That’s interesting! I got the same answer but I visualized it differently. (Imagine, for each possible subpattern, i.e. “plus shape” or “dots”, considering which items it appears in. In each case the answer is four, forming a rectangle. Two of the rectangles should extend into the ninth item, the one we’re looking for.)
This is a better answer than XOR, in a sense: it describes the pattern more narrowly. If the “true pattern” were XOR, it would be possible to have a shape or subpattern occur 6 times (if it is missing once from each row and column, e.g. if it is present everywhere except in one of the diagonals). Since this does not occur for any of the six shapes, this provides some evidence that XOR is not the “true pattern”.
(Similarly, this is very strong evidence that “just have 4 of each shape” is not the true pattern: there are 126 ways to place a shape in 4 cells, and only 9 of them make a rectangle shape. The case against XOR, where we notice that only 9 of the 15 XOR patterns are used, is much weaker, but I still believe it.)
Of course, if the goal is to just solve this particular problem, then any method works. But if we were studying the appearance of many matrices with this pattern, then you would get twice as many research points as anyone else :)
The relationship between this approach and the XOR approach is interesting, I think. Thinking in XOR terms requires fancier mental infrastructure—you need to have seen something like the idea of XOR before, and to be able to notice slightly subtle relationships between different parts of the figure. On the other hand, spotting that particular features tend to occur in rectangles involves spotting simpler things but paying more global attention to the whole figure.
It feels like these play to different aspects of cognitive ability; spotting complicated patterns versus spotting large ones, so to speak. I guess the latter is closely related to working memory size, which I know is generally thought to be a large contributor to measured IQ. The former seems like an important aspect of intelligence too, and strikes me as more likely to be trainable than working memory size.
I did it even more simply than that: Count things. Most have four iterations. Some have three iterations. The ones with three, make four. Less than 10 seconds for me. Same answer as the rest of everyone.
This is how I did it. My first instinct was to decompose the problem into the shapes {dots, circles, diamonds, square, +, X} and then plot which cells the shapes appear in. It’s pretty easy to see the rectangles after that. Though, I didn’t make the connection to XOR.
That’s also interesting… I think the two ways of looking at it are equivalent, i.e. any pattern that satisfies one should also satisfy the other. (Only because the XOR pattern works both vertically and horizontally.)
The way I solved the problem hasn’t been mentioned here by anyone, which is slightly bugging me out.
The way I solved it was looking at the whole puzzle as a single picture. The two bottom rows (except for the middle column) have pluses. Thus the solution must have a plus. The two right columns (except for the middle row—a transposed pattern from the previous pattern) have squares; the solution must have a square. There’s only two answers with both a square and a plus; I picked the one that seemed most intuitively correct.
Similarly, I go the same answer, but only by process of elimination. I knew it didn’t have dots, I knew it didn’t have a diamond, I knew it didn’t have an x, by just extrapolating from the “cut offs” in the problem. That left me with 2, but it felt...wrong. It didn’t feel intuitively right. If I had to pick on without thinking about it, number 2′s the last one I’d pick.
I only understand the pattern in a cohesive way from looking at the comments. Now it makes sense, instead of being deduced from bits of dis-unified information.
Possibly of interest: I worked out the correct answer in a minute or so, but wasn’t sure it was correct until I identified it as an exclusive or pattern, which I didn’t figure out until after I had the answer.
I note that the missing piece fits a xor pattern both across and down. I’m trying to figure out if that has to happen—that is, if the first two rows are xor across, and the first two columns are xor down, and the missing piece fits xor in at least one direction, is it required to also fit xor in the other direction?
I’m trying to figure out if that has to happen—that is, if the first two rows are xor across, and the first two columns are xor down, and the missing piece fits xor in at least one direction, is it required to also fit xor in the other direction?
