"I guess if you really feel the question is so confused as to be answerless, I'll accept that. I would still challenge you to fill in plausible assumptions and state a preference."

Remember the story, "The Lady and the Tiger"? The question was carefully formulated to be evenly balanced, to eliminate any reason to choose one over the other. Anything that got used to say one choice was better, implied that the story wasn't balanced quite right.

We could do that with your story too. If 3^^^3 people is enough to say it's better to torture one person, we could replace it with a smaller number, perhaps a googleplex. And if that's still too many we could try just a google. If people choose the specks we could increase the number of people, or maybe increase the number of specks.

At some point we get just the right number of specks to balance the torture for a modal number of people, and we're set. The maximum number of people will be unable to choose, because you designed it that way.

You did not say what happens if I don't choose. This is a glaring omission.

OK, let me tell one. You and your whole family have been captured by the Gestapo, and before they get down to the serious torture they decide to have some fun with you. They tell you that you have to choose, either they rape your daughter or your wife. If you don't choose which one then they'll rape them both. And you too.

Do you choose? If you refuse to choose then that's choosing for both of them to be raped. And you too.

But then, if you do choose, they rape them both anyway. And you too.

What should you do?

I make it a habit to learn as little as possible by rote, and just derive what I need when I need it.

Do realize that you're trading efficiency (as in speed of access in normal use) for that space saving in your brain. Memorizing stuff allows you to move on and save your mental deducing cycles for really new stuff.

Back when I was memorizing the multiplication tables, I noticed that

9 x N = 10 x (N-1) + (9 - (N-1))

That is, 9 x 8 = 70 + 2

So, I never memorized the 9's the same way I did all the other single digit multiplications. To this day I'm slightly slower doing math with the digit 9. The space/effort saving was worth it when I was 8 years old, but definitely not today.

I think something else is going on with the 2 4 6 experiment, as described. Many of the students are making the assumption about the set of potential rules. Specifically, the assumption is that most pairs of rules in this set have the following mutual relationship: most of the instances allowed by one rule, are disallowed by the other rule. This being the case, then the quickest way to test any hypothetical rule is to produce a variety of instances which conform with that rule, to see whether they conform with the hidden rule.

I'll give you an example. Suppose that we are considering a family of rules, "the third number is an integer polynomial of the first two numbers". The quickest way to disconfirm a hypothetical rule is to produce instances in accordance with it and test them. If the rule is wrong, then the chances are good that an instance will quickly be discovered that does not match the hidden rule. It is much less efficient to proceed by producing instances not in accordance with it.

I'll give a specific example. Suppose the hidden rule is c = a + b, and the hypothesized rule being tested is c = a - b. Now pick just one random instance in accordance with the hypothesized rule. I will suppose a = 4, b = 6, so c = -2. So the instance is 4 6 -2. That instance does not match the hidden rule, so the hypothesized rule is immediately disconfirmed. Now try the following: instead of picking a random instance in accordance with the hypothesized rule, pick one not in accordance with it. I'll pick 4 6 8. This also fails to match the hidden rule, so it fails to tell us whether our hypothesized rule is correct. We see that it was quicker to test an instance that agrees with the hypothetical rule.

Thus we can see that in a certain class of situations, the most efficient way to test a hypothesis is to come up with instances that conform with the hypothesis.

Now you can fault people on having made this assumption. But if you do, then it is still a different error from the one describe. If the assumption about the kind of problem faced had been correct, then the approach (testing instances that agree with the hypothesis) would have been a good one. The error, if any, lies not in the approach per se but in the assumption.

Finally, I do not think one can rightly fault people for making that assumption. For, it is inevitable that very large and completely untested assumptions must be made in order to come to a conclusion at all. For, infinitely many rules are consistent with the evidence no matter how many instances you test. The only way ever to whittle this infinity of rules consistent with all the evidence down to one concluded rule is to make very large assumptions. The assumption that I have described may simply be the assumption which they made (and they had to make some assumption).

