Monthly Archives: May 2009

Senket — Puzzle of the Week #5 solution

This is the solution to the Senket puzzle of the week #5.

 

At the start: mutual life in the upper right.

At the start: mutual life in the upper right.

First, here is how the score breaks down before making any moves.

Blue in the upper left is worth 20^2 = 400 points (apologies for showing posts but counting area).

Blue in the lower left is worth 1^2 = 1 point, as is Red in the upper right.

Red’s main territory is worth 30^2 = 900 points. Red is substantially ahead at the moment.

Blue has two main options: make territory in the upper right at 9-9, or extend the territory in the lower left at 7-1.

 

Red is toast.

Red is toast.

Looking at 9-9 first, the results are dramatic as shown here. Previously the red and blue fences in the upper right enjoyed what might be called mutual life: the small red territory in the upper right invalidated the blue fence, and the blue territory in the upper left invalidated the red fence, leaving much of the upper right as unclaimed territory.

By making territory at 9-9, Blue now has Red’s fence trapped with no way to make territory or connect to any. Thus Red is now prisoners, and much of the previously neutral territory is now Blue’s.

Blue’s largest territory now contains 17 more squares and 4 prisoners, so it is worth 20 + 17 + 4 = 41^2 = 1681 points. Red 2 adds just 3 squares to his main territory, increasing it to 30 + 3 = 33^2 = 1089 points.

Blue wins handily.

 

Blue makes the wrong choice.

Blue makes the wrong choice.

If Blue plays at 7-1 instead, as shown here, he takes an extra 10 squares and 3 prisoners, but Red 2 at 9-9 then blocks Blue from making territory there. Blue can salvage an extra 3 squares, but the results isn’t enough:

Blue: 23^2 = 529    &    (11 + 3)^2 =  196 for a total of 725.

Red: 30^2 = 900    &    1^2 = 1 for a total of 901

Senket — John’s Go Page

John’s Go Page describes Go in ten statements. It uses some pretty clever language to squeeze the description into so few rules, and it describes area scoring, where I would have preferred territory scoring, but I think that would have definitely required an extra statement or two.

Here is an attempt to unambiguously describe Senket in as many sentences:

  1. Senket is played on a 31 x 31 square grid of points, by two players called Black and White.
  2. Each point on the grid is either empty, or may be colored black or white.
  3. Two points on the grid of the same color may be connected by a fence if they are opposite corners of a 1×2 rectangle, and if the fence would not cross any existing fences.
  4. Starting with an empty grid, the players alternate turns, starting with Black.
  5. A turn is either a pass, or coloring an empty point one’s own color and adding as many valid fences of one’s own color as are desired.
  6. The game ends after two consecutive passes.
  7. Territory is any enclosed section of the board (including the edges) that does not contain opposing territory.
  8. Rule 7 applies recursively, working out from unambiguously valid territories.
  9. Each set of territories connected by fences scores points equal to the square of the sum of the area contained within the territories and the number of opponent’s posts contained with the territories.
  10. The player whose territories sum to the greater point total at the end of the game is the winner. Equal scores result in a tie.

Senket — Puzzle of the Week # 4 Solution

This is the solution to Puzzle of the Week #4.

Blue learns the hard way.

Blue learns the hard way.

Red can start in the corner, threatening to make territory immediately. If Blue is foolish he’ll try to kill Red. He starts with Blue 2. Red 3 threatens at 4, so Blue blocks there. Then Red 5 threatens at 6, so Blue plays there as well. Then Red 7 connects to the outside, completely ruining Blue’s territory. Blue 8 takes 4 squares of territory and blocks Red from taking 4 of his own, but Blue is much worse off than when he started.

 

 

 

Blue does better.

Blue does better.

Blue would do much better to simply wall Red off in the corner with Blue 2. If Red wants to live, he needs to make territory with Red 3. Then Blue 4 maximizes his territory.

 

 

 

 

 

 

 

 

Blue goes outside, so Red takes another bite.

Blue goes outside, so Red takes another bite.

Blue is slightly less successful if he protects the outside first, as shown here. Red makes more territory inside before Blue manages to seal him in.

 

 

 

 

 

 

 

 

Red is dead inside.

Red is dead inside.

Note that Blue has two ways to connect outside and Red can’t block both, so there’s no way for Red to succeed by starting outside first.

Torture and the Trolley Car

The Problem

The Trolley Problem is a thought experiment in ethics that can easily start an argument. The original form goes something like this:

A mad philosopher has sent a trolley car racing down the track toward five people tied to the rails. The people will surely die if the trolley hits them. There is a switch that will send the trolley down a different track, where the philosopher has tied only one person. You are the only person able to throw the switch, saving the five but killing the one. Would you do it?

Think about it a bit before you answer. Purely from a numerical standpoint, throwing the switch makes sense: one for five. This is the utilitarian view. But there are many who refuse, often saying something like, “I don’t know the people. That one person could be a better person than the other five.” This reflects incommensurability

There are many variants to the trolley problem. Some omit the mad philosopher, making the whole thing an accident. This can shade the scenario because there is no one to blame but you; in the first example, you can say that the death of the five people is the mad philosopher’s responsibility for setting up the problem in the first place.

Another variant is the so-called “fat man” scenario. In this one there is no switch, but you are standing on a bridge over the tracks next to a fat man, whose body could stop the trolley (yours won’t) so you have to decide whether to push the fat man over the railing onto the tracks.

You are Inconsistent

If you think you have a firm answer for the trolley problem, there are variations aplenty to give you cause to doubt yourself.

If you say you would never throw the switch, consider that every year some (small) number of people die in accidents caused by police and fire response teams answering 911 calls. So if there is a fire, you are changing your answer to the trolley problem if you dial the phone. Another example is driving an SUV — you and your passengers are safer, the people you have an accident with are less. There are other examples.