That is:
A⊕B=C (1)
D⊕E=F (2)
G⊕H=I (3)
and
A⊕D=G (4)
B⊕E=H (5)
We want to know if it is true that:
C⊕F=I
We begin with our goal, and substitute out C and F using (1) and (2):
(A⊕B)⊕(D⊕E)=I
Now we ask Wikipedia if ⊕ is associative and commutative, and the answer is yes, allowing us to rearrange that as (this is actually multiple steps, condensed):
(A⊕D)⊕(B⊕E)=I
Now we substitute using (4) and (5):
G⊕H=I
This is (3), and thus we have our proof. (Perhaps a more natural way is to start at (3) and work forward to our desired formula, but I like working backwards.)
As a side point, I believe it is the case that most (all?) Raven’s patterns are applied both horizontally and vertically.
I think the proof is simplified by the observation that (+ meaning XOR) a+b=c is the same as a+b+c=0. So if all rows have the XOR property, we find that the XOR of all entries is 0. If two columns have the XOR property, the XOR of their entries is 0, leaving 0 for the XOR of the entries in the last column, and we’re done.
As a side point, I believe it is the case that most (all?) Raven’s patterns are applied both horizontally and vertically.
The actual Advanced Progressive Matrices test isn’t in the public domain, but the most difficult items on clones are sometimes not “what comes next?” type items at all, but instead involve picking an item that completes the pattern in a broader sense. For example, I came across one where the pattern can only be seen by identifying opposite edges and viewing the grid as a torus.
For example, I came across one where the pattern can only be seen by identifying opposite edges and viewing the grid as a torus.
If you mean what I think you mean by a torus, that will maintain the vertical and horizontal symmetry. The claim I am confident in is that I don’t think any Raven’s test has two potential answers, one of which is more sensible if you perceive the pattern horizontally and another of which is more sensible if you perceive the pattern vertically. I am not sure whether that is accomplished by there being two equally reasonably concluding items, one of which is not included in the potential answer set, or by there never being two equally reasonable concluding items in the set of all possible items.
The weaker claim, that is mostly speculation, is that the description of the pattern is the same both ways. For example, consider this possibility:
1 0 1
1 0 1
1 0 ?
The answer is obviously 1, but is it because it’s an xor, adding, or multiplication? The first two work horizontally but not vertically, and the latter only works vertically. I don’t think there are many (any?) test patterns that look like that.
The most plausible pattern for that one is exclusive or; an element is only in the third item if it is in exactly one of the preceding two items.
That’s interesting! I got the same answer but I visualized it differently. (Imagine, for each possible subpattern, i.e. “plus shape” or “dots”, considering which items it appears in. In each case the answer is four, forming a rectangle. Two of the rectangles should extend into the ninth item, the one we’re looking for.)
This is a better answer than XOR, in a sense: it describes the pattern more narrowly. If the “true pattern” were XOR, it would be possible to have a shape or subpattern occur 6 times (if it is missing once from each row and column, e.g. if it is present everywhere except in one of the diagonals). Since this does not occur for any of the six shapes, this provides some evidence that XOR is not the “true pattern”.
(Similarly, this is very strong evidence that “just have 4 of each shape” is not the true pattern: there are 126 ways to place a shape in 4 cells, and only 9 of them make a rectangle shape. The case against XOR, where we notice that only 9 of the 15 XOR patterns are used, is much weaker, but I still believe it.)
Of course, if the goal is to just solve this particular problem, then any method works. But if we were studying the appearance of many matrices with this pattern, then you would get twice as many research points as anyone else :)
The relationship between this approach and the XOR approach is interesting, I think. Thinking in XOR terms requires fancier mental infrastructure—you need to have seen something like the idea of XOR before, and to be able to notice slightly subtle relationships between different parts of the figure. On the other hand, spotting that particular features tend to occur in rectangles involves spotting simpler things but paying more global attention to the whole figure.
It feels like these play to different aspects of cognitive ability; spotting complicated patterns versus spotting large ones, so to speak. I guess the latter is closely related to working memory size, which I know is generally thought to be a large contributor to measured IQ. The former seems like an important aspect of intelligence too, and strikes me as more likely to be trainable than working memory size.
(I did it with XOR.)
I did it even more simply than that: Count things. Most have four iterations. Some have three iterations. The ones with three, make four. Less than 10 seconds for me. Same answer as the rest of everyone.