Furthermore, it doesn't matter what assumptions people make (and they must make some, because of the nature of the problem), a clever scientist can learn what assumptions people tend to make and then violate those assumptions. So no matter what people do, someone can come along, construct an experiment in which those assumptions are violated, and then say, "gotcha" when the majority of his test subjects come to the wrong conclusions (because of the assumptions they were making which were violated by the experiment).

I think something else is going on with the 2 4 6 experiment, as described. Many of the students are making the assumption about the set of potential rules. Specifically, the assumption is that most pairs of rules in this set have the following mutual relationship: most of the instances allowed by one rule, are disallowed by the other rule. This being the case, then the quickest way to test any hypothetical rule is to produce a variety of instances which conform with that rule, to see whether they conform with the hidden rule.

I'll give you an example. Suppose that we are considering a family of rules, "the third number is an integer polynomial of the first two numbers". The quickest way to disconfirm a hypothetical rule is to produce instances in accordance with it and test them. If the rule is wrong, then the chances are good that an instance will quickly be discovered that does not match the hidden rule. It is much less efficient to proceed by producing instances not in accordance with it.

I'll give a specific example. Suppose the hidden rule is c = a + b, and the hypothesized rule being tested is c = a - b. Now pick just one random instance in accordance with the hypothesized rule. I will suppose a = 4, b = 6, so c = -2. So the instance is 4 6 -2. That instance does not match the hidden rule, so the hypothesized rule is immediately disconfirmed. Now try the following: instead of picking a random instance in accordance with the hypothesized rule, pick one not in accordance with it. I'll pick 4 6 8. This also fails to match the hidden rule, so it fails to tell us whether our hypothesized rule is correct. We see that it was quicker to test an instance that agrees with the hypothetical rule.

Thus we can see that in a certain class of situations, the most efficient way to test a hypothesis is to come up with instances that conform with the hypothesis.

Now you can fault people on having made this assumption. But if you do, then it is still a different error from the one describe. If the assumption about the kind of problem faced had been correct, then the approach (testing instances that agree with the hypothesis) would have been a good one. The error, if any, lies not in the approach per se but in the assumption.

Finally, I do not think one can rightly fault people for making that assumption. For, it is inevitable that very large and completely untested assumptions must be made in order to come to a conclusion at all. For, infinitely many rules are consistent with the evidence no matter how many instances you test. The only way ever to whittle this infinity of rules consistent with all the evidence down to one concluded rule is to make very large assumptions. The assumption that I have described may simply be the assumption which they made (and they had to make some assumption).

Furthermore, it doesn't matter what assumptions people make (and they must make some, because of the nature of the problem), a clever scientist can learn what assumptions people tend to make and then violate those assumptions. So no matter what people do, someone can come along, construct an experiment in which those assumptions are violated, and then say, "gotcha" when the majority of his test subjects come to the wrong conclusions (because of the assumptions they were making which were violated by the experiment).

I think something else is going on with the 2 4 6 experiment, as described. Many of the students are making the assumption about the set of potential rules. Specifically, the assumption is that most pairs of rules in this set have the following mutual relationship: most of the instances allowed by one rule, are disallowed by the other rule. This being the case, then the quickest way to test any hypothetical rule is to produce a variety of instances which conform with that rule, to see whether they conform with the hidden rule.

I'll give you an example. Suppose that we are considering a family of rules, "the third number is an integer polynomial of the first two numbers". The quickest way to disconfirm a hypothetical rule is to produce instances in accordance with it and test them. If the rule is wrong, then the chances are good that an instance will quickly be discovered that does not match the hidden rule. It is much less efficient to proceed by producing instances not in accordance with it.

I'll give a specific example. Suppose the hidden rule is c = a + b, and the hypothesized rule being tested is c = a - b. Now pick just one random instance in accordance with the hypothesized rule. I will suppose a = 4, b = 6, so c = -2. So the instance is 4 6 -2. That instance does not match the hidden rule, so the hypothesized rule is immediately disconfirmed. Now try the following: instead of picking a random instance in accordance with the hypothesized rule, pick one not in accordance with it. I'll pick 4 6 8. This also fails to match the hidden rule, so it fails to tell us whether our hypothesized rule is correct. We see that it was quicker to test an instance that agrees with the hypothetical rule.

