I’d just finished reading Ian Stewart’s book on **The Beauty of Numbers in Nature** (I’ll post the review on Friday), when I saw the above photo in a friend’s Facebook post. This is exactly the kind of thing Stewart discusses in the section on chaos, and time symmetry in dynamic systems. Read the entire article, and check out the rest of the photos Daniel Biber took in the series.

## All posts for the month **January, 2018**

# Starlings in Formation

*Posted by chh01 on January 31, 2018*

https://threestepsoverjapan.wordpress.com/2018/01/31/starlings-in-formation/

# Colossal Gardner, ch. 35

We now enter **Game Theory** with **A Matchbox Game-Learning Machine**. The first couple pages look at learning machines as a whole, and more recent computers as well, as threats to the human race. We first get Ambrose Bierce’s excellent **Moxon’s Master** (1899), about a chess playing machine that loses a game to its creator and kills him in a rage. Then there’s Norber Wiener, math professor at MIT, who expected computers to take on more and more complex tasks and push us into a suicidal war (don’t need a computer for that). Arthur Samuel was the first to program an IBM 704, in 1959, to review past games of checkers and learn from its mistakes. When this chapter was first written in 1962, computer chess programs were just showing up on the horizon, and some chess masters were convinced they could never be defeated by a machine, including American expert Edward Lasker in 1960. My, how things have changed.

(All rights belong to their owners. Images used here for review purposes only. Photo hosted on Flickr of a matchbox computer for ticktacktoe.)

But, the focus of the chapter is on making a matchbox computer. It was invented by Donald Michie, a biologist at the University of Edinburgh. He wrote about it in **Trial and Error**, an article that appeared in *Penguin Science Survey 1961*, vol. 2. It was a ticktacktoe learning machine called MENACE (Matchbox Educable Naughts and Crosses Engine), using 300 matchboxes. A drawing of a possible ticktacktoe position is pasted on each matchbox (because the machine always makes the first move only the patterns for the odd moves are required). Glass beads with colors representing possible machine plays from each pattern are placed in the boxes. Donald placed a V-shaped cardboard fence at the bottom of each box so that when you shake it, the beads randomly roll into the corner of the V to determine the next play. First-move boxes contain 4 beads of each color; third-move boxes have 3 beads of each color; fifth-moves have 2 of each color; and seventh-moves have one each. The machine’s move is determined by shaking and tilting a box, opening the drawer and looking at the color of the bead in the corner of the V. Boxes used during the game are left open until the game ends. If the machine wins, it is “rewarded” with three beads to each open box of the color used for that play. If the game is a draw, the “reward” is one bead per open box. For a loss, it is “punished” by removing the bead in the corner of the V. Eventually, the machine repeats winning plays and shuns losing ones. There’s no self-analysis of past games, just a rudimentary form of memory. (There’s a C++ program version at the codeproject.com.

Gardner didn’t expect anyone else to use 300 matchboxes to make their own machine, so he created a game he called Hexapawn, using only 24 boxes with a 3×3 board and 3 chess pieces on each side. Only two kinds of moves are allowed:

1) A pawn may advance straight forward one empty square.

2) A pawn may capture an enemy pawn by moving one square diagonally. The captured piece is removed from the board.

There are three winning endings:

1) A pawn advances to the third row.

2) Capturing all of the enemy’s pieces.

3) Reaching a position where the enemy can’t move.

Martin called the machine Hexapawn Educable Robot (HER), and the drawings for the boxes are below. It may be difficult to figure out the “colors” of the arrows, but there are up to four moves possible from any given position, so you’ll need 4 different colored beads (or M&Ms or Skittles) maximum in each box. The robot always makes the even-numbered move. The numbers under the boxes represent the turn number (2 possible moves for turn 2, and 11 possible moves for turns 4 and 6 each). Martin included mirror positions in the drawings; otherwise you’d only need 19 boxes.

