Now that's a paper airplane

From Harvey's Terry and the Pirates (via Smiling Jack and Friends)

The Humble Checker

Yet to calculate is not in itself to analyze. A chess-player, for example, does the one without effort at the other. It follows that the game of chess, in its effects upon mental character, is greatly misunderstood. I am not now writing a treatise, but simply prefacing a somewhat peculiar narrative by observations very much at random; I will, therefore, take occasion to assert that the higher powers of the reflective intellect are more decidedly and more usefully tasked by the unostentatious game of draughts than by all the elaborate frivolity of chess. In this latter, where the pieces have different and bizarre motions, with various and variable values, what is only complex is mistaken (a not unusual error) for what is profound. The attention is here called powerfully into play. If it flag for an instant, an oversight is committed, resulting in injury or defeat. The possible moves being not only manifold but involute, the chances of such oversights are multiplied; and in nine cases out of ten it is the more concentrative rather than the more acute player who conquers. In draughts, on the contrary, where the moves are unique and have but little variation, the probabilities of inadvertence are diminished, and the mere attention being left comparatively what advantages are obtained by either party are obtained by superior acumen.

Edgar Allan Poe -- "The Murders in the Rue Morgue"

Poe's opinion on this matter is more common than you might expect. It's not unusual to hear masters of both chess and checkers (draughts) to admit that they prefer the latter. So why does chess get all the respect? Why do you never see a criminal mastermind or a Bond villain playing in a checkers tournament?

Part of the problem is that we learn the game as children so we tend to think of it as a children's game. We focus on how simple the rules are and miss how much complexity and subtlety you can get out of those rules.

Chess derives most of its complexity through differentiated pieces; with checkers the complexity comes from the interaction between pieces. The result is a series of elegant graph problems where the viable paths change with each move of your opponent. To draw an analogy with chess, imagine if moving your knight could allow your opponent's bishop to move like a rook. Add to that the potential for traps and manipulation that come with forced capture and you have one of the most remarkable games of all time.

There have been any number of checkers variants.* You could even argue that all checkers games are variants since, unlike chess, there is no single, internationally recognized version. Here are some of the best and best-known.



Spanish Checkers

Considered by many connoisseurs to be the best of the national variants. It is distinguished from the American version by a queen's (analogous to a king) ability to jump long. This makes a queen powerful but, since captures are mandatory, easier to capture.






Dama (Turkish Checkers)

Also known as Greek checkers. In this variant, pieces move vertically and horizontally instead of diagonally. Among other things this doubles the playing field.



Lasca

This variant was invented by the second World Chess Champion Emanuel Lasker. It's worth noting that Lasker (who is considered one of the best chess players of all time) would look to checkers when designing his own game.

In Lasca, captured pieces are added to the bottom of columns controlled by whichever player has the top piece. Since jumping only captures the piece on top of the column, players have to take into account not only position but also height and position.

You can get a detailed write-up by following the link or you can see Lasker's original patent application here. He comes off as rather immodest, but he was the world's best chess player and a close friend of Einstein so I guess we can let him slide.


Misère checkers

This one you probably know as suicide or give-away checkers. The object is either to lose all your pieces or have them blocked so that you can't move.



Endless Checkers (or whatever the damned thing is called)

I'm certain I've read about this somewhere though I can't seem to find any record of it. Here, when a piece reaches the last it can move to an appropriate space on the first row (think of a board wrapped around a cylinder). There are no kings in endless checkers.


I have a feeling I'm missing something. Any suggestions?

* Including one I'm particularly fond of, but that's a topic for another post.

When I say paper chains, I mean paper chains

Another in the paper as manipulatives series (see here or here for background)

Here are the rules of today's competition. Each team is presented with a bucket and a hook four or five feet off the ground  and is given a couple of dozen sheets of ordinary typing paper. The object is to suspend the bucket at least two feet from the hook using nothing but the sheets of paper (scissors are allowed). The winner is the team with the chain that can support the most weight

This assignment should be as open as possible and when you're explaining it, you might want to stop with the paragraph above and see where the kids go from there. On the other hand, it's often useful to give them something to build on, so here are a couple of ideas to get things started.


Method 1. Slots and notches

One simple way of joining paper is to make small cuts. You can make matching cuts, each of which extend to the edge of the paper so that the cuts can interlock, or you can cut a hole in one end of a piece of paper then cut a slot so that the end is bigger than the hole you thread it through.

