Dreidel maximization
I'm not a genius at math, but I was sitting with a couple of friends who are naturally gifted. They are not Jewish, and I was explaining to them how one plays dreidel. (Or "the dreidel game," as most Googlable references seem to call it.) They started talking about how one could do best at it.
That is, how should one set about to win the most money through dreidel? (That not being the point of the game, but never mind about that for now.) Game theory generally involves maximizing expected earnings through decision making; the problem with dreidel is that precious little decision is involved. Let's try and describe the game and then see what variables one might be able to play with to tweak the expectations.
The four possible dreidel outcomes, together with what happens to one's earnings, go more or less like this:
N (nun): {null}
G (gimel): +x
H (hey): +x/2
Sh (shin): -2R
You get x (the size of the pot) if the dreidel lands on gimel, and 1/2x (half the pot) if it lands on hey; if the dreidel lands on shin, you put in some ante (usually an even number, in my experience), hence 2R.
Some simple things even people like me can see straight off. The first is that your earnings from gimel and hey depend on how much there is in the pot. That is, x changes with the progression of the game, so I really should have written it as a function of time, f(t,x). But that would be getting ridiculous.
One way of thinking would be that to maximize your earnings you should sit as far away from the luckiest (or craftiest) dreidelist as possible, so that the pot has a chance to fill up again after he or she spins his out-of-proportion gimels.
Another simple observation is that the relative "influences" on the game of gimel, hey, and shin depend on how much the ante is. If there's only a finite supply of money (or peanuts, raisins, or whatever currency you've chosen) available to the player, and 2R (the size of the ante required when Shin is spun) is a considerable fraction of that supply, then one shin can lay you low no matter how many nuns or gimels you've gotten.
One way of modifying the game might be to make the shin-ante (R) vary with the number of players, the size of the pot as it stands, or the time elapsed.
Or not. 35 days till Chanukah.
PS: Of course, a real mathematician has made some progress on the problem. He has conjectured, based on some simple simulations, that the length of a two-person dreidel game is on the order of the square of the number of nuts (or whatever tokens are used), and the length of a k-person game -- on the order of that number of tokens to the kth power.
11/20/05
11/13/05
Para-rabbis and PAs
"Totem pole": descriptive, not prescriptive.
MEDICINE MENSCH: What Little I Know About Medicine
By Zackary Sholem Berger
November 11, 2005
I thought I would title this month's column "What I Don't Know About Medicine," but my editors want 800 words, not an encyclopedia. What I do know about medicine should fit nicely into the space allotted.
The rotation I'm starting this week is known in hospital jargon as the sub-internship; the registrar's office calls it "advanced medicine." In this rotation the medical student plays the role of an intern, a first-year resident. He can write orders on the computer, decide on a course of treatment and prescribe medications — all with the co-signature of a supervisor, of course. The main thing is that the student manages his own list of patients. In other words, I won't be following a resident around anymore; I will be a resident.
More precisely, I'll be pretending to be a resident. That's the catch of this rotation — it's the adolescence of medical education, in which one is something more than a senior student but something less than an intern. How much like an intern will I be? If one reasons by similarity of activity (lack of sleep; independent decision-making on behalf of actual patients), I could very well call myself the intern. Could I walk into my patient's room and say, "I'm Dr. Berger, and I'll be taking care of you during your stay in the hospital?" The second half of this statement is definitely true, but the first part (with "Dr.") isn't quite. I won't have my medical degree until my graduation in May 2006, and I won't have a license to practice medicine until I finish my boards, which stretch from now (while I'm still a medical student) until the end of my residency.
On this rotation, I don't plan to introduce myself as "Doctor" (that would be quite a stretch), but I understand why some of my colleagues would. While many patients don't understand the difference between senior medical students, interns, residents, fellows and attending physicians, they come to appreciate something of the hierarchy during their hospital stay. One thing soon becomes clear to them: Medical students are at the bottom of the totem pole of health care providers. (Or almost at the bottom —more on that subject, below.) If a sub-intern walks into a patient's room and starts out by saying, "My name's Joseph Brighteyes, I'm a medical student, and I'm here to take care of you," the patient is likely to respond, "That's great, Joe, but who's going to be my doctor?"
