From Harvey's Terry and the Pirates (via Smiling Jack and Friends)
A blog of tips and recommendations for anyone interested in learning or teaching mathematics.
Tuesday, March 26, 2013
Sunday, March 24, 2013
The Humble Checker
Yet to calculate is not in itself to analyze. A chessplayer, 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 bestknown.
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 writeup 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 giveaway 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.
Saturday, March 23, 2013
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.
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.
Saturday, March 16, 2013
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.
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.
Friday, March 15, 2013
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 openended yet it has a welldefined 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.
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 openended yet it has a welldefined 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 bendbutnotstretch 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.
Paperbased 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
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 bendbutnotstretch 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.
Paperbased 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
Sunday, March 10, 2013
A clever probabilitybased story from Alfred Hitchcock
Mail Order Prophet, a 1957 episode of Alfred Hitchcock Presents starring E.G. Marshall and Jack Klugman
Saturday, March 9, 2013
Facade Chess
[disclaimer  there are no new chess variants. I don't know who came up with this idea first but I'm pretty sure it wasn't me.]
By most reasonable standards, there are too many chess variants out there already. A few are actually worth playing (such Gliński's hexagonal game), the rest are of interest, if it all, more as thought experiments and programming problems.
One potentially promising area for the latter is variants of imperfect information, which leads us to familiar games like kriegspiel and this variant, facade chess.
Start with a standard board and pieces. When I say standard pieces I mean that you will have one piece that moves like a king, one piece that moves like a queen, two pieces that move like bishops and so on.
The pieces' appearance, however, will not be standard. They will look like tiny replicas of those Aframe signs restaurants put out on the sidewalk, with slots for pictures of chess pieces on either side. On most of the pieces, the picture is the same on the front and the back, but on up to three (or some other agreed on number), the pictures have been switched.
Pieces are lined up in standard position based the picture in the front but they have to move in accordance with the picture on the back (like kriegspiel, this game definitely needs a referee). If for example, the queen had a rook's picture on front of it, you would put it in a corner but you could move it any distance vertically, horizontally or diagonally.
Each move has to be weighed in terms of both position achieved and information revealed  as soon as that rook moves diagonally, the other player will know something's up. In addition to deduction you can also find out the true identity of a piece by capturing it. Capturing a disguised piece also provides useful information about the disguised pieces still on the board.
I'm not sure how playable facade chess would be  players would probably tend to under utilize their pieces (moving rooks like pawns or queens like bishops so as not to give away their identities)  making for a slow game but from an analytic standpoint, the variant could still provide interesting problems. Chess strategies are complex to start with; imagine adding a layer of uncertainty and questions about how much value to put on concealing information.
By most reasonable standards, there are too many chess variants out there already. A few are actually worth playing (such Gliński's hexagonal game), the rest are of interest, if it all, more as thought experiments and programming problems.
One potentially promising area for the latter is variants of imperfect information, which leads us to familiar games like kriegspiel and this variant, facade chess.
Start with a standard board and pieces. When I say standard pieces I mean that you will have one piece that moves like a king, one piece that moves like a queen, two pieces that move like bishops and so on.
The pieces' appearance, however, will not be standard. They will look like tiny replicas of those Aframe signs restaurants put out on the sidewalk, with slots for pictures of chess pieces on either side. On most of the pieces, the picture is the same on the front and the back, but on up to three (or some other agreed on number), the pictures have been switched.
Pieces are lined up in standard position based the picture in the front but they have to move in accordance with the picture on the back (like kriegspiel, this game definitely needs a referee). If for example, the queen had a rook's picture on front of it, you would put it in a corner but you could move it any distance vertically, horizontally or diagonally.
Each move has to be weighed in terms of both position achieved and information revealed  as soon as that rook moves diagonally, the other player will know something's up. In addition to deduction you can also find out the true identity of a piece by capturing it. Capturing a disguised piece also provides useful information about the disguised pieces still on the board.
I'm not sure how playable facade chess would be  players would probably tend to under utilize their pieces (moving rooks like pawns or queens like bishops so as not to give away their identities)  making for a slow game but from an analytic standpoint, the variant could still provide interesting problems. Chess strategies are complex to start with; imagine adding a layer of uncertainty and questions about how much value to put on concealing information.
Labels:
board games,
chess,
imperfect information,
math games
Alexandria Puzzles for SAT vocabulary building
The following puzzles contain words from SAT vocabulary lists.
SAT Words Group 6  



 
Thursday, March 7, 2013
The Exact Chaos Spreadsheet Game
John D. Cook points out the following interesting fact:
The first player picks a to number from 1 to 99 with the only condition being that the number can't have a five in it (with a little trial and error you should be able to figure out why).
The second player can either steal the first player's number or come up with one of his or her own. If the number is stolen, the first player has to come up with another number.
Both numbers are divided by one hundred and put into the first row of a spreadsheet. Under the first type
=4*A1*(1A1)
Under the second
=4*B1*(1B1)
Now copy the formulas going down twenty rows. The copy function should produce an iterative function, with =4*A1*(1A1) followed by =4*A2*(1A2) and so on.
Have the students make a line graph of the two series. The winner is the one who ends up with the largest number.
