Monthly Archives: March 2009

Learning to Kiteboard — It’s Not Like the Video Said

I bought a practice kite to learn to kiteboard. It’s a smaller version of the kites used for real kiteboarding. It comes with a dvd that covers how to attach the kite to the line, where to practice flying (wide open spaces), how to launch and steer the kite, and how to pack the kite. It also talks about the “power zone” which is roughly sixty degrees to either side of the wind.

What it doesn’t say is how the kite flies.

That may seem obvious. If you’ve ever flown a kite, you know how a kite flies, of course — and you’d be wrong about this kite. A kiteboarding kite is not designed to fly like the traditional diamond-shaped kite I used to buy at the 7-11 when I was nine. I tried that yesterday with this kite. I launched it, ran a bit to get it up in the wind, and then watched as it gently fell to the ground. I was frustrated because it seemed there just wasn’t enough wind.

I tried several times. As I was walking up the beach in search of better wind, a woman encouraged me not to give up. I smiled, but it wasn’t looking good. Then I stumbled onto the secret.

The kite has two lines, which come down to a bar about two feet across. Angling the bar turns the kite: pull in the right line and the kite steers to the right; pull in the left line and the kite steers to the left. That much is straightforward. What isn’t immediately obvious is that the kite is much more wing-like than old-fashioned kites: it develops far more lift when it is moving across the wind than when it is steady.

The first time I noticed this, it changed the way I was flying the kite. With no change in wind I went from repeatedly watching the kite limply settle to the ground to muscling it across the wind, working against at least five or ten pounds of tension on the lines. As long as I was working the kite back and forth, it was pulling on me hard. As soon as I let it stand still, it was limp again.

So for anyone who’s just starting, here’s my one hour’s worth of experience for you: in light winds, work the kite through the wind.

Senket Introduction and Rules

Background

Senket is a board game I invented several years ago. My goal was to come up with rules as simple as possible, but game play as deep as possible.

Go was my inspiration. The rules of Go are very simple, but there are several complexities:

  • The ko rule can be confusing for beginners, especially since it is implemented/described in several different ways. Snapbacks have to be distinguished from kos.
  • The concept of life and death — simplified, that two eyes live and one eye dies — is so fundamental to the game that, while not part of the rules, no description of how to play is complete without it. Seki further complicates the issue.
  • There are different methods of scoring, although they are nearly equivalent.

That said, Go has perhaps the deepest strategy of any board game ever, and certainly the highest ratio of game depth to rule complexity of any game I know.

Senket’s rules are even simpler than Go’s, as you’ll see below. There is only one type of move, with no special cases. Nothing is ever removed from the board, or moved once placed. At the end of the game, territory is simply defined.

The question of depth of strategy remains open. From a sheer numbers standpoint, Senket is more complex than most other games, but just because the board is bigger doesn’t necessarily mean much. In gameplay, Senket seems deeper than Checkers, and perhaps as deep as Chess, but until/unless world-class games of Senket are played, it is impossible to say.

Description

Senket is played on a grid. Grids as small as 11×11 will work, but 17×17 makes for a reasonable game, and 31×31 is the official size. The goal is to surround more territory with your fences than your opponent. The overall concept is similar to Go. The mechanics of game play are similar to Twixt.

Rules

  • Players take turns making moves.
  • A move consists of placing a post on any unoccupied intersection of the grid (including the borders), and then making as many valid fences as you wish.
  • A fence connects two same-color posts across an open diagonal of a 1×2 rectangle.
  • Once placed, posts and fences cannot be moved or removed.
  • Play continues until both players pass.

Scoring

Territory is any region of the board your fences surround that does not contain any territory of your opponent. At the end of the game each player’s score is the sum of the values of each separate territory. Each territory’s value is the square of the sum of:

  • The area the territory’s fences surround (see below for examples).
  • The number of opposing posts captured within the territory.

Scoring does not take place until the end of the game. You can play inside your opponent’s territory if you think you can make territory there.

How to Move

The blue fences are valid, the red fences are not.

The blue fences are valid, the red fences are not.

In this diagram, all the blue fences are valid. All the red fences are invalid:

  • None of the red fences in the upper right are 1×2 diagonals.
  • The red fence in the lower left does not connect two posts.
  • The two red fences in the lower right are crossed; whichever was drawn second is incorrect.
  • The red fence in the bottom center connects a red post and a blue post.

