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Open-Ended Math Questions?

Ben Tilly said:
An advanced being shows you two boxes. In box A there is $1000. In box B there is $0 or $1000000. You are allowed to take either box B or both boxes, but the being will put the million in box B only if it thinks you will only take box B. Assuming that you have never seen the entity make a mistake on this problem, what should you do to make as much money as you can?

This is called Newcomb's problem. Most people find the answer obvious. They just disagree on what the answer is. In fact the answer is not well-defined. (For the record I'd only take box B.)
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As phrased, the answer is well-defined and indeed obvious:

"What should you do to make as much money as you can?" Fool the advanced being into believing that you will only take box B, but actually take both boxes. The actual method for fooling the advanced being is unfortunately left as an excercise for the student.....
 
Hard logic problem

DeviousB said:
You are visiting an island far away. On this island the natives are twice as likely to tell you lie as speak the truth (they don't like foreigners much).

One afternoon you reach a fork in the road, where two natives are sat, and ask them which road leads to the beach.

"That one." says the first native, pointing out one of the paths.

"Yeah, what he said." says the second.

What are the chances that the path pointed to goes to the beach?
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Ah yes. Doors, paths, odd restrictions...what does that remind me of.

Everyone knows the question, In front of you is a liar and a truthteller. The liar always lies, the truth teller always tells the truth. With 1 yes/no question, how do you figure out which is which? Easy, peasy.

Let's make it harder. In front of you are three people. One always lies. One always tells the truth. And one sometimes lies and sometimes does not. With 3 yes/no questions, how can you figure out which is which?

Not nearly as easy. :-)

Cheers,
Ben
 
Ben Tilly said:
Ah yes. Doors, paths, odd restrictions...what does that remind me of.

Everyone knows the question, In front of you is a liar and a truthteller. The liar always lies, the truth teller always tells the truth. With 1 yes/no question, how do you figure out which is which? Easy, peasy.

Let's make it harder. In front of you are three people. One always lies. One always tells the truth. And one sometimes lies and sometimes does not. With 3 yes/no questions, how can you figure out which is which?

Not nearly as easy. :-)

Cheers,
Ben
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Regarding the second scenario, do all three know which among them is the truthteller and which is the liar?
 
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sir drinks-a-lot said:
It depends on if you are limiting your answer to integers, or if you are looking for real numbers as well.

The first (lowest) uninteresting integer is 41, and the first uninteresting real number is 17.41. However, this is in dispute. To some, the fact that the string "41" is included in the numeral "17.41" is in and of itself interesting. To these people, the first uninteresting real number would be 17.92.
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But isn't it interesting that 41 is the first uninteresting number?
 
DeviousB said:
Well done 69dodge.
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Hmpf. I'd say the problem was badly worded. It depends on whether the exception takes precedence over the general case or not. Typically it does.

It's the difference between "B lies 2/3 of the time, unless A lies, in which case B lies 1/2 of the time" and "If A lies, B lies 1/2 of the time, but overall B lies 2/3 of the time."
 
mattlodder said:
But isn't it interesting that 41 is the first uninteresting number?
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Actually 39 is the first uninteresting number, which makes it paradoxically interesting. 41 is the first non-paradoxical uninteresting number, which is not enough to make it interesting.
 
Beleth said:
Actually 39 is the first uninteresting number, which makes it paradoxically interesting. 41 is the first non-paradoxical uninteresting number, which is not enough to make it interesting.
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I'd say that was pretty interesting, wouldn't you?

This recursive-loop set-theory is probably the only thing I remember in any detail from my highschool maths class... Mr Griffiths, damn you.
 
Ben Tilly said:
Ah yes. Doors, paths, odd restrictions...what does that remind me of.

Everyone knows the question, In front of you is a liar and a truthteller. The liar always lies, the truth teller always tells the truth. With 1 yes/no question, how do you figure out which is which? Easy, peasy.

Let's make it harder. In front of you are three people. One always lies. One always tells the truth. And one sometimes lies and sometimes does not. With 3 yes/no questions, how can you figure out which is which?

Not nearly as easy. :-)

Cheers,
Ben
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If the liar and truthteller know who the flip-flopper is
"Number 1, what would your response be to the question, 'Are you the truthteller?'?"

"Number 2, what was Number 1's response?" This question determines the identity of either the liar or truthteller.

"Hey, liar/truthteller, is Number 1 the flip-flopper?"
 
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insomneac said:
If the liar and truthteller know who the flip-flopper is
"Number 1, what would your response be to the question, 'Are you the truthteller?'?"

"Number 2, what was Number 1's response?" This question determines the identity of either the liar or truthteller.

"Hey, liar/truthteller, is Number 1 the flip-flopper?"
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Good try, but it isn't that simple. Here is what's wrong with that solution.

Let's label our three T(ruthteller), E(ither), L(iar). Then a response of "Yes" and "Yes" could happen with TEL, ETL, LET, or LTE. The either person can still be any of the three, so I'm not sure who you'd direct the third question to.


If you want the answer, here it is.

Ask the first person whether the second person is less likely to tell the truth than the third person. The person who is said to be less likely must be either the liar or truth teller. (The truth teller points you to the liar, the liar points you to the truth teller, and it doesn't matter what Mr E. says.) The second question establishes which, and the third sorts the other two.
 
