[Vampyr]

smerriman, on 2018-April-13, 15:17, said:

I wouldn't say it's impossible to figure out that if 4 bananas represents the number 4, 3 bananas represents the number 3. Sure, you could define a picture of 3 bananas to mean 28 if you really wanted to, but on the silliness scale, I'd rate that sillier than some other uses of the word silly here

But then where is the “puzzle”aspect? Though I must admit that little kids like this sort of thing. Perhaps it came from Highlights magazine.

[cherdano]

I think this is a great funny little puzzle if you get it, and an annoying silly dumb puzzle if you don't...

[barmar]

Vampyr, on 2018-April-13, 15:40, said:

But then where is the “puzzle”aspect? Though I must admit that little kids like this sort of thing. Perhaps it came from Highlights magazine.

All puzzles involving analogies and pattern matching require making somewhat arbitrary decisions about what features are relevant. But these are still considered useful tests because that's one of the main components of general intelligence and intuition. How do infants pick up on language on their own? They discern the salient features that distinguish objects, and notice the relationships to the sounds they hear around the same time. No one ever tells someone the similarities and difference between cats, dogs, and other small mammals, we just notice them intuitively.

So while you could make arbitrary mappings, that's not natural. The point is to look for salient features.

In this puzzle, pointing out the wrong answers forces the puzzler to look more carefully for additional details. The initial formulas look like it's just "a picture represents a number arbitrarily". When we're told that this simple mapping doesn't produce the right answer, we have to look for some other relationship. And in this case, you notice that there's something numeric in each of the pictures that corresponds directly to the initial mappings (e.g. bunch of bananas = 4, and there happen to be 4 bananas in each bunch), and realize that this isn't just a coincidence, it's the whole point.

On the other hand, assuming that coincidences are significant is also a common cause of falacious reasoning. We often give too much significance to coincidences, because we assume that everything happens for a reason. If you're thinking about someone at the time they call you, you feel that something "spooky" has happened, because you forget about all times that you get calls from someone you weren't thinking about at the time.

But an important step in critical thinking is figuring out when correlations are significant versus just happenstance. So in this puzzle, you might first think that the banana relationship is just a coincidence. But then you notice the same relationship with the clock. Finally, it takes a little more work, but you can find the same relationship with the geometric figures. And then you put this together with the general understanding that puzzles aren't random (because they'd be insoluble if you could just make up any rule you want), they're intended to make you recognize patterns, so it becomes clear that these correlations are obviously the solution.

And if you can't solve it yourself, the fact that you kick yourself for not seeing it when someone explains it to you is also an indication that this relationship is somewhat natural.

[kenberg]

I agree with the above. Now it is probably (well, at least probably) a mistake to make too much of this but Barry's point about insolubility has a lot going for it. If we agree that the three banana bunch is at least different from the two banana bunch, and the clock saying 2 is different from the clock saying 3, and the three nested polygons are different from the two polygon figures then, and we don't need any deep mathematics to agree with this, we have way too many unknown quantities for us to able to work this out simply through mathematics. Basically, it isn't mathematics, it is something else. So what is it? Of course the answer has to be a bit of a guess.

Barry earlier commented that maybe his years of doing computer science actually interfered with his ability to do this. Exactly. It's the old saw that if you are a hammer everything looks like a nail. The first thing to do is to decide "I might be a hammer, but this problem is not a nail". After that, all goes well, or at least it might.

Long ago I was driving down to Roanoke with Bill and Marge Wilson, both now deceased (Kevin Wilson you will find on BBO, he is their son). We stopped for lunch, chatted with the guy making the sandwiches, disclosed we were going for a bridge tournament. When he learned that I was a mathematician he ohh-ed and ahh-ed about how useful that must be for bridge. Marge said that yes, not doubt it is, but it also helps to have a little common sense. Absolutely.

PS The problem that Zel mentions earlier about the points on a circle: That's a math problem. Points are points, regions are regions, you need to assume the points are "in general position" and then you just count (abstractly). I was joking about actually drawing it with 13 points.

