Raven Matrices 291z28

Raven’s Progressive Matrices The puzzle is very simple, and does not even require much explanation. It simply shows you a 3×3 (or 2×2) matrix of black-and-white symbols. The lower right corner is not filled in, but the rest are. You are supposed to deduce the pattern and figure out what should most logically fill the lower right corner. For example: \ | / { | } ( | ?

What would go in the “?” spot? Good, a “)”. That‟s a pretty simple pattern. They get much more complicated, but they still are all based on just a few basic rules. Please note that I made up all these . You don‟t know which kind of matrix a given matrix is, but you can figure it out pretty quickly.

Rules Momentum Look at problem 2 on the iqtest.dk site. That‟s momentum. If the first symbol and the next symbol look the same, except for one little thing moves or changes or adds to itself, and then it moves or changes or adds to itself by the same amount on the next symbol, then that‟s momentum. Just follow that. Example: ( (( ((( _ __ ___ { {{ ?

The answer: {{{. Note that this rule can become less obvious if there is what I call “carry”. That is, if the symbol itself is a little 3×3 matrix, and you “move” to the right, then some of the elements will fall out of the little matrix, so then you must “carry” them over to the next row of the litle matrix.

Set Completion

Look at problem 8 on iqtest.dk. That is simple set completion. Think of each symbol having a number of properties: size, color, shape, etc. If you can‟t sem to follow a progression like you can in Momentum, but it just looks like a bunch of random, but somewhat related things with similar properties, then the problem can be set completion. I can best show you this in an example (use your imagination about the shapes): ^ O [] O [] ^ [] ^ ?

Answer: O You need to complete the set of shapes on the last row. Notice that the last column also needs to complete the set of shapes (the diagonal too in this case, but that‟s not always the case). A common property of set-completion that makes this kind of problem much easier, is to look at the triangles made. Notice:

^ x x x x ^ x ^ x

and

x x [] x [] x [] x x

and

x O x O x x x x ?

.

Obviously, the ? should be an O. Set-completion is simple, if the first row has a red, white, and blue, and the second has one red, one white, and one blue, make sure the third has one of each. This can get more complicated because you can have multiple properties, shapes and colors etc, all compounded on each other. But, if you just find the triangle, this problem is simple.

Composition If one symbol looks like the other two put together, then it is just composition. You just have to figure out in what way it should be put together. Maybe the rule is, always put it on the inside of the first. Maybe it‟s, always put it on the outside. Whatever it is, this one is usually pretty easy. I won‟t even give an example. Question 30 uses composition.

Subtraction Subtraction is much like composition, look for one thing looks like the other two put together, but with a twist. The subtraction could be complete, just one shape minus the other. Or it could be XOR (exclusive-or). You take two symbols, and take out the lines which are in one, but not the other. Example: _|_| | __| |__| |_| |_| __| |__ ?

Answer: | |

Functions If the first symbol in a row looks like the last symbol, but the middle symbol looks weird or especially if it‟s a line or arrows or something simple, then that middle symbol might be a “function.” By function I mean something that geometrically transforms the first symbol into the third symbol. The function is not necessarily intuitive, but usually makes sense in of what the function symbol looks like. In the same example I used above, the vertical bar “|” is a function that reflects the first symbol horizontally over itself, like a mirror. \ | /

{ | } ( | ?

What‟s interesting is that you can apply one function to another function. So, you might apply a rotation function with a flipping function, flipping the rotation function, creating a function that both rotates and flips. Pretty cool.

Replacement Replacement is where they trick you. The rule might be very simple, but it becomes very hard to figure out quickly, because the elements inside the symbol change for arbitrary reasons simultaneously. Question 25 is an example of movement with replacement together.

Commonality Finally, if all the symbols look randomly chosen with a bunch of properties and possible configurations, then start to look for commonalities. Don‟t look for a 1.2.3. pattern like movement, just look for rules that each symbol has in common. For example, say that each black element should be on top of a white element in exactly one symbol in each row. This can be difficult, but is easier if you know that it‟s none of the other rules, and you are looking for a commonality, not a progression of patterns. Once you have some rules, start ruling out answers until you find a final answer. Question 26 is an example.

