Recognizing Visual Patterns

Finding patterns is at the heart of mathematics. While sometimes these patterns can lead us astray (the Greeks believed false things about perfect numbers because of patterns that didn't continue, for example), the ability to recognize and extend patterns is extremely important.

Searching for visual patterns can be as simple as identifying the change from one item in a sequence to the next. Some possible changes to look for include

• changes in color
• rotation
• vertical or horizontal translation
• changes in shape
• changes in size.

When looking for visual patterns, it is a good practice to make a hypothesis based on one or two terms and then test it against an additional item to see if your expected pattern matches the entire sequence.

Being able to recognize visual patterns will allow you to solve problems like this:

What comes next?

image

If we track the blue square, we see that it goes from the top, down to the second, and then to the third, so it will likely be in the fourth (at the bottom). Notice that there are \(2\) options with the blue square at the bottom.

If we look at any other square, say the yellow square, we see that it similarly moves downwards, and that after it is the fourth, it moves to the top. Hence, the \(2^{\text{nd}}\) square should be yellow, the first square should be green, and the third square should be red.

Thus, the answer is B. \( _ \square \)

What comes next?

image

We see that the number of circles increases, while they still have the same height. Counting the number of circles, we see that there is \(1,\) then \(2,\) and then \(4.\) This sequence doubles, so the next one should have \(8\) circles. Counting carefully, we get that the answer is D. \(_\square\)

What comes next?

ParallelAngleBisector

Looking at the first three shapes, we recognize that the order of arrangement is such that the number of sides in each polygon is increasing by one, going from left to right. The first one has \(3\) sides, the second one \(4,\) and the third one \(5.\) Thus, the forth one should have \(6\) sides, implying that \(a)\) is the answer.

Also, the colors of the first three polygons are red, orange and yellow, respectively, which is the first three colors of a rainbow. Thus, the next color should be green, which again is the answer \(a).\) \( _\square\)

What comes next?

Let's label the vertices (corners) of the pentagon as follows:

Then, the red diagonal arrow connects \(1\rightarrow 4,\) then \(2\rightarrow 5,\) and then \(3 \rightarrow 1.\) In other words, it is connecting the first vertex to the vertex which is 3 away in the clockwise direction. Thus, we expect the next arrow to connect \(4 \rightarrow 2\), so the answer is B. \( _\square\)

What comes next?

image

Looking at the images, we can recognize the numbers \(1, 2, 3, \ldots.\) However, there is something additional added to them. If we were to ignore the numbers \(1, 2, 3,\) then we realize that we still get the numbers \(1, 2, 3,\) but rotated by \(180\) degrees! Thus, this sequence is obtained by taking a number, rotating it by \(180\) degrees and then superimposing it on the original image. The next number that we should use is \(8,\) and this procedure yields \(\text{c)}\) as the answer. \(_\square\)

What is the number of squares in the last section where the question mark is?

ParallelAngleBisector

The number of boxes in the first section is \(1.\)
The number of boxes in the second section is \(1,\) which is the sum of all the numbers of boxes before that.
The number of boxes in the third section is \(2,\) which is the sum of all the numbers of boxes before that, i.e. \[1+1=2.\] The number of boxes in the fourth section is \(4,\) which is the sum of all the numbers of boxes before that, i.e. \[1+1+2=4.\] The number of boxes in the fifth section is \(8,\) which is the sum of all the numbers of boxes before that, i.e. \[1+1+2+4=8.\] Thus, the number of boxes in the sixth and last section should be \(16,\) which is the sum of all the numbers of boxes before that, i.e. \[1+1+2+4+8=16. \] Therefore, the correct answer is \(\text{c).} \ _\square\)

• What Comes Next?
• General Term Pattern Recognition