors and each color is used to draw a different complete graph (e.g. – blue for a pentagon, yellow for a 7-point star, green for a 13-point path). Next, ask the question “are there any complete graphs that, if their lines are arbitrarily six-colored, are certain to contain as a subgraph at least one of the six graphs listed above?” According to Ramsey’s theorem, for graphs above a certain number of points, all complete graphs have this property. Call the smallest graph of this infinite set of graphs “the Ramsey graph,” and the number of points for it is the “Ramsey number” for that set of subgraphs.

According to Gardner, every Ramsey graph includes a game and a puzzle. As an example, have two players taking turns picking up any colored pencil and coloring a line of the Ramsey graph. The first one to complete one of the specified subgraphs is the loser. Because it’s a Ramsey graph, you can’t have a tie. The result is called a critical coloring for one of the specified set of subgraphs. The puzzle involves a complete graph with one fewer point than the Ramsey graph. This is the largest complete graph in which the game can end in a draw. The puzzle is in how you find a coloring for the critical graph in which none of the subgraphs appears.

The rest of the chapter contains walkthroughs for example games, and the discoveries of various solutions to some of the more famous questions in Ramsey theory. The below table shows the simple graphs with known generalized Ramsey numbers.

(Simple graphs for which the generalized Ramsey number is known.)

Now, there are some problems with The Colossal Book of Mathematics, including a number of obvious typos. The biggest one is that figures that appeared in color in Scientific American are black and white in the book, and it’s almost impossible to figure out what the original colors were. That’s the problem here in this week’s puzzle.

Erdos and Andras Hajnal asked “What’s the smallest graph of any kind, not containing K6, that forces a monochromatic triangle when it is two-colored. Ronald Graham (mathematician at Bell Labs, who started out as a trampolinist, and is a former president of the International Juggler’s Association) proposed the below solution for an 8-point graph. His proof is reductio ad absurdum: assume that a two-coloring that avoids monochromatic triangles is possible, and then force such a triangle. At least two lines from the top must be one color, say, gray. The graph’s symmetry allows the two outside lines to be gray with no loss of generality. The end points of the two lines must be colored something else to prevent the formation of a gray triangle.
This week’s puzzle: Complete the argument.

More Escher-related cartoons.

From Bizarro. Which is pretty Escher-like in its own right.

Colossal Gardner, ch. 32

Paper folding.
Martin starts out by mentioning Stanislaw Ulam’s question of how many ways there are to fold a rectangular map. The wiki article on map folding doesn’t say anything about Ulam, but does indicate that this is still an unsolved combinatorial theory problem.  The map is precreased along horizontal and vertical lines to create a matrix. The folds can only be made along these lines, and the final result must be a packet with any rectangle on top and the others below it. To make the question precise, we number the rectangles, and then count all the possible permutations of the n cells, reading from top to bottom. Cells are numbered the same on both sides so it doesn’t matter if they are face up or face down. Either end of the packet can be “up” or “down,” so every fold has two permutations, one the reverse of the other.

(All rights belong to their owners. Images used here for review purposes only. Possible fold for a 4 stamp strip.)

The simplest case is the “strip of stamps,” i.e. – a 1xn rectangle. Even today, the only solution is to use a recursive formula, and there’s no known non-recursive approach. All six permutations are possible with a strip of 3 stamps (3!=6), but only the above 16 folds can occur with n=4 (4!=24). It’s 50 folds for 5 stamps, 144 for 6 stamps, and 16,861,984 for 16 stamps. The problem is that the number of calculations increases exponentially as n increments, but (quote) “heuristic methods from physics can be used to predict the rate of exponential growth of this sequence.”

For a 2×2 square, 8 of the 4!=24 permutations can be folded, half of which are reversals of the other half. For a 2×3 rectangle, things get more confusing because you can tuck corners of the map into pockets created by other cells. 6!=720 permutations, but Gardner claims to only have been able to fold 60 of them. An alternative is to put letters on the map’s cells and see what words you can spell. His examples include “ill-fed” and “filled,” and “squire” changed to “risque.” Dudeney, who appeared in the chapter on The Calculus of Finite Differences, mentions on page 130 of his 536 Puzzles & Curious Problems, that there are 40 ways to fold a 2×4 rectangle into a packet with #1 on top.