No Forced Moves

A fence does not have to be drawn just because it can be. The lone blue post on the left can connect to either of the two posts above it. The blue player will decide whether to connect this post on a future move or not. Usually there is no reason not to connect all posts that can be, but there are a few specific circumstances where it is beneficial (or even necessary) not to draw all possible fences.

Scoring Example

A completed game.

A completed game.

This diagram shows a completed game, with each player’s territory shaded in his color.

The area in the lower right marked in yellow is not Blue’s territory. Even though Blue surrounds it, the red territory inside it makes the area inside the blue fence but outside the red fence neutral. It counts for neither player.

The blue post in the upper left magenta territory counts as a prisoner for Red. The blue and red posts in the yellow area are not prisoners, and count for no one.

Red scores:

  • Upper left area: 27
  • Upper left prisoners: 1
  • Upper left value: (27 + 1)^2 = 28^2 = 784
  • Lower left area: 10
  • Lower left value: 10^2 = 100
  • Lower right area: 2
  • Lower right value: 2^2 = 4
  • Total score: 784 + 100 + 4 = 888

Blue’s areas are connected. Blue scores:

  • Upper right area: 19
  • Lower left area: 12
  • Total score: (19 + 12)^2 = 961

Red actually surrounded more territory and prisoners (40 to 31) but Blue connected his territory and won.

Basic Strategy

It’s easier to make territory in the corners, but it’s also important to connect your territory. As shown in the example above, Red took much more territory, but lost because Blue connected and he didn’t. On larger boards this is even more important.

Consider the value of different available moves. As play winds down in one location, there will probably be bigger plays elsewhere that offer a larger reward. This is a careful balancing act that continues throughout the game.

Senket is not a kill-or-be-killed game (as compared to Chess or Checkers). Every game will likely end with both sides securing at least some territory, so look for advantageous ways to divide territory being contested, rather than a way to take your opponent prisoner, unless that’s a real possibility. Think “how can I get more out of this than my opponent?” rather than “how can I destroy my opponent?”

I’ll follow up with examples as I prepare them for the web.

Freeman Dyson Followup Puzzle — solution

In Freeman Dyson Puzzle Solution I asked the follow up question, “What is the smallest integer where you can move the first (most significant) digit to the right, and the result will be exactly half the original number?” The answer is below…

 


 


Maybe it’s a bit of a cheat, but the answer is 105263157894736842. Note that by moving the 1 to the right, you also lose a leading zero, hence the “bit of a cheat” aspect.

Gilligan’s Island Economics — Inflation

Inflating the Money Supply

In Gilligans Island Economics — Auto Bailout, I described what’s wrong with simply throwing money at an economic problem. Now let’s talk about where that money comes from. Remember, the Skipper and Gilligan have been making a tidy living building huts for everyone, but now the market for huts is slowing down (everyone has one) and the Skipper is worried about where Gilligan will find the money to buy coconut cream pies.

Thurston Howell III proposes a bailout. He’ll take an extra $100 out of his trunk of money and give it to the Skipper and Gilligan, and that way they have extra time. Suppose that the Skipper is “generous” and Gilligan gets $30. Gilligan immediately runs to Mary Ann and offers her $10 for every pie she makes for the next week (Gilligan is skinny, but he can really eat). The Professor also likes coconut cream pies, so where he normally gives Mary Ann $2 for pies during the week, he ups his offer to $4. Mary Ann ends up making more during the week, and she’s able to offer more money to Lovey Howell for the shell necklaces she makes, forcing Ginger to pay more as well.

You can see where this ends up. The castaways started their economic system fairly: Mr. Howell gave $100 to each castaway, so the total economy on the island was $700. Now with the bailout there is an extra $100, for a total of $800, floating around. Prices will go up as people find they have more money to pay with.

The Ripple Effect

So who benefits and who suffers? Gilligan and the Skipper obviously benefit — they get an extra $100 of free money to split between them. Everyone on the island suffers to the extent that they are holding dollars. If the Professor happens to have $200, because he gets paid infrequently and needs to have cash reserves, while Mary Ann keeps only $20 on hand because she makes money on pies every day, then the Professor suffers ten times the damage that Mary Ann does.

