# Complete this cross sequence

What is the next element of this sequence and why?

Possible choices:

Hint 1

There are more than one answers, but only one of the listed options is correct.

Hint 2

Consider each cross as two Cartesian axes with the same scale.

Hint 3

(Apparently this puzzle is really, really hard) Consider the lines that are not crosses as the graph of a function $$y=f(x)$$. Draw a grid (or the ticks in the axes). Any scale will be fine but some scales are more convenient than the others.

Hint 4

There is a progressive pattern in the sequence. That means that the answer does not have something in common with all the elements in the sequence: it really continues the sequence in some sense. Also, $$[-3;3]$$ is a good scale for the two axes.

Hint 5

Using $$[-3; 3]$$ as scale write the value of $$f(x)$$ for each diagram and for each $$x \in [-3; 3]$$. Now it should be clear what is the pattern

C

Reasoning

Following the hints, if we consider the axes to have the same scale with displayed range as $$[-3,3]$$ for both axes, then the four functions in order seem to be $$f(x) = x-2 \,\,\,,\,\,\, f(x) = 3-x \,\,\,,\,\,\, f(x) = x \,\,\,,\,\,\, f(x) = 3$$ Now consider the value of $$f(2)$$. Reading in order we have $$f(2) = 0,1,2,3$$
This suggests that the next graph in the sequence will have $$f(2) = 4$$. As we can see, graph C is the only such candidate and, indeed, the graph appears to be $$f(x) = 2x$$ so this is the next in the sequence.

The answer might be this: Considering that the picture can be of any scale in all diagrams i.e. : each cross (axes) may have different scales as scales or markings are not mentioned.

First diagram : $$\fbox{1}x - 1y = \text{ (some constant say } \fbox{2})$$ i.e. $$\fbox{1}x - 1y = \fbox{2}$$ Second $$\fbox{1}x + 1y = \fbox{1}$$ Third: $$\fbox{1}x-1y=\fbox{0}$$ [Note: the right hand side constant and the coefficient of $$x$$ form a base 3 number system. i.e first : $$12$$, second: $$11$$, third: $$10$$, fourth: $$02$$. The plus and minus alternate]. Fourth: $$\fbox{0}x + 1y = \fbox{2}$$, Fifth: $$\fbox{0}x-1y = \fbox{1}$$ Thus option E.

• considering the pictures as diagrams is a ggod starting point. The issue is that this solution works only if the diagrams have different scales – melfnt Jun 1 at 6:35

Given the recent hints about Cartesian coordinates here's a new attempt:

E

Since

Suppose the range is [3,3] for each diagram (per hint 3), we could make a chart regarding the slope, y-intercept, and integral of each one. There is no obvious pattern, still. However, there is a clear link between the signs of y-intercept and integral. Per hint 1 which states that there could be more than 1 possibilities compatible with the existing sequence, E is the only option that combines negativity and the relation between y-intercept and integral

Ι say it is

figure B.

The reason is:

the top set has 4 figures with an even number of angles, namely 12,6,6,6. The bottom set has four figures with an even number of angles and one figure with an odd number of angles. Figure B has 9 angles.

• I really don't get the point of this answer. If your reasoning is correct the answer should have an even number of angles too – melfnt May 31 at 17:28
• @melfnt. When you draw geometric figures with straight lines you should have in mind two fundamental aspects: a) the number of straight lines b) the number of angles. – Vassilis Parassidis May 31 at 18:02
• Sure, maybe your reasoning about the number of angles is corrected. The issue is: since you are finding the next element of a sequence you should add an item with the same parity of the pre-existing elements, not with the opposite parity. – melfnt May 31 at 20:53
• The reason I put 9 is all numbers on the first group have a common factor. In addition to that the second group has one odd number and I thought you did that intentionally. These are the reasons why I chose B.. – Vassilis Parassidis May 31 at 23:44