(WWII paper map.)

In 1942, a company printed an advertising premium for a 3×3 square, where the idea was to fold the map so that Hitler, Mussolini and Tojo each appeared behind the bars. The image above is wrong, in that Hitler and Mussolini are on one side of the sheet, with the window cut out of the bars in the top left corner. Tojo and the second, lower, window are supposed to be on the flip side of the sheet. I’ve tried solving this puzzle, but the paper tears too easily if it’s too thin, and doesn’t bend at all if it’s too thick.

(The Devil’s Fold. Put the same letters on the back side of the sheet.)

Other paper folding games include Robert Neale’s Sheep and Goats and Beelzebub (see how many spelling variants of Beelzebub you can find). In the addendum for this chapter, Gardner mentions that there’s a pattern you can follow for Sheep and Goats that you can use as a magic trick, separating the sheep from the goats while folding the map under a table.

No puzzles this week.

Gakken Mini Printing Press Kit

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

Well, it looks like Gakken really has dropped down to a once-a-year release schedule for their Adult Science line. I attribute this to the rising costs for making the kits, and a drop in sales due to the weak economy and increased sales tax. There’s also the fact that it does take a massive amount of research to write up the science elements of the magazines, and there is a big challenge in finding ideas for new kits in the $30-$40 dollar range that haven’t been done to death already. Still, I like making these things, and I would be happier if Gakken at least went back to 2 a year.

(Examples of old Japanese printing presses.)

Anyway, this time we have the mini printing press. I like this on several levels, not the least of which is nostalgia-related. When I was in junior high, we had a graphic arts class which included carving designs in linoleum-topped blocks and then making prints by applying ink to the block and then pressing the paper down on it. Later, when I was in Junior Achievement in high school, my group operated motorized presses similar to the one this kit is based on, for making personalized stationery and business cards. Since everyone else hated cleaning the ink off between batches, they focused only on the jobs with larger print orders. I didn’t mind the cleaning, though, and I tackled the smaller projects, often having to change inks 3-4 times in one night. And I still have all my fingers intact.

The mini printing press is based on the design of the Mizuno Foot press (1880’s), which in turn was originally developed and patented by Stephen Ruggles in New York in the mid-1800’s, as the Ruggle’s Footpress. The idea is that the stirrup handle was foot-operated when pressing the type against the paper. The kit consists of roughly 40 pieces, including the screws and other small parts. Gakken suggests a 30-minute assembly time, but it took me 1 hour to build it, largely because I was being careful, and taking lots of photos for giving the assembly instructions in English. There’s nothing really tricky about putting the printer together, but it does help to position the linkage arms first before trying to mount them on the steel rods, and there are two points where you need to understand what you’re doing really well to avoid getting frustrated (one is when you put the platen plate in place – you need to keep the body assembly jiggly to squeeze the platen between the frame halves. The other is when you snap the top ink plate in place – it snaps on and off in order to let you tuck the ends of the felt ink sheet in when you get ready for printing. That is, the top ink plate opens and closes when you want to remove the felt sheet for washing.

(The letter tray, and a small sampling of the letters I’d cut apart so far. There are 96 letters and characters for the alphabet set, and there’s a lot of flash to trim. I’m not done with that part yet.)

The kit is very sturdy, and bigger than it looks in the photos in the magazine. The stirrup handle is about 6″ tall, and it has maybe a 4″x4″ footprint. The box comes with one small tube of black ink, but the magazine recommends using any commercial water-based paints. Suggestions include’s Mat Multi paint tubes, and Pentel’s Ef Water Paints. I haven’t tried printing anything yet simply because I’ve been really busy. But, the kit comes with two sets of letter blocks, one for the hiragana character set, and the other for the English alphabet (upper and lower case letters, the numbers 0-9, and a handful of punctuation symbols. You only get one of each letter, so if you’re printing something complicated, like a business card, you have to do it in multiple passes, removing the letters and moving them around for the next pass. The press is sized for printing to business card blanks, but you can use larger pieces of paper if you fold them up first. Additionally, the magazine suggests taking a rubber eraser that’s 5.5 mm thick, and carving that up to make different designs to print from. If the eraser isn’t thick enough, you can use double-sided tape, or felt sheet backing to make up the additional spacing. If you want more letter sets (750 yen), or replacement rollers (500 yen), you can buy them from Gakken. I am planning on getting 2 more letter sets, if I can come up with ideas for things to print.

