Turn them into binary, so for example for the first row with blue = 1, white = 0, you'll get :

1101-0110-1011

They follow the XOR truth table: http://en.wikipedia.org/wiki/Exclusive_or i.e the first & second columns are the input and the third column is the output

It applies to the second row of the question and if you do this for the third row you'll get the answer A.

Look at rows 1,2, and 3. Overlapping the three circles in Row 1 would create a FULLY white circle. Overlapping the circles in the second row again makes a full white circle so the third should also form a white circle so the answer is A

I'll leave my solution seeing as nobody did what I did or at least posted it.

From the first two rows, I noticed that you get the third column by checking whether the particular level has the same colors on the first two rows. If so, that particular level will be white on the third column, otherwise (meaning it will have different colors on the first and second column), it will be blue.

Looking at the third row:

• Column 1 has WBWB (starting from the innermost level outward)

• Column 2 has BWWB therefore

• Column 3 has BBWW which is A.

Hopefully I helped.

Horizontally similar colors gives white circle , and diffrent gives blue , like that

blue + white = blue

blue + blue = white

white + white = white

My logic was much simpler then anybody else's. Just notice that each row or column has one white ring in the center, one in the middle and one on the outer edge. If you see the third row or column the only one which was missing was the outer white ring. And A was the only one with the white ring on the outer edge. Just took me less than a minute to solve this one. Cheers!

THE MOST SIMPLIFIED SOLUTION

In every row there is a white centered circle and 2 blue circles;

and a white peripheral ring and 2 blue peripheral rings

The 3rd row requires a blue centered circle with a blue peripheral ring

So the answer is: $\boxed A$ :D

Solution using Sets, Set Intersections, Set Differences, and Set Unions.

question image

I started by labeling all rings that are blue as numbers in a set starting from the outermost ring being 1 and the inner most ring being 4. Here is my labeling of the sets.

Eventually I noticed that column $C$ in rows 1 and 2 are a product of $A \cup B - (A \cap B)$ which uses Set Intersection , Set Subtraction, and Set Union . Thus making row 3 column $C$ have the value $\{3,4\}$ .

This is just how I came to the solution, and in this case atleast it is logically equivalent to the xor method as they will be included in the set (i.e. turned on, made 1, made blue) by definition of my sets by if they are blue, then they will be excluded from the set (i.e. turned off, made 0, made white) by being either being both white which excludes them from being added in the first place by $A \cup B$ , or they will be excluded from the set by both being blue thus being excluded by subtracting the set $A \cap B$ .

Picture the left square and middle square on top of one another. The one on the left has the second ring blue and the middle on has the first ring blue so we know that the right one has to have the first and second rings blue. They both have the outermost ring blue so the right one needs to have the outer ring blank.

I just picked the one with white on the outside cuz the other rows had one with white on the outside and two with blue.

I got it with mixed reasoning (I pretty much guessed it). At first, I tried turning it into binary but I couldn't find the logical sequence there :3

What I thought of was the changing colours of the circles:

The white will move to inner levels if: - moving left to right - moving up to down

The white will move to outer levels if: -moving right to left - moving down to up

The only problem with this is the center of the circle. I just decided the center should be blue since the other two rows only had 1 white center :33 Any thoughts?

My answer was kinda simple. Basically, if all the white segments from the row are combined, they need to complete a white circle

Also, white may overlay with another white, so that'd leave A as the only answer

Each row needs to create a full withe circle. In the third row there are white circle on the second, third and fourth, but not for the first. So A is the only one with a withe circle in the first circle.

Seeing this complicated explanations I am a bit uncertain of mine. But here it is: I was looking for a circle with two whites because the lower left corner has it. This logic eliminates all but A and B. Then I eliminated B as it already apears in the middle in the bottom. Which left me with A as solution. Please comment if this was a pure coincidence or if it is a way to go.

I will explain my solution even though there is still one unexplained phenomena from my pov. However, I did get the answer correct.

First arrange the squares by their rings using a table of rows and columns: Row 1 - 8

Ring3 - 1B 2W 3B 4W 5B 6B 7B 8B

Ring2 -1B 2B 3W 4B 5W 6B 7W 8W

Ring1 - 1W 2B 3B 4B 5B 6W 7B 8W

Center 1B 2W 3B 4B 5W 6B 7W 8B

Now, it is possible to draw diagonal lines through each successive pair of repeating colors. For example, notice the long diagonal connecting blue rings from Ring 3, R1 - Center, R4 (There's 4 blues in a row)

You can see the pattern more clearly now. It took work but what's going on is that the rings bounce from Ring3 towards the Center and back. Take the first square for example, it has a blue center and white ring nearest the center. So, the next square has a white center and blue nearest the center. This is because the blue center is moving away and the white ring is seeking the center. Again, Ring3 and Ring 2 in the first square are blue, so they will seek the center together and descend. This is why the 2nd square shows them still together but one ring closer to the center.