Pick a number x between 0 and 1. Then repeatedly replace x with 4x(1x). For almost all starting values of x, the result exhibits chaos. Two people could play this game with starting values very close together, and eventually their sequences will diverge.Cook's on to something here. This does have the makings of a simple game that ties in to math and computing concepts like functions, iterations, spreadsheets and chaos. Here are the rules:
The first player picks a to number from 1 to 99 with the only condition being that the number can't have a five in it (with a little trial and error you should be able to figure out why).
The second player can either steal the first player's number or come up with one of his or her own. If the number is stolen, the first player has to come up with another number.
Both numbers are divided by one hundred and put into the first row of a spreadsheet. Under the first type
=4*A1*(1A1)
Under the second
=4*B1*(1B1)
Now copy the formulas going down twenty rows. The copy function should produce an iterative function, with =4*A1*(1A1) followed by =4*A2*(1A2) and so on.
Have the students make a line graph of the two series. The winner is the one who ends up with the largest number.
Labels:
Excel,
functions,
iteration,
mathematical games,
spreadsheet
Wednesday, March 6, 2013
Kriegspiel and Dark Chess
I've been talking about games of perfect information and I have another post on the subject coming up so this seems like a good time to mention two of the best known chess variants of imperfect information.
From the nice people at Wikipedia:
From the nice people at Wikipedia:
Kriegspiel (German for war game) is a chess variant invented by Henry Michael Temple in 1899 and based upon the original Kriegsspiel developed by Georg von Rassewitz in 1812.[1][2] In this game each player can see their own pieces, but not those of their opponent. For this reason, it is necessary to have a third person (or computer) act as a referee, with full information about the progress of the game. When it is a player's turn he will attempt a move, which the referee will declare to be 'legal' or 'illegal'. If the move is illegal, the player tries again; if it is legal, that move stands. Each player is given information about checks and captures. They may also ask the referee if there are any legal captures with a pawn.and
Dark chess is a chess variant with incomplete information, similar to Kriegspiel. It was invented by Jens Bæk Nielsen and Torben Osted in 1989. A player does not see the entire board, only their own pieces (including pawns), and squares where these pieces could move.I've never actually played either of these games (I have enough trouble with unvaried chess), but they raise all sorts of interesting questions about forming a strategy with incomplete information.
Sunday, March 3, 2013
Perfecting the imperfect
[disclaimer  I've only field tested the first of these, so I can't guarantee that all of the variations will play smoothly. On the bright side, there ought to be plenty of room for improvement. As with all discussions of game variants, you should probably assume that countless people have already come up with any idea presented here.]
When the subject of perfect information games comes up, you probably think of chess, checkers, go, possibly Othello/Reversi and, if you're really into board games, something obscure like Agon. When you think of games of imperfect information, the first things that come to mind are probably probably card games like poker or a board game with dicedetermined moves like backgammon and, if you're of a nostalgic bent, dominoes.
We can always make a perfect game imperfect by adding a random element or some other form of hidden information. In the chess variant Kriegspiel, you don't know where your opponent's pieces are until you bump into them. The game was originally played with three boards and a referee but the advent of personal computing has greatly simplified the process.
For a less elaborate version of imperfect chess, try adding a dieroll condition to certain moves. For example, if you attempt to capture and roll a four or better, the capture is allowed, if you roll a two or a three, you return the pieces to were they were before the capture (in essence losing a turn) and if you roll a one, you lose the attacking piece. Even a fairly simple variant such as this can raise interesting strategic questions.
But what about going the other way? Can we modify the rules of familiar games of chance so that they become games of perfect information? As far as I can tell the answer is yes, usually by making them games of resource allocation.
I first tried playing around with perfecting games because I'd started playing dominoes with a bluesman friend of mine (which is a bit like playing cards with a man named Doc). In an attempt to level the odds, I suggested playing the game with all the dominoes face up. We would take turns picking the dominoes we wanted until all were selected then would play the game using the regular rules. (We didn't bother with scoring  whoever went out first won  but if you want a more traditional system of scoring, you'd probably want to base it on the number of dominoes left in the loser's hand)
I learned two things from this experiment: first, a bluesman can beat you at dominoes no matter how you jigger the rules; and second, dominoes with perfect information plays a great deal like the standard version.
Sadly dominoes is not played as widely as it once was but you can try something similar with dice games like backgammon. Here's one version.
Print the following repeatedly on a sheet of paper:
Each player gets as many sheets as needed. When it's your turn you choose a number, cross it out of the inverted pyramid then move your piece that many spaces. Once you've crossed out a number you can't use it again until you've crossed out all of the other numbers in the pyramid. Obviously this means you'll want to avoid situations like having a large number of pieces two or three spaces from home.
If and when you cross off all of the numbers in one pyramid you start on the next. There's no limit to the number of pyramids you can go through. Other than that the rules are basically the same as those of regular backgammon except for a couple of modifications:
You can't land on the penultimate triangle (you'd need a one to get home and there are no ones in this variant);
If all your possible moves are blocked, you get to cross off two numbers instead of one (this discourages overly defensive play).
I haven't had a chance to field test this one, but it should be playable and serve as at least a starting point (let me know if you come up with something better). The same inverted pyramid sheet should be suitable for other dice based board games like parcheesi and maybe even Monopoly (though I'd have to give that one some thought).
I had meant to close with a perfected variant of poker but working out the rules is taking a bit longer than I expected. Maybe next week.
In the meantime, any ideas, improvement, additions?
Originally posted in West Coast Stat Views
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