(Printer with the letter tray in place.)

As for the magazine, that’s not quite as action-packed as it has been in the past. It’s advertised as being 56 pages, but that depends on whether you count the extra 4 sheets of paper stock at the front. (Yes, you’re expected to cut the stock sheets out and chop them up into business card-shaped rectangles to print on.) After that, we have 4 pages of artwork gallery, 2 pages with the cover model (artist and model, Natsume Mito) and several pages of printing suggestions from professional graphic artists. These suggestions include rotating the paper to make circular image patterns, making gift tags, and using rubber bands to hold longer strips of paper in place vertically when you print on them. Other suggestions include mixing vertical and horizontal text, and using combinations of other characters to fake what looks like the alphabet (i.e. – LEET). To print graphics, take a rubber eraser that’s 5.5 mm thick or a bit thinner and transfer your image to one side (you can either draw directly on the eraser with a pen or pencil (right-left flipped), or draw normally on tracing paper in soft pencil and use the “silly putty effect” to left-right flip the image to the eraser. Either way, once the image is on the eraser, use a cutter knife to cut around the lines and remove the parts you don’t want to print to a depth of about 0.5-1mm. When you make the cuts around the lines, it’s ok to hold the knife at a 45 degree angle to make a bit of a shoulder on either side of the line to give the printing surface more physical strength. If the eraser isn’t 5.5mm thick, you can use double-sided tape or a thin sheet of felt to make up the difference.

We get 6 more pages of example cards with images in different colors, instructions on applying the ink to the roller (if you need to, press the roller against the ink sheet with both hands), and suggestions for using more than one color ink on the roller at a time. There’s 6 pages of the history of printing presses, from the Gutenberg press up to the ones of 1927. This is followed by a visit to Koedo with author Sanae Hoshio. Sanae has written 4 short stories in the Letterpress Print Shop Mikazuki-dou setting, revolving around a small printing press company in Saitama prefecture, and the article here features members of the shop that may have inspired her stories. One of the short stories is reprinted in the magazine.

(The main target is business card stock.)

There’s 4 pages of the history of Japanese font types and hot metal typesetting (which my grandmother used to do for a living), a couple pages of ads, and finally 8 pages of assembly instructions for the kit. There are two more suggestions which could be useful – one is to bolt the printer down to a sheet of wood to make it more stable as you do the printing. The other is to take a small sheet of foamcore, rule horizontal and vertical lines on it, punch holes at the intersections, and use that for holding your plastic letter pieces when you’re not printing with them.

(A view of the back, and part of the linkage system that controls the movement of the roller.)

The kit’s fun to build, and it makes a great conversation piece. If you have a 3D printer, you might want to use that to make more letters, or just buy 2-3 letter sets from Gakken. But, I think it’s more impressive to make graphics out of erasers, and print up those as specialty gift tags or customized business cards. It’s up to you. Regardless, I recommend this kit if you can avoid the import mark-ups.

You can watch the official Gakken video at youtube.

Colossal Gardner, ch. 31

Just as famous as the Soma Cube, I would say, is John Horton Conway‘s Game of Life. Conway’s primary work, as of the writing of this chapter, had been in pure mathematics. In 1967, he discovered what’s now known as the Conway Groups and which ties into John Leech’s dense packing of unit spheres in 24-D space, where each sphere touches 196,560 others. But, Conway also likes recreational math, and one of his simulation games involves three simple rules.

Initially, you’d use a checkerboard and small checkers, or graph paper and a pencil. Life now runs primarily as software. You start by placing a few of the counters on the grid, one each per “cell”. Each cell has 8 neighbors, four orthogonally, four diagonally.

1) Survivals – Every cell with two or three neighboring counters survives for the next “generation.”
2) Deaths – Each counter with 4 or more neighbors dies from overpopulation (is removed). Each counter with 1 or no neighbors dies from isolation.
3) Births – Every empty cell adjacent to exactly 3 neighbors is a birth cell. A counter is placed in that cell in the next move.