[gwnn]

I like the principle espoused here:
http://www.flyingcol...ion-genius-sic/

Quote

The resolution I would like you to keep is: start calling these things fake maths. Right there, in the comments. Instead of saying "It's 9, you morons!" or "It's 1, you morons!" or "It's ambiguously written, you morons!", say "This is fake maths."

I love you MrAce but I hate these "genius" puzzles.

[barmar]

From that site:

Quote

In most cases, they are either a) flat-out wrong, b) not uniquely determined, or c) attempts to get hold of your details on Facebook. Often all three.

Very little in real life is "uniquely determined". Language is incredibly ambiguous, yet we generally get along fine with it. Puzzles like these exercise the reasoning methods that allowed humans to become the dominant vertebrate on the planet (was going to say "dominant species", but it can be argued that some insects and bacteria are comparable due to their quantity and ubiquity).

[kenberg]

The original post describes it as a riddle. Mathematics was not mentioned. Yes there are plus signs, equal signs and so on, but it is a riddle.

But a variety of things come to mind as this discussion goes on.

I basically don't use Facebook so I was not aware of what seems to be an ongoing debate. There was a recent NYT article describing how I can have FB download the data that they have on me. I did this and it was laughably meager. My view of FB is such that I don't entirely trust that this is all that they have on me, but even if we triple the amount it would be very meager. So I am just out of it as far as any ongoing aggravation with FB is concerned.

But issues about how questions can, intentionally or unintentionally, be ambiguous or misleading are of interest. Trigger warning: Slight brag ahead. Long ago I took the GREs. Who cares what the letters stand for, you take them as you finish college. One of them was on general scientific knowledge. I believe I got every question except one correct. That one I did not answer. I was sure that I knew what they wanted but I thought the question was so badly phrased that there really was no decent answer to it and so I refused to answer it. I can be stubborn. I think stubborness can be a positive trait, but not always.

But here is a modern update. The AP exams for Calculus are in mid-May and it crossed my mind to volunteer at the local high school to help students who may be preparing for them. I looked at some of the questions and there are some that I really like, I think they are very good questions and I think I could help students with some of the subtleties. But then I started looking over some of the instructions. On the non-multiple choice questions you get one point for the correct answer and the rest of the points, however many, are for how you write up the answer. Uh oh. I got the feeling I might be doing more harm than good if I come in and guide them to the solution but I don't know just what they are supposed to write down. My younger daughter is now 50, but when she was young she was having trouble in her algebra course. I was a little wary of butting in but I decided I had to. But I took my time. I did not show her how I would work the problem, I had her explain as best she could what her teacher had said and then I helper her to understand what her teacher was getting at. This worked, it worked very well, but you can see the point. The teacher was not saying anything wrong but the teacher's approach might well have been different from my approach. I accepted the teacher's approach and we worked with that. With the AP, getting at just what they have in mind might be more demanding than I am up for as a volunteer.

Anyway, language indeed has ambiguities.Speaking to someone face to face, a misunderstanding often is noted and corrected at once. In written materials with no allowance for questions as to just what is meant, this can be a much trickier situation.

Back to the puzzle. Yes, it can be frustrating. That was its intent. In no sense should it be considered an intelligence test or a mathematics test or anything like that. It's a riddle.

PS I am still working on the fruit problem where the sum is 4. Originally it did not come up clearly on my browser and I thought the sum was to be 1. That's impossible. But 4? It's spring and I have some yard work to do. That's my excuse. But I will see what I can do.

[gwnn]

Don't overdo it Ken (it=the other fruit puzzle).