Ben Tilly said:
First of all an amusing piece of trivia. In the 50's in a random sample of 5th grade teachers, more than half could not correctly decide whether 2/3 is larger than 3/5. (1) Based on anecdotal evidence, I don't think that teachers today would do any better than they did 50 years ago. So innumeracy is not a shock for the general public.
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Anecdotal evidence is usually worthless; at best it is a good starting point for designing a serious study. I know a building full of teachers, K-5, who know that 2/3 is larger than 3/5 - however, that is another piece of anecdotal evidence. Please don't make generalized statements about a group of professionals. I don't mean to be touchy, but there is a lot of bunk being passed off as truth about education (especially in the United States).
 
This puzzle is no longer a derail

Ben Tilly said:
Good try, but it isn't that simple. Here is what's wrong with that solution.

Let's label our three T(ruthteller), E(ither), L(iar). Then a response of "Yes" and "Yes" could happen with TEL, ETL, LET, or LTE. The either person can still be any of the three, so I'm not sure who you'd direct the third question to.


If you want the answer, here it is.

Ask the first person whether the second person is less likely to tell the truth than the third person. The person who is said to be less likely must be either the liar or truth teller. (The truth teller points you to the liar, the liar points you to the truth teller, and it doesn't matter what Mr E. says.) The second question establishes which, and the third sorts the other two.
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The conclusions we draw from the responses are more important than the questions. My solution also identifies either the liar or truthteller in two questions and sorts the others with the third. I can't elaborate further until tomorrow morning.
 
donq said:
Anecdotal evidence is usually worthless; at best it is a good starting point for designing a serious study. I know a building full of teachers, K-5, who know that 2/3 is larger than 3/5 - however, that is another piece of anecdotal evidence. Please don't make generalized statements about a group of professionals. I don't mean to be touchy, but there is a lot of bunk being passed off as truth about education (especially in the United States).
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Speaking of anecdotal evidence, you may wish to actually ask those teachers that question before claiming that they will all get that answer correct. When I first ran across the study from the 50's, I expressed my astonishment to a grade 1 teacher that I knew well, only to be shocked when she got the answer wrong.

Now they may get that answer correct. However your building is just as anecdotal as my experiences. And the study done 50 years ago (which I encountered 20 years ago and no longer have) is by now irrelevant. You'd need to do the study again. While I won't admit to confidence that over half would get it wrong, a significant proportion will.

Most of us have difficulty accepting that what we consider common sense is not common sense for everyone that we know. I have collected a number of examples over the years. My favorite example is very simple. Draw two cups, one upright and one tilted by 30 degrees. Present them to a person, say that the cups are supposed to be half full of water and ask that person to draw the water lines in.

It looks like a trivial test. And it is. But the funny thing is that most adult women will get it wrong. (If I remember correctly, about 70% of women and 10% of men can't do this.) In my experience very few men think this is possible until they actually see it in action. You are probably one of them. If so, then I suggest trying this on several women you know before you respond. You may get a surprise.

Some background on this. This is called the Piaget water-line test. The background is that Piaget discovered that many cognitive skills are linked to specific developmental stages in children, until you hit that developmental stage you do not have that skill. Examples of clearly linked skills include the ability to understand that not everyone has seen what you have and knowledge that water does not change its volume when it changes its shape.

Anyways this skill is one which people aquire during puberty. Then most men develop it and most women do not. There are some corresponding verbal skills that women develop and men do not. (I read about them at the same time that I read about this. I got that question wrong, forgot the article, remembered it a month later then was astonished when my mother and girlfriend got this question wrong. Ever since I've wanted to find more references on this, but I haven't succeeded.) There is no obvious correlation between these abilities and general intelligence, education, or success in life.

Cheers,
Ben
 
insomneac said:
The conclusions we draw from the responses are more important than the questions. My solution also identifies either the liar or truthteller in two questions and sorts the others with the third. I can't elaborate further until tomorrow morning.
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I am obviously unable to find the hole in your reasoning until your reasoning is presented. However I have demonstrated to my satisfaction that your first two questions do not suffice.

Cheers,
Ben
 
Ben Tilly said:
I am obviously unable to find the hole in your reasoning until your reasoning is presented. However I have demonstrated to my satisfaction that your first two questions do not suffice.

Cheers,
Ben
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No need for that, I found the error myself. It's the same one you pointed out earlier in this thread. No sleep and no food make insomneac look stoopid.
 
A variation of the "uninteresting number" paradox: what is the smallest integer that cannot be unambiguously defined by any English phrase of seventeen words or fewer?

Also, I realize that the previous discussion of 0.999... = 1.0 is very old. But I'm surprised that nowhere in it did anyone suggest that for all the fretting about infinite decimal expansions of real numbers, there's actually no such thing as a finite decimal expansion of a real number.

"1.0" represents an expression consisting of a 1, a decimal point, and unlimited repetition of the symbol "0". For any decimal expansion that ends in a digit rather than ellipses or a "repeating" bar over the rightmost digit(s), the appending of unlimited repetition of "0" is assumed, and is represented by the white space on the page to the right of the numerals just as clearly as in e.g. "0.333..." the ellipsis represents unlimited repetition of the final digit printed. Just because we drop the zeros and omit the ellipsis in the special case of repeating zeros, doesn't mean they don't belong there. In that light, the expression ".999..." becomes no more mysterious than "1.000..." or (to abbreviate) "1.0".

At least, that's what I remember being taught in HS algebra class. Was I misled?

Respectfully,
Myriad
 
Myriad said:
A variation of the "uninteresting number" paradox: what is the smallest integer that cannot be unambiguously defined by any English phrase of seventeen words or fewer?
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Suppose f is a function from {1,2,3,...N} to the positive integers. What is the smallest positive integer K for which there is no x such that f(x)=K? What if you knew that for n<N, f(n)=n?
 
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