Putting it all together Skeptical that the answers are so easily based on the rules above? They really are. What makes the difference between an easy and a hard problem is that a hard problem will use multiple rules together. Fortunately, using multiple rules together usually doesn‟t make it much harder to figure out as long as you systematically think through these possibilities. If you are having trouble with a problem, you should stop, take a deep breath, look back at the matrix as a whole, and then think through each of these rules and rule them out or use them as appropriate. All matrices follow one or more of these rules. On the iqtest.dk site, the lowest score you can get is “below 79″. The highest you can get is “above 145.” The answers to all of the questions I put below, along with an explanation referencing the rules used to get the answers: D Come on (momentum?) F Momentum

B Momentum G Set completion (angle of line) A Momentum (size and column) H Momentum B Momentum (notice that the squares hide each other) E Set completion H Set completion / Momentum ? A Subtraction C Application of function (enlargement along axis) F Set-completion (angle and number lines) B Momentum D Subtraction H Subtraction E Composition and set-completion with replacement F Momentum (one example has carry) C Subtraction E Momentum with carry D Set-completion (angle and number black/white) G Momentum A Oppositing? (a bit like subtraction but from sets of attributes of platonic ideal) B Set-completion (1. small ball color, 2. big outside shape, 3. inside v. outside) H Set-completion (1. flat bottom, 2. widening, 3. partially-closed top) B Non-repetition? Set-completion? (Movement and replacement?) A Commonality (180 deg rotational symmetry and middle pegs always covered) H Subtraction G Set-completion E Function-application (and function-application on other functions!) A Composition D Subtraction yields line which is a function you apply which is reflection and delete line E Movement G Set-completion? G Valuation (attach negative integer for ball inside circle, positive outside) then add C Function application (with a bit of spatial reasoning) F Set-completion (big-stack color, two-stack color, bar-chart position) H Movement and replacement based on progression (replace as hidden by dark square) F Function application (functions on functions) B Movement with carry and replacement

Identify puzzle elements. These are the questions you need to ask to identify the puzzle elements: 1) What are the large shapes involved? 2) What are the small shapes involved? 3) What are the symbols involved? 4) What are the lines doing? 5) What are the colours doing? 6) What are the dots doing? 7) In 3x3 within 3x3 problems: Check if the answers lies in the fact that the columns in each 3 x 3 successive matrix shift one column to the right, and in doing so, the shapes change each time according to a logical rule? 8) Can you stack the boxes on each other? 9) Is there a numerical sequence?

What are the large shapes involved? In general, the third row must always contain the same large shapes as rows 1 and 2, in the same proportion. For easier tests, this usually applies to columns also. For harder tests, it may only apply to rows or only to columns. Let's start off with a very basic puzzle. Look at this example:

This one is easy. ABC and DEF both have 3 circles. Therefore GHI must also have 3 circles. Therefore I must be a circle in order to complete the sequence. Note that ADG and BEH also have 3 circles each. Therefore CFI must also have 3 circles, so I needs to be a circle. Look at the next one. For now, please ignore the lines, and just look at the circles:

Ignoring the lines, we still know that in I there has to be a circle, for the same reason as the previous example: if the first two rows have 3 circles, then I has to be a circle to follow the rule. Next example. Ignore the lines and the small circle. Just look at the large shapes (the triangles):

Both ABC and DEF have two triangles pointing up and one down. Note that the same rule applies for ADG and BEH. In order for GHI and CFI to match the other two, then I must be a triangle pointing down in order not to break the pattern. The last example. Ignore the lines and focus just on the shapes for now:

You'll see that the top two rows and left two columns both contain one square, one triangle and one diamond each. Therefore I must be a square in order not to break the sequence. My approach to these Raven's test is to always first look at the big shapes (ignoring other elements) and first work out what big shape needs to go into I. Once the large shape is identified (or if there are no large shapes to work with), the next question to ask is: What are the small shapes involved? Much as for large shapes: in general, the third row must always contain the same small shapes as rows 1 and 2, in the same proportion. For easier tests, this usually applies to columns also. For harder tests, it may only apply to rows or only to columns. Have a look at this example:

We have a sequence of large shapes with smaller shapes within them. The first rule as taught about is to ask 'What are the large shapes involved?' and to figure which large shape will be in I (in this example it will be the square). The next question is 'What are the small shapes involved?' Looking at ABC and DEF, we get a small dark square and a small dark circle in each row. The columns ADG and BEH follow the same pattern. So in order to continue the sequence, GHI and CFI should both only have one small dark circle and one small dark square. However, these conditions are already satisfied, so actually no small shape is required in I. So the final shape in I is simply a large square with nothing in it. Let's go back to the triangle example from above (we can ignore the lines for now):