Thus we can see that in a certain class of situations, the most efficient way to test a hypothesis is to come up with instances that conform with the hypothesis.

Now you can fault people on having made this assumption. But if you do, then it is still a different error from the one describe. If the assumption about the kind of problem faced had been correct, then the approach (testing instances that agree with the hypothesis) would have been a good one. The error, if any, lies not in the approach per se but in the assumption.

Finally, I do not think one can rightly fault people for making that assumption. For, it is inevitable that very large and completely untested assumptions must be made in order to come to a conclusion at all. For, infinitely many rules are consistent with the evidence no matter how many instances you test. The only way ever to whittle this infinity of rules consistent with all the evidence down to one concluded rule is to make very large assumptions. The assumption that I have described may simply be the assumption which they made (and they had to make some assumption).

Furthermore, it doesn't matter what assumptions people make (and they must make some, because of the nature of the problem), a clever scientist can learn what assumptions people tend to make and then violate those assumptions. So no matter what people do, someone can come along, construct an experiment in which those assumptions are violated, and then say, "gotcha" when the majority of his test subjects come to the wrong conclusions (because of the assumptions they were making which were violated by the experiment).

I think something else is going on with the 2 4 6 experiment, as described. Many of the students are making the assumption about the set of potential rules. Specifically, the assumption is that most pairs of rules in this set have the following mutual relationship: most of the instances allowed by one rule, are disallowed by the other rule. This being the case, then the quickest way to test any hypothetical rule is to produce a variety of instances which conform with that rule, to see whether they conform with the hidden rule.

I'll give you an example. Suppose that we are considering a family of rules, "the third number is an integer polynomial of the first two numbers". The quickest way to disconfirm a hypothetical rule is to produce instances in accordance with it and test them. If the rule is wrong, then the chances are good that an instance will quickly be discovered that does not match the hidden rule. It is much less efficient to proceed by producing instances not in accordance with it.

I'll give a specific example. Suppose the hidden rule is c = a + b, and the hypothesized rule being tested is c = a - b. Now pick just one random instance in accordance with the hypothesized rule. I will suppose a = 4, b = 6, so c = -2. So the instance is 4 6 -2. That instance does not match the hidden rule, so the hypothesized rule is immediately disconfirmed. Now try the following: instead of picking a random instance in accordance with the hypothesized rule, pick one not in accordance with it. I'll pick 4 6 8. This also fails to match the hidden rule, so it fails to tell us whether our hypothesized rule is correct. We see that it was quicker to test an instance that agrees with the hypothetical rule.

Thus we can see that in a certain class of situations, the most efficient way to test a hypothesis is to come up with instances that conform with the hypothesis.

Now you can fault people on having made this assumption. But if you do, then it is still a different error from the one describe. If the assumption about the kind of problem faced had been correct, then the approach (testing instances that agree with the hypothesis) would have been a good one. The error, if any, lies not in the approach per se but in the assumption.

Finally, I do not think one can rightly fault people for making that assumption. For, it is inevitable that very large and completely untested assumptions must be made in order to come to a conclusion at all. For, infinitely many rules are consistent with the evidence no matter how many instances you test. The only way ever to whittle this infinity of rules consistent with all the evidence down to one concluded rule is to make very large assumptions. The assumption that I have described may simply be the assumption which they made (and they had to make some assumption).

Furthermore, it doesn't matter what assumptions people make (and they must make some, because of the nature of the problem), a clever scientist can learn what assumptions people tend to make and then violate those assumptions. So no matter what people do, someone can come along, construct an experiment in which those assumptions are violated, and then say, "gotcha" when the majority of his test subjects come to the wrong conclusions (because of the assumptions they were making which were violated by the experiment).