To play, put inside a single bead for each colored arrow on the box. You make your first move, then pick up the box with the matching diagram, shake it, close your eyes, open the drawer and take out one bead. Close the box, put it down, put the bead on top of the box, and open your eyes. Check the color of the bead, and move the machine’s pawn as indicated on the diagram of that box. Repeat until the game ends. If the machine wins, replace all the beads and play again. If it loses, remove only the bead for its last move, replace the other beads and play again. If a box is empty, the machine has no non-fatal move and it resigns. In this case, remove the bead of the preceding move. The below chart shows HER’s learning curve for the first 50 games.

(HER learning curve over 50 games.)

Martin suggests that you could develop your own rules for rewarding HER, and you could construct a Hexapawn Instructable Machine (HIM) with new rules to play against HER (you’d need to draw new diagrams for the odd-numbered plays for both sides). Stuart Hight, director of research studies at Bell Labs in Whippany, NJ, built a machine called NIMBLE (Nim Box Logic Machine) for playing Nim with three piles of three counters each. Other choices include simplified versions of Go (on the intersections of a 2×2 checkerboard), or minicheckers (below). Martin was of the opinion that the below right mini-chess board is still too complicated to implement, but that it’s within the realm of a computer program.

(Minicheckers and Minichess board layouts.)

In the addendum, Martin mentions a math club that built 2 hexapawn machines, and then told two students that they’d be playing against each other. The two students were put in separate rooms, and the box computers were in a third. The students couldn’t believe who their opponents really were when they were led to the third room at the end of the experiment.

Martin also mentions how many chess players got angry at the idea of machines beating them, and talks about Fritz Lieber’s SF story of a machine tournament, **The 64-Square Madhouse** (*If*, 1962).

**Puzzle:** None.

**Challenge:** Make your own machine for a 5×5 checkerboard game.

*Posted by chh01 on January 29, 2018*

https://threestepsoverjapan.wordpress.com/2018/01/29/colossal-gardner-ch-35/

# Hatoyama Elk Puzzle

For Christmas, I received the Elk puzzle by Hatoyama. It’s rated at difficulty level 6, and after having spent an hour on it, I can say that yes, it’s hard. I knew there had to be a specific trick to it, and I could see how the ending would go if I could just make one specific twist to get the pieces to align the right way. The problem was that I couldn’t figure out how to make that missing move. So, yeah, I cheated and went to youtube. The video I watched wasn’t edited all that well, so even when I followed the moves suggested, I had trouble getting it right. The thing is, the two pieces are marked “Hatayama” and “NOB”, and that’s actually a hint that alignment is everything. It took practicing the moves over and over for an hour before I finally got to where I understood what’s going on, and I can now assemble and disassemble it from memory in less than 30 seconds. It is a challenging puzzle, and worth tackling if you like 3D metal puzzles. It’s also well-made, solid and the plating doesn’t fleck off.

*Posted by chh01 on January 25, 2018*

https://threestepsoverjapan.wordpress.com/2018/01/25/hatoyama-elk-puzzle/

# 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.”

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.

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.

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.

*Posted by chh01 on January 22, 2018*

https://threestepsoverjapan.wordpress.com/2018/01/22/colossal-gardner-ch-34/

# Proof, The Science of Booze, comments

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.

*Posted by chh01 on January 18, 2018*

https://threestepsoverjapan.wordpress.com/2018/01/18/proof-the-science-of-booze-comments/

# 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.

*Posted by chh01 on January 17, 2018*

https://threestepsoverjapan.wordpress.com/2018/01/17/ramsey-answer/

# 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.

*Posted by chh01 on January 15, 2018*

https://threestepsoverjapan.wordpress.com/2018/01/15/colossal-gardner-ch-33/

# More Escher-related cartoons.

*Posted by chh01 on January 11, 2018*

https://threestepsoverjapan.wordpress.com/2018/01/11/more-escher-related-cartoons/

# 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.

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.

*Posted by chh01 on January 8, 2018*

https://threestepsoverjapan.wordpress.com/2018/01/08/colossal-gardner-ch-32/

# Gakken Mini Printing Press Kit

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 Craypas.com’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.

*Posted by chh01 on January 2, 2018*

https://threestepsoverjapan.wordpress.com/2018/01/02/gakken-mini-printing-press-kit/