We already know that in I we should have an upside down large triangle. But should it have a circle in it? Looking at the rows, we see each row has only one circle; furthermore each column also has only one circle. Putting a circle in I would violate this rule of only one circle so there are no small shapes in I, so no small shape is required in I. As you can see, for large shapes and small shapes the rules tend to be quiet logical. Now we start moving on to trickier territory: What are the symbols involved? Various symbols may be present, like multipliers, plusses, minuses, division symbols, etc. Their behaviour can be odd, sometimes they act as „rules‟, altering the behaviour of things around them Here's an easy enough example to start off with:

First we can work out that the large shape is a square. Note that the rows and columns are operating with different rules for large shapes. The rows are operating on a 'one-of-each' principle while the columns are 'three-of-a-kind' for large shapes. There are no small shapes, so we can skip that question. The symbols consist of a diamond, a star, and a plus-sign, and follow the inverse rule to the large shapes : the rows are three-of-a-kind, and the columns are one-ofeach. In I we get a square with a plus-sign inside of it. Now let's go on to a puzzle where the symbols start developing weird rules of their own:

There are shapes here, but they at first seem to be doing their own thing. But actually they are being transformed by the symbols. Symbols often have rules which you have to dissect. Consider ADG. The large square suddenly becomes a smaller square. It appears that a plus-sign causes a shape to decrease in size - in other words, ADG can be read as 'Large square in A is diminished in size by the symbol in D, causing a small square to result in G.' BEH does not have a plus-sign though, it has a diagonal line. Rather than having an effect on size, it appears to have an effect on orientation, causing objects to revolve by 45 degrees. So BEH can be read as 'S-shaped line in B is forced to revolve by diagonal line in E, resulting in an s-shape that has been nudged 45 degrees in H'. Consider also DEF as 'Plus sign in D is forced to revolve by diagonal line in E, resulting in a plus sign that has been nudged 45 degrees in F'. Look at ABC, the s-line appears to transform shapes without changing shapes or orientation i.e 'The square in A becomes transformed by the s-shape line in B, resulting in a clover shape in C.'

So let's solve for I now. Based on the observations above, we know that: 1) the plus sign causes the top shape to diminish into a smaller bottom shape 2) the s-shaped lines cause the shape on the left to transform into something different on the right hand side. 3) diagonal lines change orientation Following the rules and looking at CFI, the large clover needs to be come diminished into a smaller clover in I; likewise looking at GHI, the small square needs to be transformed into a small clover in I; however both the plus sign in F is skew, and now has diagonal lines, so we can assume the small clover changes in orientation. This gives our answer as a small clover with a slight orientation change i.e. answer 'e' Once you've sorted out the large shapes, small shapes and symbols, the next question to ask is: "What are the lines doing?" Lines typically delete or add on to one another. Sometimes a line or a part of it will rotate. Here's a good example of deletion:

Much like in the above examples from symbols, the blocks in the above sequence work almost like an equation. For ABC, all the lines in A and B which are held in common will be deleted, leaving only the leftover lines to form C. The same rules hold for DEF, ADG, BEH. Therefore both CFI and GHI yield the same result - a short diagonal line in the left hand corner. In the next puzzle, the lines do something else:

The lines are rotating, one small segment at a time. First in A, the lower right hand line segment rotates 90 degrees. This forms a new picture which is presented in B. Then the upper left hand segment (i.e. the segment directly opposite) in B rotates 90 degrees to form the picture in C. Exactly the same rule applies to DEF. The columns follow a similar rule, except it involves first the lower left segment, then moves on to the opposite line in the upper right segment. You can solve for I either by looking at the rows or columns, but in either case the answer is the same: it will be 'e.' So in summary, lines can be used in a number of ways, but for most of the time in Raven's tests they are deleting one another. Often this deletion will work almost like a formula eg (lines in A) - (lines in B) = C If the lines are not deleting, then most of the time some segment of line is rotating. Having figured out what the lines are doing, the next step is to ask: What are the colours doing? Colours can expand, shrink, or move, or otherwise change according to a logical rule.