Method 2. Tubes

Take a sheet of paper and roll it into the tightest tube you can manage. Bend the tube in three places to form a triangle (the bends also serve to keep the tube from unraveling), then insert one end of the tube into the other (you may need to pinch the end you're inserting to get it started). Jam it as snug as you can without tearing or crumpling.

The resulting triangle, like most of our paper manipulatives, is surprisingly sturdy, particularly when force is applied in certain ways. You can do some interesting construction projects using these triangles and little if anything else, but right now we're interested in using them as links in a chain.

As with most things, the strength of your chain will depend on how much of the force is borne by the strongest part. In this case that will be one of the two sides that don't include the join.


There are, of course, other ways of making paper chains (and there's no reason some kind of a paper rope couldn't be used instead). It's entirely possible that the kids will come up with something we hadn't even thought of.

Making this a very successful assignment indeed.

Paper Models

The always fun Low Tech Magazine has a very cool post on the rediscovery of paper models. As mentioned before, paper can only be folded into a certain family of shapes, but modelers have found some clever ways around the limitations over the years.

These do it yourself paper toys used to be quite popular. Now they're making a comeback online. The Low Tech article links to a wide array of historical models (most of which also provide opportunities for teaching across the curriculum). You can download simple ones as individual activities...

Or bigger, more complex ones as group projects.

Paper Towers

As mentioned before, paper is a excellent medium for engineering projects because

1. It's cheap and cheap is good for experimentation

2. Paper has a number of interesting and useful properties from a material science standpoint

3. From a pedagogical standpoint, these properties take on extra value because the material is so exceptionally ordinary. We want students to get in the habit of looking for the interesting in uninteresting places.

My recent fascination with the subject started with this idea for a class project/competition:

The task: build a platform at least 6" tall to support a 1' by 1' board (or piece of cardboard) and some weights that will be placed on it. (best of three tries).

Materials: 20 sheets of standard typing paper and 20 1" pieces of Scotch tape.

Objective: to support the most weight possible

In addition to the previously mentioned benefits of using paper for manipulatives, this project has a few other big selling points.

The students will probably be surprised at how strong an arrangement of paper tubes and cones can be and this surprise might help feed their curiosity.

The project is largely open-ended yet it has a well-defined metric of success. The better tower is the one that supports the most weight. There's little question about whether one design is better than another.

The project can employ some surprisingly sophisticated engineering.

The project can be used, with little or no modification, for students ranging from fourth graders to first year engineering students. There is enough potential for discovery that a ten year old and a twenty year could both learn something from giving it a try.

Two variations to this project immediately suggest themselves.

First is to use a fixed weight (let's say five pounds) and make the winner the highest structure that can support the weight.

Second is to use some function of weight and height. I'll let you think about that one on your own.

Reseeing Paper

We don't talk about it as much as we should but one of the fundamental goals of education is cultivating alertness, producing students who notice and think about everything around them. One way of achieving this is to show kids new ways of looking at the ordinary.

There are few things more familiar and mundane than a blank sheet of paper, but from a material science standpoint, that paper has some cool properties. Paper doesn't like to stretch or compress. Normally that would make for an object that didn't like to bend. Bending usually entails stretching because the outside of a curve has more area than the inside due the thickness of the material being bent. A sheet of paper, though, is so thin that this difference is negligible. Think about a pipe; the circumference of the inside is noticeably less than that of the outside. Now think about a tube made by lining up opposite edges of a piece of paper. The circumference of the inside is less than that of the outside but the difference is very small.

This bend-but-not-stretch property means that paper can only be bent into a certain family of shapes. You can make a tube or a cone or even an extruded sine curve but you can't, for example, make a dome.

These properties of paper also make it a great medium for discussing the properties of shape. In sheet form, paper is almost synonymous with flimsy, but if you form a sheet of paper into a tube or a cone* it can support a surprising amount of weight. Sandwich the previously mentioned extruded sine curve between two other sheets and you get a very strong board relative to its weight. Add another layer with the extrusions at right angles and the performance is even more impressive.

Paper-based manipulatives and projects are a great way of teaching a number of concepts in geometry, physics and engineering. More importantly though, they make the point that the most unexceptional class of things imaginable is, in fact, pretty exceptional.

Here are some project ideas. If you like what you see check back because I'll be updating this post.

The Platform Contest

Old Time Paper Models

Paper Chains

Paper bridges

A clever probability-based story from Alfred Hitchcock

Mail Order Prophet, a 1957 episode of Alfred Hitchcock Presents starring E.G. Marshall and Jack Klugman