# Uncle's mathematics puzzles after dinner

I was at my grandmother's home for a dinner. With my siblings, after the dinner party, watching a prime time show. One of my uncles, who is a mathematics professor, didn't like and asked us to sit down beside him for some puzzles. We solved many formations of Fibonacci and Premium Educational series except one which is posted below:

212  224  246  278  310  322

He asked us to tell the previous two and next two numbers of this series. Can someone help me to answer my uncle, who left this as a challenge for us?

Hint 1:

It is based on the formations of the unique numbers.

Hint 2:

One of my siblings reminded me that the first number, but not the answer, of this formation series is -7, as told him by Uncle personally.

Hint 3:

My grandmother's home is at the 5th floor in a 7 story building.

• Is it on purpose that the last number in the series reads 'minus 322' and not 'hypen, space, 322'? Jul 5, 2017 at 12:31
• Also, what does "formations of the unique numbers" mean? Would it be equivalent to say "no two numbers in the series are the same"? Jul 5, 2017 at 13:02
• It's very suspicious that all the numbers we know are even (and in fact f(n+1) = f(n)+2 (mod 10) seemingly) but -7 is odd (a debatable point, of course, but certainly "not even".) Jul 5, 2017 at 13:56
• First and last 2 numbers in series might be: (148 180) 212 224 246 278 310 322 (344 376) Jul 5, 2017 at 16:58
• I’m not sure what archaephyrryx meant when they asked “Is the rule recursive?”, but I believe that your response was misleading.  I believe that a recursive rule would mean that it’s easy to compute $U_{n}$ as a function of $U_{n-1}$, and, in fact, that that’s the easiest way to compute $U_{n}$.  But actually, it’s straightforward to compute $U_{n}$ without knowing $U_{n-1}$.  In fact, you can’t compute $U_{n}$ as a function of $U_{n-1}$ unless you also know the value of ${n}$ (or $n-1$), as far as I can see. But thanks for a fun, challenging puzzle. Jul 9, 2017 at 18:46

The requested numbers are

$$\mathbf{158,~~ 190},~~ 212,~~ 224,~~ 246,~~ 278,~~ 310,~~ 322,~~ \mathbf{354,~~ 376}$$

The formula for the n th Uncle number, $U_n$, is

$$U_n = 5\times p_n + 2\times n - 19$$ where $p_n$ is the n th prime.  The numbers shown in the question are $U_{13} = 5\times41 + 2\times13 - 19 = 212$  through  $U_{18} = 5\times61 + 2\times18 - 19 = 322$

For completeness, I could explain that
$U_{1} ~~= 5\times2~ ~+ 2\times1~ ~- 19 = 10 + 2 - 19 = -7$

How did I figure that out?

• We have a monotonically increasing sequence of integers that begins with an odd number, but it appears to settle into a groove of being all even numbers.  That made me think: that’s one bit off from a monotonically increasing sequence of integers that begins with an even number, but settles into a groove of being all odd numbers.

• Hint #3 mentions some parameters regarding grandmother’s home.

And, having solved the problem and written the solution, I took another look and noticed that

• the question body mentions the word “prime”:
… watching a prime time show.
Looking at the question’s edit history, I see that Hint #1 originally said, “Is in the first para.  Tricky and you need to look around.” but that was edited out.

• Good work! and this is absolutely correct. Jul 9, 2017 at 11:50

I believe the last two numbers are:

344 and 376

And the first two numbers are:

158 190

Why ?

Because the difference between the first and second number is 12, now lets add 10 to that and then another 10 now lets subtract 10 and then another 10 till you reach 12 now lets add 10 and another and so on.....

• How do you reconcile that with the given information that the series begins with −7? Jul 7, 2017 at 22:10
• Well I don't know Jul 7, 2017 at 22:14
• @PeregrineRook it starts with -7 but it is not the answer maybe if you continue the series you will get there Jul 7, 2017 at 22:17