Number Sequences Puzzle - Solution

The Puzzle:

Number Sequences
There are infinitely many formulas that will fit any finite series.

Try to guess the next number in each sequence using the simplest mathematical operations or ideas (you may get a different answer than my solution though):

· 8723, 3872, 2387, ?
· 1, 4, 9, 18, 35, ?
· 23, 45, 89, 177, ?
· 7, 5, 8, 4, 9, 3, ?
· 3, 8, 15, 24, 35, ?
· 2, 4, 5, 10, 12, 24, 27, ?
· 1, 3, 4, 7, 11, 18, ?
· 99, 92, 86, 81, 77, ?
· 0, 4, 2, 6, 4, 8, ?
· 1, 2, 6, 24, 120, ?
· 5, 7, 12, 19, 31, 50, ?
· 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, ?
· 126, 63, 190, 95, 286, 143, 430, 215, 646, 323, 970, ?
· 4, 7, 15, 29, 59, 117, ?
· 4, 4, 341, 6, 4, 4, 6, 6, 4, 4, 6, 10, 4, 4, 14, 6, 4, 4, 6, 6, 4, 4, 6, 22, 4, 4, 9, 6, ?




Our Solution:

You may have come up with other good and well-reasioned answers, but here are the ones I have:

· 8723, 3872, 2387, => 7238 (moving of numerals)
· 1, 4, 9, 18, 35, => 68 (×2+2, ×2+1, ×2+0, ×2-1, ×2-2)
· 23, 45, 89, 177, => 353 (×2-1)
· 7, 5, 8, 4, 9, 3, => 10, 2 (two series - every second number: 7, 8, 9, 10 and 5, 4, 3, 2)
· 3, 8, 15, 24, 35, => 48 (+5, +7, +9, +11, +13)
· 2, 4, 5, 10, 12, 24, 27, => 54, 58 (×2, +1, ×2, +2, ×2, +3, ×2, +4)
· 1, 3, 4, 7, 11, 18, => 29 (a+b=c, b+c=d, c+d=e)
· 99, 92, 86, 81, 77, => 74 (-7, -6, -5, -4, -3)
· 0, 4, 2, 6, 4, 8, => 6 (+4, -2, +4, -2, +4, -2)
· 1, 2, 6, 24, 120, =>720 (×2, ×3, ×4, ×5, ×6)
· 5, 7, 12, 19, 31, 50, => 81 (a+b=c, b+c=d, c+d=e)
· 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, => 322, 161 (×3+1, /2, ×3+1, /2 ?)
· 126, 63, 190, 95, 286, 143, 430, 215, 646, 323, 970, => 485, 1456 (/2, ×3+1, /2, ×3+1 ?)
· 4, 7, 15, 29, 59, 117, => 235 (×2-1, ×2+1, ×2-1)
· 4, 4, 341, 6, 4, 4, 6, 6, 4, 4, 6, 10, 4, 4, 14, 6, 4, 4, 6, 6, 4, 4, 6, 22, 4, 4, 9, 6, => 4, 4


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