The rest of the chapter is then an exploration of some of the different configurations of cells that were known at that time, including guns, battleships, spinners and gliders. The wiki article adds that Conway had been working on John von Neumann’s attempts to find a machine that could build copies of itself. von Neumann developed a complicated set of rules that worked on paper, and Conway simplified them.

The addendum talks about the popularity Life has had with computer hackers, and mentions a few more objects that can be made with the counters. There are a few puzzle questions regarding how many cycles it takes for a particular structure to die out, but it’s much more fun trying to find starting patterns that last forever without turning into boring spinners or flashers.

Instead, try playing with two of the better-known implementations, and see what you can come up with.

Note: I do just want to mention here that back in the mid-80’s, PCs were just getting accepted in business offices, and I was the main go-to guy for supporting the machines the company had at one factory location. As part of my support duties, I had to write a defect tracking database system (dBase was the only product on the market then), and I needed some support tools for it. So, I bought a copy of Turbo Pascal, which I consider to have been the best programming language available. I taught myself programming using Turbo, and the defect tracking system worked great (it was eventually discontinued because of office politics from the mainframe IT group, which never managed to implement a replacement system before the company was carved out of existence via an M&A). Anyway, I loved using Turbo, and one of my side projects was writing a version of Conway’s Life. (Another was a crypto program for helping me solve crypto-quips in the newspaper.) I am now happy to have discovered Free Pascal, an open source Pascal compiler that apparently supports Turbo-style programming. When I finish these blog entries on Martin Gardner’s book, I’ll check it out, and maybe write another Life game.

Printing Press Assembly Photos

My original intention was to run the instructions for building the printing press next week, after the review of the magazine. Instead, I’m uploading two blog entries in one day, today. If you haven’t read the one on the Newton magazine article on Penrose Tiles, click the “previous” button.

There’s about 40 pieces to the Gakken mini printing press kit, plus the 2 letter sets, ink bottle, screwdriver, and eye dropper. The only other tool you’ll need is a scissors or diagonal cutter for cutting the letters out of the mold frame and trimming off the flash. Figure 30-45 minutes for building the kit, and up to another hour for cutting out the letters. You may want to have a tray ready for holding the letters to avoid losing them; or take a sheet of foam core, rule a 9×9 grid on one side, punch holes at the intersections and use that for holding the letters in the same positions as in the mold frame. You don’t really need to check if you’re missing any pieces, but laying everything out flat on a table may make it easier to find the parts you need for each step.

(All rights belong to their owners. Instructions from the magazine used here for assembly purposes only.)

Ok, getting started. Get the left and right arms, plus one of the regular screws. Position both arms so the ends of the 2 cross beams for the right arm fit into the matching left arm cross beams, and hold them together by tightening down the screw inside the middle cross beam.

Take the left and right frame pieces, and fit them together as shown in the photo.
I messed up the numbering. Put screws in the cross beams marked #1 and #2, plus the unmarked one, and tighten them down snugly. Don’t put a screw in the cross beam marked #3 – you’ll need that free to squeeze the arms assembly in next.

Stand the frame up as shown in the photo. You can see the screws in place in what’s now the back side of the frame. Pull the cross beams at point #1 (numbering from the previous photo) apart just enough to let you slide the arms assembly into the frame. The two nubs on the ends of the arms will go into the matching wells in the frame. When you’re done, the arms should be able to rotate up and down inside the frame.

Now, put a regular screw into cross beam #1 of the frame and tighten that down snugly.

Take the letter tray holder (my kit had the the letter tray already in position in the holder; you can take the letter tray out if you like) and flip it over to be face down, with the tabs pointing to the frame. See the two little tabs on the frame the lower arrows are pointing at? Take the letter tray holder and hook the tabs at the end of the holder behind the tabs on the frame, and rotate the holder to an upright position. The screw holes at the back of the holder will line up with the matching tab holes on the frame (see below photo).

Like this.

Turn the frame around, and take two screws and screw them down into the tray holder until they’re snug.

Locate the three metal shafts, and the shaft board. You’ll need the shorter and middle-length shafts right now.

With the back of the frame facing you, position the shaftboard so that the two pins at the side point to your right. Put the far end of the shaft board inside the frame such that the short shaft will run through the middle hole of the frame, through the shaft board, and then out the other side of the frame, as shown above. (Note, I have the screwdriver propping up the shaftboard just to help me get a better photo.)