Is qualification defined more by knowledge or by capability? This larger question comes up when talking about the thousands of people whose elbow grease makes a big hospital run as smoothly as it can. Even lower on the pole than the medical student is the physician assistant. P.A.s train for fewer years than medical students, so they often get less respect from folks in medical academia. But in every rotation where, as a medical student, I've worked together with P.A.s or P.A. students, they have seemed more on the ball than the medical students — not as over-theoretically concerned with fascinating disease entities and more knowledgeable about what might immediately help the patient. Part of the reason is that the P.A. student spends more time in her early training (most P.A.s I've met are women) familiarizing herself with the way things actually are in the uncompromisingly practical universe of the hospital or clinic. Perhaps P.A.s can't name 12 different kinds of small-vessel disease. (I'm making up that number; I certainly can't name them, either — not off the top of my head.) But such encyclopedic knowledge is sometimes transcended, or rendered irrelevant, by practical considerations. On the other hand, you don't want to miss the rare diseases when they do pop up. So both kinds of training (practical and theoretical encyclopedic) are essential, even complementary. The trick is to appreciate both kinds of thinking at the same time, even though most people find one of them more congenial than the other.
A similar divide is evident in the Jewish tradition, especially with regard to Torah study. In previous generations — and even today in some circles — memorizing vast tracts of the Talmud is a prized skill. On the other hand, if you've dipped into any of the Talmud, you know that it takes considerable exegetical ingenuity to derive laws applicable to everyday life from the rich stew of magic, folklore, intellectual speculation, casual yeshiva talk and campfire tales about the destroyed Temple. (Even the strict constructionists, those who actually bury their fingernail clippings according to the magical practice of the rabbis, still have to exercise creativity in deriving everyday religious practice from Babylonian-Jewish digression.) The two skills — encyclopedic knowledge of the Talmud and the ability to apply the Talmud to everyday religious life — are complementary, but rarely are found in the same person. Many of the greatest talmudic scholars have famously refused to adjudicate Jewish law, protesting that they could see all sides of any issue.
Today's Jewish communities rely very much on what some people call "para-rabbis" — people who aren't religious scholars but are reliable, even expert, in the details of daily practice. You might have met some of them: the guy who knows everything about building a sukkah; the woman who is obsessed with reading Torah; the member of the local synagogue who knows exactly how to help a recently bereaved family.
In other words, Jews (at least religious Jews) depend on "rabbis' assistants." In the same way, a hospital must cobble together a vast number of people who have different abilities. Some sit in offices and absorb themselves in the study of disorders affecting 10 people in the entire world, while others, down amid the gore in the E.R., can contradict those same scholars with absolute certainty if they see in front of them a practical problem they know how to solve quickly and effectively.
Bit by bit, I hope to amass both kinds of knowledge till I can be considered a doctor (a learned person, according to the word's historical meaning) in at least one of these categories. But no matter what, there always will be plenty of material for my upcoming masterwork, "What Zack Berger Doesn't Know." I look forward to doing the research.
Zackary Sholem Berger is becoming a doctor, hour by sleepless hour. Feel free to write him about it at doctor@forward.com.
"Totem pole": descriptive, not prescriptive.
MEDICINE MENSCH: What Little I Know About Medicine
By Zackary Sholem Berger
November 11, 2005
I thought I would title this month's column "What I Don't Know About Medicine," but my editors want 800 words, not an encyclopedia. What I do know about medicine should fit nicely into the space allotted.
The rotation I'm starting this week is known in hospital jargon as the sub-internship; the registrar's office calls it "advanced medicine." In this rotation the medical student plays the role of an intern, a first-year resident. He can write orders on the computer, decide on a course of treatment and prescribe medications — all with the co-signature of a supervisor, of course. The main thing is that the student manages his own list of patients. In other words, I won't be following a resident around anymore; I will be a resident.