Have a look at this example:

For rows ABC and DEF, a new colour is added to the sequences in a clockwise directions. For the columns a different rule applies. In ADG and BEH, the colours move in a clockwise direction one segment at a time. Colours are generally not too much of a challenge in Raven's tests, they are usually added merely to add complexity or as a red herring (an example of colour as a red herring will be shown below). Sometimes colours do follow bizarre rules, but they usually expand/shrink/move, so just decide which one is the most applicable and take it from there. The next question is: What are the dots doing? Dots are the most likely element in Raven's tests to have peculiar random placements in order to confuse the puzzle solver, although there is usually some logic to their movements. However, if you cannot find any underlying logic in their movements, try assuming that the placement is random and see if there is any other logical element behind the dots. Have a look at this one:

Looking at the large shapes first (and ignoring the dots), it's clear that we have to have a triangle in I. As for the dots, in ABC we have: an inner dark dot, an inner white dot, and an outer white dot. Same for DEF. The columns follow the same rule. Not that the positioning of the dots is otherwise random and does not follow any logical rule. Since we already have two white dots in GHI and CFI, I must have an inner black dot. The answer for this puzzle will therefore be any triangle with a dark dot in it, therefore it will be 'b' The next example is a bit more complicated, but shows that dot movement isn't necessarily random:

Basically the two circles move one square to the right in the next box. If they are at the end of the row, that circle will instead move on to the first square of the next row. After entering a black area, the circles change colour: a white circle will become black, and a black circle will become white. Following the movements with the circles, we find that the answer will end up as 'h' Follow the above steps, look at each puzzle aspect in isolation and you should be able to solve most Raven problems - most Raven problems are really just combinations of simpler puzzles but you have to be able to look at the individual elements in isolation to see this. However, there is a particularly brutal class of Raven problem consisting of 3x3 grids within the 3x3 grid, which deserves some discussion in and on its own: The grids-within-a-grid puzzle The problem with these grids-in-a-grid puzzles is that they are often so complex that there can logically be multiple correct answers, and which is the actual correct answer is really up to the

whimsy of the tester. This is the point where a Raven's test can become less a measure of intelligence of the test-taker and more of a measure of the tester's predisposition. I still struggle with these, but based on the few examples I could find on the net, here is a potential rule to try out: Check if the answers lies in the fact that the columns in each 3 x 3 successive matrix shift one column to the right, and in doing so, the shapes change each time according to a logical rule. Look at this one:

Look at A. See that leftmost column that forms cross-triangle-cross? This column shifts one column to the right in B, and also changes shapes to circle-cross-circle, and then shifts columns once more in C and becomes triangle-circle-triangle. If you look at the other columns in all the rows, you'll see that all the columns shift space and change shape in this similar manner, with cross always becoming a circle, a triangle always becoming a cross, and a circle always

becoming a triangle. This makes it suddenly easy to find I, which must be 'b', because that is the only option that will allow H to shift columns/change shapes without violating the rule set. Now have a look at this one:

In this one the columns shift to the left, rather than to the right, but they also change symbols

according to specific rules. For example, the right most column in A shifts to the left in B, with clubs changing to hearts, and hearts to diamonds. Another thing to check for, if you can't figure out the answer to the puzzle, is to check for this rare puzzle: Can you stack the boxes on each other? Have a look here:

Basically, ABC can be put next to each other to make continuously running lines. DEF can be stacked on top of one another to make continuous lines. Then GHI goes back to having lines next to each other. Note that there is no logical rule for the columns, this is a rows only puzzle. The last question to ask is: Is there a numerical sequence? Some puzzle just follow a 1-2-3 sequence. If you are struggling to see any sequence upon asking the earlier questions, quickly check whether you are not confusing yourself where a simple 1-2-3 relationship exists.

An example:

Now let's go through a few examples of what I consider to be 'layered' puzzles, where multiple puzzle elements are combined in a way intentionally meant to confuse the puzzle solver. Example 1:

So this is a combination of a large shapes test with a lines test, but the trick here is that the side of the shape is used as a line. Each row/column has to have a partial line, a blank line, and a full line. To complete the sequence we need to use answer 'h' Example 2:

This actually consists of 2 seperate line puzzles. The inside lines (touching the dot) and the outer lines are two separate components to consider. Inner lines will delete whatever line is held in common, whereas outer lines will only preserve whatever line is held in common. So for the inner lines: lines in A + lines in B = lines in C, but removing the lines that are held in common. For the outer lines: lines in C = lines held in common by A and B Following the same rules for GHI, the answer will be '2' That's it for my tutorial on how to tackle a Raven's progressive matrix. The key to solving them successfully is to understand all the different tools that Raven used to build the puzzles in the

first place, and to dissect the puzzle into it's most basic elements. Practicing the test athttp://www.iqtest.dk/main.swf will help you get used to going through the steps.