Turn the frame around again, and run the medium shaft through the holes in the middle of the arm assembly.

Now, this is a slightly tricky part. You want the two linkage arms and two retaining caps. If you play with the linkage arms, you’ll see that they can be folded into different shapes, some more useful than others. Also, the arms are marked “1” and “2”.

With the back of the frame facing you, put linkage 1 to the right side and linkage 2 to the left. Fold the arms to match the photos. I’m calling the two joints in the middle of the linkage “elbow 1” (e1) and “elbow 2” (e2). The end of the linkage that just has the hole is the “tail” (t), and the piece with the spring and rotation thing is the roller holder.

Turning the frame counterclockwise a little so you’re looking at the left side, take linkage 2 and hold it so that the tail is aimed down at the table, and the spring roller holder is kind of tucked in between the other two arms. Slide the linkage arm onto the medium and short shafts, with the medium shaft going through e1, and the short shaft fitting into the end cap of e2.

Turn the frame to get to the right side, and mount linkage arm 1 onto both shafts in the same way. Then take the two retaining caps and put them on either end of the medium shaft. You don’t have to push them on very far. Just enough that the linkage arm elbows don’t shimmy around too much as the linkage moves back and forth. You can adjust the retainers later when the press is fully assembled (see below photo).

Take two of the regular screws and put them in the screw holes of the mounting caps at the elbows “e2” of the linkage and tighten them down so the linkages are firmly secured to the short shaft. (Push the linkages together so they’re all the way on the short shaft before tightening the screws.)

These guys here (shown before being tightened down).

You now want the long metal rod, the stirrup, the stirrup shaft, the collar and the two remaining retainer caps. Slide the pin of the stirrup into the matching slot on the stirrup shaft (doesn’t matter which side of the stirrup faces forward or backward).

Slide the long metal shaft through the linkage tail (t) on the left side of the frame, through the collar (with the thicker disk end closest to the linkage arm), through the lower holes of the shaft board, then out the other side of the shaft board and through the tail (t) of the right-side linkage arm.

Hold the shaft in place by sliding a retainer cap onto the left end of the shaft.

The caps are going to look like this on the left side of the frame.

The stirrup shaft will then fit over the right end of the metal shaft, and the two plastic pins of the shaft board below. Hold the right side of the metal shaft in place with the other retainer end cap.

Notice that the tail of the right-side linkage arm is located between the shaft board and the stirrup shaft. You can push the back retainer caps so that they fit snugly on either side of the press frame.

Take the washer-head screw and use it to fix the stirrup shaft in place on the shaft board. Tighten it down snugly, but be careful to not strip out the plastic threads.

Like this.

Get the ink plate and set it face down. Take the stopper and attach it to the plate with a regular screw.

This way. Tighten down the screw until it is snug and the stopper doesn’t wiggle at all.

Almost done.

Turn the ink plate right side up, and position the two fingers at the front of the plate so that they fit under the top edge of the letter plate. Rotate the back edge of the ink plate downward.

Looking at the back of the frame, you’ll notice that the stopper kind of hits the top cross beam of the frame. The idea is that you’re going to put a piece of cloth over the ink plate when it comes time to start printing, and the edges of the cloth are going to be tucked under the side edges of the ink plate, so that when you push the plate down, the stopper will snap past the frame cross beam and hold the ink plate firmly in place, simultaneously trapping the cloth sheet in place. This will make more sense later, when you get ready to start printing. The point is that to clean the cloth afterward, you’re going to need to pull the back of the ink plate up, pulling the stopper off the cross beam and letting you fully remove the ink plate from the press.

For right now, push the ink plate down so the stopper snaps into place below the crossbeam, as shown in the photo above.

Find the roller, and the two roller end caps. Push the end caps firmly into place at either end of the roller.

Like this.

Push the stirrup down to bring the roller hand ends of the linkage arms up to where you can see them. It will help to have the hands rotated to be pointing up and to the back of the press as shown in the above photo. Snap the roller into place in the hand pieces.

Like this. Note that the roller end caps have two small spacers that slide back and forth on the roller shaft. Make sure those spacers are between the roller hand pieces and the roller itself. But really, there is only one way the roller will easily fit onto the linkage arm hands.

