Person B saying “there is a solution” provides person A with useful information.
Little details, such as the speed at which another person finds the solution (and the fact that they found it at all) gives clues as to what type of problem it is—divergent or convergent thinking, overall hardness, etc.
The fact that a specific person x was able to find the solution narrows the space to “things that person x would be good at solving”.
Finally, the resources which another person put into finding the solution provide a rough upper bound to how many resources the seeker will have to devote to find it for himself, reducing the risk involved in the investment.
All of these effects are social in nature, which means that it is not unlikely that we humans have in-built mechanisms to use this information without being able to consciously articulate what exactly the information we have gained is.
That someone found the solution cannot be relevant in cases where it’s known that there is a solution, where this effect seems to still apply. I don’t see how one could extract anything about divergent or convergent thinking, since you don’t know how they solved it or usually how long they took; if you knew how long it took and you knew whether they tended towards convergent thinking, then you could infer whether you should focus harder on convergent or divergent thinking, but if you know neither...?
I think my explanation of my thoughts is lacking, let me give a specific example of what I mean.
Imagine a teacher with a penchant for pointless questions ask non-mathematics students the following question:
“What is 6+7+8+9+...+347”?
Most of the students in the classroom will begin dutifully adding the numbers up. Some of them won’t even bother—they’ve estimated the time it will take and it isn’t worth the effort to solve such uninteresting busywork.
Of course, someone will take about five seconds to shout out that they have an answer.
Now the other students know that there is a way to solve the problem that doesn’t involve investing a large amount of time. They’ll get out of “let’s tediously add all the numbers” mode and go into “let’s find a quick shortcut to solving this” mode.
Everyone knew a solution existed, but they didn’t imagine it would be the quick, clever sort of solution until someone actually solved it quickly. The fact that someone found the answer without investing large amounts of time and resources into the problem gave them vital information about the best method for finding the answer.
One could also appeal to the story about Gauss as a child adding up 1..100 by a clever trick, and none of his classmates figuring it out despite clearly seeing that Gauss must’ve done something clever.
But notice how your example does not fit my points: “since you don’t know how they solved it or usually how long they took”; in this case, you have a very good estimate of how long it will take them to use the O(n) summation algorithm from all your past sums, and since you were all assigned the problem at the same time, you also know precisely how long it took them.
In the Shannon anecdote, you know nothing about how long it took the brother to answer it nor, given how heterogenous puzzles can be, how long it might take him to solve it, nor is there even any ‘brute force’ approach for most puzzles which you could compare against a ‘clever’ approach and so choose to look for a clever approach rather than spend more time executing the brute force approach.
Similarly for web searching, there’s typically no brute force approach at all: if Google spits out a list of 10 hits total for the paper title and you look at all 10 and they fail, then what? What’s the dumb brute force approach in searching? You simply have to try another ‘clever’ approach, because you’ve exhausted all your available data.
Sorry, you’re right, I didn’t read your previous post carefully enough.
I agree that if this phenomenon is real, in order to explain it in terms of a rational agent you do need to either know something about the person who solved it, or how long they took, or some other detail about them in order for this to be helpful in any way.
In the real world, however, a declaration of having solved the problem always leaves some sort of knowledge. In the web search case that just unfolded in this thread, by posting a solution you leaked the information that a solution existed and that it didn’t take an unreasonable amount of time to figure out, which provided Benja additional incentive to start looking for a clever approach.
I’ll agree that it does seem like there is more than simple information gain going on here though. Perhaps there are other factors, such as the insertion of an element of competition?
I’ll agree that it does seem like there is more than simple information gain going on here though. Perhaps there are other factors, such as the insertion of an element of competition?
Certainly seems possible. I admit I tend to announce the time it took to find something that someone failed to as part of showing off and elevating myself, so it would be no surprise if the recipient felt shamed and inflamed into looking better—the difference between peak and average performance might explain the differential.
Person B saying “there is a solution” provides person A with useful information.
Little details, such as the speed at which another person finds the solution (and the fact that they found it at all) gives clues as to what type of problem it is—divergent or convergent thinking, overall hardness, etc.
The fact that a specific person x was able to find the solution narrows the space to “things that person x would be good at solving”.
Finally, the resources which another person put into finding the solution provide a rough upper bound to how many resources the seeker will have to devote to find it for himself, reducing the risk involved in the investment.
All of these effects are social in nature, which means that it is not unlikely that we humans have in-built mechanisms to use this information without being able to consciously articulate what exactly the information we have gained is.
That someone found the solution cannot be relevant in cases where it’s known that there is a solution, where this effect seems to still apply. I don’t see how one could extract anything about divergent or convergent thinking, since you don’t know how they solved it or usually how long they took; if you knew how long it took and you knew whether they tended towards convergent thinking, then you could infer whether you should focus harder on convergent or divergent thinking, but if you know neither...?
I think my explanation of my thoughts is lacking, let me give a specific example of what I mean.
Imagine a teacher with a penchant for pointless questions ask non-mathematics students the following question:
“What is 6+7+8+9+...+347”?
Most of the students in the classroom will begin dutifully adding the numbers up. Some of them won’t even bother—they’ve estimated the time it will take and it isn’t worth the effort to solve such uninteresting busywork.
Of course, someone will take about five seconds to shout out that they have an answer.
Now the other students know that there is a way to solve the problem that doesn’t involve investing a large amount of time. They’ll get out of “let’s tediously add all the numbers” mode and go into “let’s find a quick shortcut to solving this” mode.
Everyone knew a solution existed, but they didn’t imagine it would be the quick, clever sort of solution until someone actually solved it quickly. The fact that someone found the answer without investing large amounts of time and resources into the problem gave them vital information about the best method for finding the answer.
One could also appeal to the story about Gauss as a child adding up 1..100 by a clever trick, and none of his classmates figuring it out despite clearly seeing that Gauss must’ve done something clever.
But notice how your example does not fit my points: “since you don’t know how they solved it or usually how long they took”; in this case, you have a very good estimate of how long it will take them to use the O(n) summation algorithm from all your past sums, and since you were all assigned the problem at the same time, you also know precisely how long it took them.
In the Shannon anecdote, you know nothing about how long it took the brother to answer it nor, given how heterogenous puzzles can be, how long it might take him to solve it, nor is there even any ‘brute force’ approach for most puzzles which you could compare against a ‘clever’ approach and so choose to look for a clever approach rather than spend more time executing the brute force approach.
Similarly for web searching, there’s typically no brute force approach at all: if Google spits out a list of 10 hits total for the paper title and you look at all 10 and they fail, then what? What’s the dumb brute force approach in searching? You simply have to try another ‘clever’ approach, because you’ve exhausted all your available data.
Sorry, you’re right, I didn’t read your previous post carefully enough.
I agree that if this phenomenon is real, in order to explain it in terms of a rational agent you do need to either know something about the person who solved it, or how long they took, or some other detail about them in order for this to be helpful in any way.
In the real world, however, a declaration of having solved the problem always leaves some sort of knowledge. In the web search case that just unfolded in this thread, by posting a solution you leaked the information that a solution existed and that it didn’t take an unreasonable amount of time to figure out, which provided Benja additional incentive to start looking for a clever approach.
I’ll agree that it does seem like there is more than simple information gain going on here though. Perhaps there are other factors, such as the insertion of an element of competition?
Certainly seems possible. I admit I tend to announce the time it took to find something that someone failed to as part of showing off and elevating myself, so it would be no surprise if the recipient felt shamed and inflamed into looking better—the difference between peak and average performance might explain the differential.