A Puzzle

Figure 1 shows the problem that confronted me over the breakfast table this morning.

Figure 1

As I've said before, I dislike this format and what we should say is that we say that we have a function $f(x)$ that maps $x$ to $y$:

• $f(1)=3$
• $f(2)=3$
• $f(3)=5$
• $f(4)=4$
• $f(5)=4$
• $f(6)= \, \, ?$

I'll make a small adjustment here so that the graph has $x$ coordinates that correspond. It looks like this with the answer of $f(6)=6$ added which I'll explain and Figure 1 shows the result:

• $f(0)=1$
• $f(1)=3$
• $f(2)=3$
• $f(3)=5$
• $f(4)=4$
• $f(5)=4$
• $f(6)= 6$

Figure 1: permalink

As can be seen from the graph the pattern repeats once $(5, 4)$ is reached. Figure 2 is the pattern expanded.

Figure 2: permalink

What is the formula to represent the numbers as part of a recurrent series given by $a(n)$ = ... ? Can the graph be expressed in terms of $y=f(x)$ for some function $f$? These are two interesting questions that I should try to answer. Once I crack it, I'll post the information here and maybe a link to a new blog post about how I cracked it.