Every ring does this no matter the color, the rings seek the center then move away once they've arrived there. Sometimes colors descend and ascend into the same ring, when this happens I have not systematically found a solution to that final mystery. I think that if a color ascends or descends and there's no color there to take its place, then the abandoned ring will change its color to the opposite of what it was prior. Nevertheless, the bouncing fact alone is enough to take a good guess at the answer once the pattern is written out and understood.

This is how I visualized this:

1. Put figure 1 on top of figure 2 in each row.
2. White on top of white - of course result is White.
3. Blue on top of white OR White on top of Blue- You always will see the DARKER color - Blue in this case.
4. Blue on top of blue - They 'cancel' each other out and give White.

For those who want a 'logical/binary' answer:

The above logic, in essence, is what XOR actually means: Here Blue is 1 and White is 0.

• (0 OR 0) => 0
• (1 OR 0) => 1
• (0 OR 1) => 1
• (1 OR 1) => 0

I simply looked at the blank, or white space and decided on whether it was (inner, outer, or middle) and used that to decide. The top row was inner, outer, middle. The second row was outer, middle, inner. So the bottom row must be middle, inner, OUTER.

I didn't remember the XOR truth table so I use a more simplistic approach and I guess a less smart way to solve it but this is how i did it.

I saw the patterns for row number one and two by seeing it as outer ring, middle ring, inner ring and the center circle, I did thought of it like it was the solar system.

The first row I saw it like this: The first square has the inner ring white; the second square has the outer ring and center circle white; the third square has the middle ring white which was the last ring left to become white. So I applied that logic to the second row the last ring left to become white was the inner ring.

The third row was interesting because the second square has the middle and inner ring white so it repeat the middle ring from the first square but I knew that the one left was the outer ring although I did doubt it.

I thought that the answer was the letter E because I thought that it was a different case from the first two rows and in this case the center circle would be white because in row number one and number two the squares one and three the center circles are blue so I saw the third row as an inverted case but even if that were the case it would break the pattern in row number one and two and the other answers were illogical according to that pattern so I choose letter A mainly because the outer ring follow that pattern.

This pattern can be applied from left to right, right to left, up to down and down to up.

Similar to Chee's answer, you can think of the circle colors like light switches. Blue (flipping switch on) + Blue (flipping switch again) turns light off. Blue + White or White + Blue turns light on. White + White leaves light off.

Every row has at least one white ring at the bottom, A is the only option with a white ring at the bottom.

My 10 year old son worked this out ... I still don't get it.

The white is always where there was no white before.

XOR truth table (https://en.wikipedia.org/wiki/Exclusive_or)

• 0, false (WHITE)
• 1, true (BLUE)

When the image that has the white circle in the middle and a ring at one point in the outer circle is shown, the second one after that will have the outer ring selected of the one it was. And the next tie this happens the ring before and now will be selected.

I turned them into binary and it gave me this :

ROW 1 (10-11 01-10 11-01) ROW 2 (11-10 01-01 10-11) ROW 3 (01-01 10-11 _ - _)

So I checked the relationships and noticed: If you add the first number of the first two columns of ROW 1, the result matches the first number in the third column of that same row. If you subtract the second number of the the second column of ROW 1 it matches the second number of the third column in that row. But in the ROW 2 it works if you inverse which pair you add and which pair you subtract. Ie you subtract the first numbers (01 from 11), and add the second numbers (10 and 1). And that makes it match the third column of that row. So in the third row I used the pattern that was bound to come again, ie that of the first row. It matched answer A. I thought it was a good time to give up since it had already worked! It's probably similar to the explanation below about XOR tables

Just look at the rows and see if it contains a white outer ring, if not, there’s your answer.

Easiest way to think of this is to consider white=1 and blue= --1 Moving left to right in each row, compare the corresponding rings in the first two circles, and multiply their values to get the values of the corresponding rings of the third circle.

Starting with the outer rings of the top row, We have blue then white. --1(blue) ×1(white)= --1(blue)

Now let's try the bottom rows outer rings --1(blue) × --1(blue) = 1(white)

As you can see, all the circles follow this pattern so it's easy to confirm the answer is A.

The remaining rings which should change into white would be the ones that did not change color in the the last two models. (Looking at 3 individual horizontal lines. Therefore, the answer is A.

Pattern follows two rules:

R1 A tiles shares exactly 1 shaded region with its neighbour

R2 Every tile is painted in 2 or more regions.

all circles in each horizontal line fills all the blue with white when joined

so the third one on the last row should fill the mmmm it's circle line on it's edge

very very and very simple i hardly took 15 seconds! look the inner most circle in each row(horizontal) are 2 blue and 1 white and like this each row outer most circles are 2 blue and 1 white according to this for third row option a follows .

It's very simple: you just have to understand the pattern in each line. It's a sum between the rows of the first two houses, resulting the third one.

The pattern says:

blue + blue = white

blue + white = blue

white + white = white

Look at rows 1,2, and 3. Overlapping the three circles in Row 1 would create a FULLY white circle. Overlapping the circles in the second row again makes a full white circle so the third should also form a white circle so the answer is A