Rotate the hands forward, and now you’re ready to start printing. Push the stirrup down, and pull it up, to test the movement of the arms and the roller. If everything moves smoothly, the roller should roll back and forth across the ink plate, down over the letter plate, and to the bottom of the frame. If you keep pushing the stirrup down, the press will push against the letter plate. If you have letters in the plate, and a business card blank in place on the press, you’ll make an imprint on the card.

Find the card holder plate, the felt sheet and the two felt holder fingers.

Put the felt sheet in the indent in the card holder plate, and push the finger pieces into the holes at either side of the sheet to keep it in place. Note now that there’s a lip on the holder plate at one of the edges of the felt sheet. You’re going to set your business card blank on that lip when it comes time to do the printing, so that lip indicates “down” on the card holder plate.

Snap the card holder plate onto the medium-length rod at the front of the machine, with the lip at the “down” position.

Like this.

If the letter tray is already in the press, lift it out. Use a scissors or diagonal cutter to cut all the letters out of the mold frame. Remove any excess flash from the sides of the letters. Put the letters you want into the tray. When you’re done, slide the tray back into place in the tray holder plate.

And that’s it for the kit assembly portion. Again, pull and push the stirrup to check the roller movement across the ink plate and the letters, and that the card plate moves forward to press the card against the letter faces (the type). If necessary, loosen or tighten the retainer caps on the short and medium-length rods to prevent shimmying or avoid jamming of the arm linkages against the frame.

I haven’t had time to do any printing myself, so I don’t have instructions ready for that, yet. In the meantime:
The black “ink” that comes with the kit is actually a water-based paint. Pull the back of the ink plate up so the stopper comes off the crossbeam. Take the piece of white cloth and lay it flat on the ink plate, and tuck the edges under the plate. Put the ink plate back in place on the printer and snap it down again, keeping the cloth sheet flat on top. Use the eye dropper to wet the cloth, and squeeze ink onto the sheet. Push and pull the stirrup to run the roller over the ink until it’s evenly smeared across the sheet. This may take a couple minutes, and it may help to just push the roller against the cloth sheet directly with your hand. Put a blank card on the card holder, and push the stirrup down farther to ink the type. Run the roller over the type a few times to make sure the letters are evenly inked.  Finally, push the stirrup all the way down, to press the type against the blank card. Release the stirrup and check the card to see if it printed the way you want. If not, re-ink the type and press again, or maybe press a little harder. Don’t press so hard as to damage the press. Practice a few times until you get it right. If necessary, experiment with the amount of water you put on the ink cloth, to avoid the “ink” being too thick and too watery.

Direct akken youtube video link

To print a card using the same letters more than once, use the little extraction tool to push the letters out of the tray, and reposition them as needed. Print again, and repeat as necessary. When you’re done, unsnap the ink plate and remove the cloth sheet and the roller.  Soak the sheet and roller in a bowl of warm water to clean off the paint before it has a chance to dry.

Newton Science Magazine, Jan. 2018, part 2

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

The second article from the Jan., 2018, Newton magazine is “Making Beautiful Tiles With Mathematics”. Having written up the chapter on Penrose Tiles from the Gardner book, I was interested to see how the authors would address Penrose’s work in this article. They start out by mentioning M. C. Escher, and then covering a lot of the concepts addressed in Martin’s Scientific American article, which I highlighted back in July. Is it possible that someone at Newton read my blog and got an idea for this article? Nah, doubt it, but still…

Anyway, I didn’t go through the Japanese text really closely, so I can’t be absolutely sure, but I didn’t see Penrose’s name in there at all. Which is kind of strange because many of the tiling concepts in this article are very similar to what was in Martin’s Mathematical Recreations article on Penrose’s non-periodic tilings. The main big difference is that the illustrations in the Newton article look more like CG pictures, and they are all attributed to Japanese artists. Note that the pink and tan tiling (above, lower right corner) is the same as the one Martin used, but with different colors, while “Birds and Fishes” (above, middle left) is suspiciously like Penrose’s Tiling Chickens.

Then we get Makoto Nakamura’s “T-Ball”, which is based on something Escher had made but that fact seems to have been overlooked. Yes, Newton magazine has nice pictures and interesting articles, but there may be a failing here in giving proper attributions to the people that originated these tilings. Or, I may be wrong…