# Sequence Puzzle - First Question of 42

Professor Tera was walking on Connecticut Road, where there are some public restrooms.
Tera saw some students standing in a row. They seemed to be playing a game called Evolution.
And it looked like more and more students were joining them.
Then Tera heard a conversation between two other teachers.

One of the teachers pointed to the row of students and said:
"Look! He's the worst student in my class!"
The other teacher said:
"Why?"
"We had twenty-four math tests, and he failed all of them!"
"How?"
"He got zeroes in the first two tests. And, in the third test,
he got only three points!"
"Only six points. And he got twelve in the fifth test."
"Looks like he's getting better and better. I suspect he'd get
twenty-four in the next one..."
"No, zero!" The teacher was angry.
"Oh, no..."
"He had a cold before the sixth test and was sent to hospital.
So he missed the sixth test."
"Twenty-four. And then, forty, then twenty-four."
"Ah, forty - That's only fifteen points below the pass mark,
but his scores seem to be dropping. What about the tenth time?"
"Let me think... Ah, it slipped my mind."
The conversation continued.
"Looks like he never met my expectation..."
"I agree."
"Yeah, you know, I promised to give him a toy spaceship once he
got a hundred."

Find the missing terms in the sequence.

0 0 3 6 12 0 24 40 24 [___] 6 12 6 0 0 24 16 0 0 8 6 16 0 0 ...

Question 1: Find the missing terms in the sequence above.

And fill in the blanks in the extended version given in the hints!

Question 2: Try extending the sequence.

Will the student pass the next test? Will the student be sent to hospital another time?

Hint 1:

Pay attention to the title and the italicized words.

Hint 2:

I tried several times to find this sequence on OEIS, but failed every time.
It's probably because the sequence broke their rules.
Here's what I did:
http://OEIS.org/search?q=0+0+3+6+12+&sort=&language=&go=Search
http://OEIS.org/search?q=3+6+12+0&sort=&language=&go=Search
http://OEIS.org/search?q=0+24+40+24&sort=&language=&go=Search#found_nothing
http://OEIS.org/search?q=6+12+6+0+0&sort=&language=&go=Search
http://OEIS.org/search?q=56+12+32+16&sort=&language=&go=Search#found_nothing
http://OEIS.org/search?q=128+124+128+82&sort=&go=Search#found_nothing

Hint 3:

Extended sequence: (Please correct the mistakes, if any)
0 0 3 6 12 0 24 40 24 [___] 6 12 6 0 0 24 16 0 0 8 6 16 0 0 56
12 32 16 54 48 24 72 272 72 48 56 6 84 128 20 [___] 128 124 128 82
16 106 60 88 92 118 132 28 136 138 100* 136 128 168 188 98 172 90 144 140
124 164 164 292 148 128* 674* 152 248 336* 280 432 368* 168 348 428 588* 160 332 330*
336* 248* 372* [___] 368 628* 480* 816* 392* 360 780* 476 600* 612 936*
562 790* 452** 396 968**** 520*** 620** 552 426* 604* 406 ...

Hint 4:

Unlike this, which is actually a set, not a sequence,
my puzzle is indeed a sequence.

Hint 5:

23/3

Hint 6: (Related Tags)

(for the story line)
(for the main sequence)
(for the story line)
(for Hint 5)
(for the story line)
(for Question 1)

Hint 7: (to make Hint 5 clearer)

Hint 5 shouldn't be read as "23 divided by 3" or "23 of March".
"23/3" also has nothing to do with the slang abbreviation "233"
(which basically means "LOL") in China.

• If I understand correctly, then rot13(gur dhrfgvba vf vzcbffvoyr, fvapr gur cragnqrpnguyba unf ab fgnoyr cbchyngvba - gur oynax vf hasvyynoyr) Commented Jul 25, 2020 at 19:46
• @Woofmao , Did you see the "164" in "Hint 3"? Commented Jul 26, 2020 at 3:26
• @Woofmao , Let me explain it more clearly. You said that the 10th term is impossible because the 10-bit figure is unstable. But, the 67th one is unstable too, while Hint 3 says that the term is "164" instead of "impossible". Commented Jul 31, 2020 at 2:56
• I found a (possible) sequel: puzzling.stackexchange.com/questions/101240/… Commented Aug 18, 2020 at 7:39

First blank:

$$\dfrac{314}{15}$$

Second blank in hints:

Estimated to be $$\dfrac{1004}{3}$$

Third blank in hints:

Estimated to be $$\dfrac{868}{3}$$

Reasoning:

It is the expectation final cell count of the 1 by n rectangle in Conway's Game Of Life. If the result is not an integer, take the average cell count in each phase.