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This sequence goes on forever, but here are the first 7 terms. The X's are just there to hide the actual digits. Can you find all the correct digits of the 8th entry?

1024, 90X1X, 25023X, 490364, X1031X0, XX1041X4, X690XXXX, ________?

Hint:

No complicated calculations needed. In fact, it's based on a very simple rule. I'm sure this sequence would've been solved within minutes if I hadn't replaced the digits with X's.

More specific hint:

Squares involved

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  • $\begingroup$ Does the use of 'X' represent the sameness of all digits? $\endgroup$ Commented Jun 4, 2020 at 17:03
  • $\begingroup$ @JohnBrookfields No, it doesn't. $\endgroup$ Commented Jun 4, 2020 at 17:06
  • $\begingroup$ @Jarvis In order to find the two digit number between the squares, you have to think of other numbers. What could they possibly stand for? $\endgroup$ Commented Aug 3, 2022 at 9:46

3 Answers 3

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To wrap this one up at last, @sedrick and @JohnBrookfields were almost there - they had cracked the beginnings and endings of each term, just not the middle. Here's the complete rule - the sequence comprises:

ascending pairs of square numbers, sandwiching the number of prime numbers that fall between them.

Add in the missing square numbers and see that part of the pattern:

1024, 90X1[6], 25023[6], 490364, [8]1031[0]0, [12]1041[4]4, [1]690X[196]

Now add in the missing prime counts and see that part of the pattern:

1024, (2,3)
90[2]16, (11,13)
250236, (29,31)
490364, (53,59,61)
8103100, (83,89,97)
12104144, (127,131,137,139)
1690[5]196 (173,179,181,191,193)

This is all coming together nicely, so we can be pretty confident that we have found the full pattern now.

All that remains is to find the next term, which would be made up of...

the numbers 225 (15 squared) and 256 (16 squared), sandwiching the number '06' (primes 227,229,233,239,241,251), i.e. 22506256.

There's a little hint towards this in the title of the puzzle:

'Look at me' suggests looking at the OP's username: Prim3numbah - this suggests that prime numbers are involved somehow!

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  • $\begingroup$ When number sequences were still kinda popular :D. This is correct ofc. Well done! $\endgroup$ Commented Aug 11, 2022 at 12:36
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    $\begingroup$ Thanks @Prim3numbah - ended up in a sidebar rabbithole and rediscovered this one! $\endgroup$
    – Stiv
    Commented Aug 11, 2022 at 12:41
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The pattern is

1024, 90X1X, 25023X, 490364, X1031X0, XX1041X4, X690XXXX
1024, 90216, 250236, 490364, 8103100, 12104144, 16904196

Hence, the next number is

22505256

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    $\begingroup$ Why 05 in the middle vs. 04? $\endgroup$
    – El-Guest
    Commented Jun 4, 2020 at 17:08
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    $\begingroup$ Yes, that's also something that confused me. It seems like it increases every two numbers, but then 02 occurs thrice, so I'm not sure anymore. $\endgroup$
    – sedrick
    Commented Jun 4, 2020 at 17:10
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    $\begingroup$ @sedrick well done, but the mid number is incorrect. The mid numbers represents something else $\endgroup$ Commented Jun 4, 2020 at 17:15
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The fist few digits represent the odd number squares and the last few digits represent the even number square. ($1^2$ and $2^2$, $3^2$and $4^2$ and so on.) So, the next number will be $22504256$. The numbers given are: $1024, 90216, 250236, 490364, 8103100, 12104144, 16904196$ and finally $22504256$

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  • $\begingroup$ Good work, but the mid number is not correct. Take a look on the comment section above $\endgroup$ Commented Jun 4, 2020 at 17:26

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