5
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What is the next number in the sequence?

22 , 15 , 1 , 14 , 16 , 3 , -2 , 10 , 9 , 7 , ?

I will be adding more hints later if required :)

HINT 1:

The required number is the last in the sequence.

HINT 2:

Adding 100 to each number would make more sense

HINT 3:

enter image description here

I am new here, any improvements on the question are welcome!

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  • 1
    $\begingroup$ hint required... $\endgroup$ – Abhishek Patel Jun 10 '16 at 12:26
  • 2
    $\begingroup$ @AbhishekPatel added another hint :) $\endgroup$ – Gintas K Jun 10 '16 at 13:38
10
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The next number is:

11

Because:

When 100 is added (hint 2) these are the decimal ASCII codes for the letters zsertgbnmk which follow a squiggly pattern (confirmed by hint 3) across a qwerty keyboard.. the next and final letter is o which is ASCII code 111. Taking 100 off this leaves 11.

Observations:

In general I enjoyed the puzzle. I may have added Hint 3 as part of the question either as is or referenced a 'tilde' that the pattern resembles. This would have worked well as part of a story, as the pattern adds an element of confusion (could mean a lot of things.. wave, approximately, matrices). In my opinion Hint 2 rather gave the answer away, and could have been more subtle. Reference to shifting, rather than 'add 100', could have been used with Hint 3 'Shift me and I become a wave' or something more creative. But thanks for the ride ;)

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0
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The next number is:

444

because:

the Lagrange interpolation of the points $(1, 22)$, $(2, 15)$, $(3, 1)$, $\ldots$, $(10, 7)$ is: $$\frac{23}{120960}x^9 - \frac{187}{20160}x^8 + \frac{1391}{6720}x^7 - \frac{1337}{480}x^6 + \frac{28247}{1152}x^5 - \frac{136123}{960}x^4 + \frac{15670483}{30240}x^3 - \frac{5593949}{5040}x^2 + \frac{509699}{420}x - 480$$ and setting $x=11$ (the next $x$ value) produces 444.

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