I did it this way too. I can’t help feeling like the xor way is smarter.
This is how I did it. My first instinct was to decompose the problem into the shapes {dots, circles, diamonds, square, +, X} and then plot which cells the shapes appear in. It’s pretty easy to see the rectangles after that. Though, I didn’t make the connection to XOR.
That’s also interesting… I think the two ways of looking at it are equivalent, i.e. any pattern that satisfies one should also satisfy the other. (Only because the XOR pattern works both vertically and horizontally.)
The way I solved the problem hasn’t been mentioned here by anyone, which is slightly bugging me out.
The way I solved it was looking at the whole puzzle as a single picture. The two bottom rows (except for the middle column) have pluses. Thus the solution must have a plus. The two right columns (except for the middle row—a transposed pattern from the previous pattern) have squares; the solution must have a square. There’s only two answers with both a square and a plus; I picked the one that seemed most intuitively correct.
Similarly, I go the same answer, but only by process of elimination. I knew it didn’t have dots, I knew it didn’t have a diamond, I knew it didn’t have an x, by just extrapolating from the “cut offs” in the problem. That left me with 2, but it felt...wrong. It didn’t feel intuitively right. If I had to pick on without thinking about it, number 2′s the last one I’d pick.
I only understand the pattern in a cohesive way from looking at the comments. Now it makes sense, instead of being deduced from bits of dis-unified information.
Do I know my IQ now?
I got the four, but not the rectangle—I just noticed that two elements only appeared three times.
Also how I did it. FWIW I know it took me more than a minute, but definitely less than five.
I thought about the pattern completely differently: every element is present in a 2x2 subarray.
Possibly of interest: I worked out the correct answer in a minute or so, but wasn’t sure it was correct until I identified it as an exclusive or pattern, which I didn’t figure out until after I had the answer.
I note that the missing piece fits a xor pattern both across and down. I’m trying to figure out if that has to happen—that is, if the first two rows are xor across, and the first two columns are xor down, and the missing piece fits xor in at least one direction, is it required to also fit xor in the other direction?
That is:
and
We want to know if it is true that:
We begin with our goal, and substitute out C and F using (1) and (2):
Now we ask Wikipedia if ⊕ is associative and commutative, and the answer is yes, allowing us to rearrange that as (this is actually multiple steps, condensed):
Now we substitute using (4) and (5):
This is (3), and thus we have our proof. (Perhaps a more natural way is to start at (3) and work forward to our desired formula, but I like working backwards.)
As a side point, I believe it is the case that most (all?) Raven’s patterns are applied both horizontally and vertically.
I think the proof is simplified by the observation that (+ meaning XOR) a+b=c is the same as a+b+c=0. So if all rows have the XOR property, we find that the XOR of all entries is 0. If two columns have the XOR property, the XOR of their entries is 0, leaving 0 for the XOR of the entries in the last column, and we’re done.
Agreed; my proof doesn’t make use of the fact that C⊕C=0, and if you use that fact you get there quicker.
The actual Advanced Progressive Matrices test isn’t in the public domain, but the most difficult items on clones are sometimes not “what comes next?” type items at all, but instead involve picking an item that completes the pattern in a broader sense. For example, I came across one where the pattern can only be seen by identifying opposite edges and viewing the grid as a torus.
If you mean what I think you mean by a torus, that will maintain the vertical and horizontal symmetry. The claim I am confident in is that I don’t think any Raven’s test has two potential answers, one of which is more sensible if you perceive the pattern horizontally and another of which is more sensible if you perceive the pattern vertically. I am not sure whether that is accomplished by there being two equally reasonably concluding items, one of which is not included in the potential answer set, or by there never being two equally reasonable concluding items in the set of all possible items.
The weaker claim, that is mostly speculation, is that the description of the pattern is the same both ways. For example, consider this possibility:
The answer is obviously 1, but is it because it’s an xor, adding, or multiplication? The first two work horizontally but not vertically, and the latter only works vertically. I don’t think there are many (any?) test patterns that look like that.
Yep, I also got this.