More precisely, I'll be pretending to be a resident. That's the catch of this rotation — it's the adolescence of medical education, in which one is something more than a senior student but something less than an intern. How much like an intern will I be? If one reasons by similarity of activity (lack of sleep; independent decision-making on behalf of actual patients), I could very well call myself the intern. Could I walk into my patient's room and say, "I'm Dr. Berger, and I'll be taking care of you during your stay in the hospital?" The second half of this statement is definitely true, but the first part (with "Dr.") isn't quite. I won't have my medical degree until my graduation in May 2006, and I won't have a license to practice medicine until I finish my boards, which stretch from now (while I'm still a medical student) until the end of my residency.
On this rotation, I don't plan to introduce myself as "Doctor" (that would be quite a stretch), but I understand why some of my colleagues would. While many patients don't understand the difference between senior medical students, interns, residents, fellows and attending physicians, they come to appreciate something of the hierarchy during their hospital stay. One thing soon becomes clear to them: Medical students are at the bottom of the totem pole of health care providers. (Or almost at the bottom —more on that subject, below.) If a sub-intern walks into a patient's room and starts out by saying, "My name's Joseph Brighteyes, I'm a medical student, and I'm here to take care of you," the patient is likely to respond, "That's great, Joe, but who's going to be my doctor?"
Is qualification defined more by knowledge or by capability? This larger question comes up when talking about the thousands of people whose elbow grease makes a big hospital run as smoothly as it can. Even lower on the pole than the medical student is the physician assistant. P.A.s train for fewer years than medical students, so they often get less respect from folks in medical academia. But in every rotation where, as a medical student, I've worked together with P.A.s or P.A. students, they have seemed more on the ball than the medical students — not as over-theoretically concerned with fascinating disease entities and more knowledgeable about what might immediately help the patient. Part of the reason is that the P.A. student spends more time in her early training (most P.A.s I've met are women) familiarizing herself with the way things actually are in the uncompromisingly practical universe of the hospital or clinic. Perhaps P.A.s can't name 12 different kinds of small-vessel disease. (I'm making up that number; I certainly can't name them, either — not off the top of my head.) But such encyclopedic knowledge is sometimes transcended, or rendered irrelevant, by practical considerations. On the other hand, you don't want to miss the rare diseases when they do pop up. So both kinds of training (practical and theoretical encyclopedic) are essential, even complementary. The trick is to appreciate both kinds of thinking at the same time, even though most people find one of them more congenial than the other.
A similar divide is evident in the Jewish tradition, especially with regard to Torah study. In previous generations — and even today in some circles — memorizing vast tracts of the Talmud is a prized skill. On the other hand, if you've dipped into any of the Talmud, you know that it takes considerable exegetical ingenuity to derive laws applicable to everyday life from the rich stew of magic, folklore, intellectual speculation, casual yeshiva talk and campfire tales about the destroyed Temple. (Even the strict constructionists, those who actually bury their fingernail clippings according to the magical practice of the rabbis, still have to exercise creativity in deriving everyday religious practice from Babylonian-Jewish digression.) The two skills — encyclopedic knowledge of the Talmud and the ability to apply the Talmud to everyday religious life — are complementary, but rarely are found in the same person. Many of the greatest talmudic scholars have famously refused to adjudicate Jewish law, protesting that they could see all sides of any issue.
Today's Jewish communities rely very much on what some people call "para-rabbis" — people who aren't religious scholars but are reliable, even expert, in the details of daily practice. You might have met some of them: the guy who knows everything about building a sukkah; the woman who is obsessed with reading Torah; the member of the local synagogue who knows exactly how to help a recently bereaved family.
In other words, Jews (at least religious Jews) depend on "rabbis' assistants." In the same way, a hospital must cobble together a vast number of people who have different abilities. Some sit in offices and absorb themselves in the study of disorders affecting 10 people in the entire world, while others, down amid the gore in the E.R., can contradict those same scholars with absolute certainty if they see in front of them a practical problem they know how to solve quickly and effectively.
Bit by bit, I hope to amass both kinds of knowledge till I can be considered a doctor (a learned person, according to the word's historical meaning) in at least one of these categories. But no matter what, there always will be plenty of material for my upcoming masterwork, "What Zack Berger Doesn't Know." I look forward to doing the research.
Zackary Sholem Berger is becoming a doctor, hour by sleepless hour. Feel free to write him about it at doctor@forward.com.
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