The Real World

The situation is the same in the real world economy. Banks create money the way Mr. Howell did every time they issue a loan: the borrower has the money to buy a house (for example) while the person who deposited the money in the first place still has access to their account. The amount of money is tracked in terms of the actual money floating around and the money that is available in one way or another. A better description is in the wikipedia article on money supply.

The money supply can increase (somewhat) harmlessly as the economy increases — if an extra hundred castaways washed up in the lagoon the island economy would experience a significant decline in prices as the 700 circulating dollars distributed themselves among 107 people. In a circumstance like that it makes sense for Mr. Howell to pull more money out of his trunk.

Last week the Fed increased the money supply by over 1 trillion dollars. To misquote Everett Dirksen (who apparently never actually said it), “a trillion here, a trillion there, and pretty soon you’re talking about serious money.” Actually, a trillion is serious money by itself. It’s about the same as the above scenario on the island. We should expect to see prices increase significantly as that extra money hits.

The Fed has the ability to increase the money supply arbitrarily. There is no one beneficiary, but a range of them, in the form of banks and financial institutions. Certainly not you or me. The Fed have been going to the trunk and pulling out extra money for nearly one hundred years, and often spending it on nothing more useful than coconut cream pies.

Freeman Dyson Puzzle Solution

The puzzle from Freeman Dyson is Very Smart is:

  • Is (there) an integer where, if you take its last digit and move it to the front, turning, say, 112 to 211, it’s possible to exactly double the value. Dyson will immediately say, “Oh, that’s not difficult,” allow two short beats to pass and then add, “but of course the smallest such number is 18 digits long.”

My reasoning toward an answer goes like this:

The number should end in 2 (and start with 1) to make it the smallest possible such number. Then the number and its double must look like this:

  1nn...nn2
x         2
-----------
  21nn...nn

2 x 2 is 4, so:
  1nn...nn2
x         2
-----------
  21nn...n4

That means the second-to-last digit has to be 4, so:

  1nn...n42
x         2
-----------
  21nn...n4

2 x 4 is 8, so:

  1nn...n42
x         2
-----------
  21nn...84

That means the third-to-last digit is 8, so:

  1nn...842
x         2
-----------
  21nn...84

2 x 8 is 16, so:

        1
  1nn....842
x          2
------------
  21nn...684

That means the fourth-to-last digit is 6, so:
        1
  1nn...6842
x          2
------------
  21nn...684

Now it changes because of the carried 1. 2 x 6 + 1 is 13, so:
        11
  1nn....6842
x           2
-------------
  21nn...3684
Following on like that ends up with:

   1 1  1111 1 11
  105263157894736842
x                  2
--------------------
  210526315789473684

So the answer is 105263157894736842. That’s the smallest number that works for this puzzle, but 210526315789473684 is not the smallest number where the reverse can be done: what is the smallest integer where you can move the first (most significant) digit to the right, and the result will be exactly half the original number?

Freeman Dyson is Very Smart

In an article in the New York Times, a puzzle is described that Freeman Dyson supposedly solved in his head in a few seconds.

  • Is (there) an integer where, if you take its last digit and move it to the front, turning, say, 112 to 211, it’s possible to exactly double the value. Dyson will immediately say, “Oh, that’s not difficult,” allow two short beats to pass and then add, “but of course the smallest such number is 18 digits long.”

I finished reading the article this was in, then as I was falling asleep I thought of how to find the answer, but couldn’t keep the digits straight in my head. If I’d been thinking only of the length of the solution perhaps I would have been able to find it. In any case, a few moments with a pencil and paper the next day was enough to figure it out.

It’s a really nice puzzle. I’ll post the answer separately.

John Conway Agrees With Me

Okay, maybe I agree with John Conway. He did come up with the idea first.

Mr. Conway is giving a series of lectures at Princeton regarding free will. From the article, Conway and his colleague Simon Kochen say they have proven that, “if humans have free will, then elementary particles — like atoms and electrons — possess free will as well.”

That sounds pretty much in line with my post The Illusion of Free Will, where I proposed two identical boxes filled with identical arrangements of atoms and said, “there are two possibilities: the two boxes will remain identical as the molecules go about their business, or some randomizing factor — quantum or otherwise — will cause the boxes to fall out of sync.”

I did have the hubris to cast doubt on the second possibility and prefer the first. I don’t know what Conway’s thoughts on the likelihood of the two are.