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.
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.