[barmar]

kenberg, on 2018-April-17, 09:28, said:

But here is a modern update. The AP exams for Calculus are in mid-May and it crossed my mind to volunteer at the local high school to help students who may be preparing for them. I looked at some of the questions and there are some that I really like, I think they are very good questions and I think I could help students with some of the subtleties. But then I started looking over some of the instructions. On the non-multiple choice questions you get one point for the correct answer and the rest of the points, however many, are for how you write up the answer. Uh oh. I got the feeling I might be doing more harm than good if I come in and guide them to the solution but I don't know just what they are supposed to write down. My younger daughter is now 50, but when she was young she was having trouble in her algebra course. I was a little wary of butting in but I decided I had to. But I took my time. I did not show her how I would work the problem, I had her explain as best she could what her teacher had said and then I helper her to understand what her teacher was getting at. This worked, it worked very well, but you can see the point. The teacher was not saying anything wrong but the teacher's approach might well have been different from my approach. I accepted the teacher's approach and we worked with that. With the AP, getting at just what they have in mind might be more demanding than I am up for as a volunteer.

I think you may have been getting too worked up about this.

Yes, there's some subjectivity, but it's unlikely they require the student to come up with a specific answer. The purpose of this it to determine whether the student really understands the underlying process. If there are multiple ways of approaching it, any of them would likely be acceptable.

I don't remember having to do that when I took the AP Calc exam. I guess at that time they just assumed that if you got enough of the answers right, you must understand what's going on, so there was no need to complicate it. But there's probably reasons why they've gone the extra mile these days.

[kenberg]

gwnn, on 2018-April-17, 11:43, said:

Don't overdo it Ken (it=the other fruit puzzle).

I appreciate your warning and followed the implicit advice and read/scanned the article you indicated. I had converted it to problem involving a cubic in two variables. I have heard a lecture or two about the relationship of this to cryptography but I can't say I either followed them or much remembered them, so I went back to a more primitive approach. It's one of those things where until you arrive you have no idea if you are making any real progress. After reading your note I decided maybe I should surrender. A good choice.

This is in fact a very good problem. Some time back (well, 20 some years back) on a road trip to a regional my car mates asked me to explain about Fermat's Last Theorem. No, they did not want me to give them the proof (whew!) they wanted to understand what the problem was. I think by the time we arrived they more or less understood the point. This problem is similar in spirit. With a little thought, people can understand what is to be done. It is very very simple to understand what is to be done. But then doing it. That's hard. It may then be harder to appreciate why anyone would care, but that's a personal thing.

Mathematics has a great many aspects to it, but one of them is that a problem that is very easy to state can be very difficult to solve.

[barmar]

kenberg, on 2018-April-17, 14:35, said:

Mathematics has a great many aspects to it, but one of them is that a problem that is very easy to state can be very difficult to solve.

Right. The "even better puzzle" looks like it should be simple algebra. The 95% number seems reasonable in that regard, since most people don't remember how to do this (if they even understood it when they were in high school). But it turns out to be very advanced mathematics that could probably only be solved by a college math major. And many of the math theorems that stumped the pros for centuries are very simple to state.

Just looking at the formulas, it's hard to see why the Pythagorean Theorem is trivial to prove, while Fermat's Last Theorem took so long and required application of many different areas of advanced math (most of which didn't even exist when Fermat formulated the problem). As much of a genius as he was, he was almost certainly mistaken when he wrote "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain."

[cherdano]

barmar, on 2018-April-18, 09:24, said:

Right. The "even better puzzle" looks like it should be simple algebra. The 95% number seems reasonable in that regard, since most people don't remember how to do this (if they even understood it when they were in high school). But it turns out to be very advanced mathematics that could probably only be solved by a college math major.

You are quite wrong. I would guess that among research mathematicians (I am one) 90% would not know how to solve this problem. My own field is quite close to number theory, so I know enough of the theory - but it would still take me quite a long time to do the computations (in particular playing around to find the first solution).

[barmar]

cherdano, on 2018-April-18, 09:35, said:

You are quite wrong. I would guess that among research mathematicians (I am one) 90% would not know how to solve this problem. My own field is quite close to number theory, so I know enough of the theory - but it would still take me quite a long time to do the computations (in particular playing around to find the first solution).

Good point, although I could point out that I didn't say "any math major" or even "most math majors". A mathematician should know the difference between "necessary" and "sufficient" conditions.