If you say you would throw the switch, consider the following scenario: a surgeon has five patients waiting for transplants, and their outcome doesn’t look good. The surgeon discovers that another of his patients is a perfect match for the other five. Hypothetical, yes, but from a utilitarian viewpoint equivalent to the fat man variant. So would you have the surgeon kill the one patient and give kidneys, liver, heart, etc. to the others?

Don’t feel bad. As far as I can tell, philosophers (who do this for a living, how cool is that?) don’t have a firm answer for the trolley problem.

The Trolley Problem is Everywhere

It isn’t cited all that often, but there are many scenarios that map back to the trolley problem. There’s the famous “If you had a time machine, would you kill Hitler?” question. As an aside, here’s a really funny short story on using time machines to kill historical mass-murderers. The 911 phone call question and SUVs are others. Any time you balance one person’s safety against another’s, you’re participating in a form of the Trolley Problem.

The question of torture is another form of the Trolley Problem. Suppose the mad philosopher has put five people in the way of the trolley but the alternate track has no one on it. However, the philosopher has put a padlock on the switch. Only the philosopher knows the combination to the lock, and you have only a minute to get the answer. What would you be willing to do to get it?

There are numerous complexities in trying to translate this to the real world, including the fact that coerced information is suspect at best, and potential torture victims are rarely so obliging as to place people in as obvious and timely a predicament as tying them to a trolley track. That said, it’s interesting to me that we often discuss this in terms of far more vague and unlikely scenarios (“a terrorist has set a dirty bomb to go off in Manhattan in one hour…”) when I think it would occur far more often in more mundane and practical scenarios: imagine a platoon of soldiers who capture one member of an enemy formation. Obviously the prisoner likely has time-sensitive information on the location, plans, and capabilities of the enemy formation that will have a high-probability impact on the survival of the platoon. In this very real situation the potential value of torture is clear, and yet we don’t authorize it.

Psychologically there’s a difference between the probabilities of a 911 call and the certainties of the Trolley Problem or of torture. People are willing to put some unknown others in slightly greater risk in order to achieve a near-certain benefit, and no one argues that. Something more concrete, like waterboarding a man, in hopes of achieving something more vague, like general information on far away enemies with indefinite plans, is a harder thing to justify. But it’s difficult to rule out as the (potential) situation moves closer to the trolley tracks. I don’t have a good answer for the surgeon variant, though.

Philosophy is hard, so it’s no surprise that this question provokes controversy.

 

Note: I’m keeping an informal list of people who have been waterboarded, and their reactions.

Senket — Scoring Resolved

I have an answer for scoring. It doesn’t involve changing the text of the rules, just careful application of them. It may result in some odd scores for some very artificial board layouts, but since they don’t seem likely to ever turn up in actual gameplay, I’m comfortable with this as a solution. Here’s an example board that poses the question, how do I score this?

 

How to score this?

How to score this?

Here’s the sentence in the rules that describes how to identify territories: Territory is any region of the board your fences surround that does not contain any territory of your opponent.” So consider how to handle the above. There are two unequivocal territories: the one point blue in the upper right, and the one point red in the lower left. It only makes sense to work out from those, and it’s only fair to do it simultaneously. So at the same time, the one point blue in the upper right precludes the red three-post fence in the upper right from claiming territory, and the one point red in the lower left precludes the long blue fence from claiming territory. So this board is scored correctly as it appears: one point each for red and blue.

In the pathological case:

 

Not as hard as it looks.

Not as hard as it looks.

The one point red in the upper left and the one point blue in the lower right are the starting points. They invalidate the blue fence that starts at 1-9 and the red fence that starts at 7-1. That means that the red fence that starts at 1-8 and the blue fence starting at 5-1 are both valid and make territory. But that in turn means that the blue fence starting at 1-7 and the red fence starting at 3-1 are both invalid. And that means that the red fence starting at 1-6 and the blue fence starting at 1-1 are valid and make territory. And that means that the blue fence starting at 1-5 and the red fence starting at 1-2 are invalid, and finally the red fence starting at 1-4 and the blue fence starting at 1-3 are valid. So each player claims the same amount of territory and prisoners.

Here’s another example:

 

Who wins?

Who wins?

EDIT: The rule is subtle, and I didn’t apply it correctly here. This is wrong:

The critical point to understand is the lower left corner. Without that, Blue is toast. But Blue’s one point territory there invalidates Red’s fence from 1-3 to 5-1. The same thing happens in the upper right, leading to a score that looks like this:

Red's mistakes in the upper right and lower left are fatal.

Red's mistakes in the upper right and lower left are fatal.

Blue’s largest territory is worth (26 + 3)^2 = 841 points, while Red’s largest is just 25^2 = 625 points. Each of them have a 1 point territory, but Blue has an additional territory worth 9^2 = 81 points. Blue wins handily. If Red had managed to make territory in the lower left, he would have won easily.

 

Corrected Version

The lower left is critical, as I said, but the upper right is tricky as well. It is bounded by two unequivocal territories: the main Red territory in the center, and the one point Blue in the upper right. Working from each of those in, the Red invalidates the Blue fence from 8-11 to 11-5 at the same time as the Blue invalidates the Red fence from 9-11 to 11-7. The result looks like this:

 

Blue wins, but he doesn't get 81 points in the upper right the way I thought he did.

Blue wins, but he doesn't get 81 points in the upper right the way I thought he did.

 

 

Blue still wins, but he earns only 1 point in the upper right instead of 81.

I should add that these territories were generated automatically. I have that coded at this point. I now need to get the territories marked for area instead of post scoring, and generate scores automatically.