7. The digits represent the number of letters in each word of the puzzle. “What” has four letters, “digit” has five letters, “should” has six letters, etc. The last word is “numbers” which has seven letters.

O S. The series contains the first letter of each month in alphabetical order (April, August, December, February, January, July, June, March, May, November). O and S represent the remaining 2 months, October and September.

54325. Rather than being a mathematical solution, the answer is derived by prepending (putting in front of) the value on the left side of the equation with the previous answer. So for 3 you take 25, the previous answer, and put 3 in front, giving you 325. Then you put 4 in front to get 4325 and lastly, put 5 in front to get 54325.

455. The numbers are from the Fibonacci sequence in groups of 3. The next numbers in the series are 34 (the 3 was already used in 213) and 55.

E (one, two, three, four, five, six, seven eight)

E. Oh say can you see, by the dawn’s early light.

3 ½.

If you break up the numbers into groups of 4, a,b,c,d, then 2 × |a – d| = |b – c|.

9 2 4 8 → 2 × |9 – 8| = |2 – 4|, or 2 × 1 = 2
4 3 7 6 → 2 × |4 – 6| = |3 – 7|, or 2 × 2 = 4, so
n 3 2 3 → 2 × |3 ½ – 3| = |3 – 2|, or 2 × ½ = 1;

I’m not sure this is the correct solution, but it’s the best I’ve found so far. Email me if you know of a better one.

91. To get the number in the fourth column, you add the numbers in column 1 and 2, then multiply by the number in column 2. f(n,m) = (n + m) * m
For example, f(2,3) = (2 + 3) * 3 = 15. Thus f(6,7) = (6 + 7) * 7 = 91

675.

The next number in the sequence is n squared minus m or f(n,m) = n² – m
f(6,9) = 6² – 9 = 27
f(9,27) = 9² – 27 = 54
f(27,54) = 27² – 54 = 675
f(54,675) = 54² – 675 = 2241

The missing numbers are 4 and 9. The list is sorted alphabetically by the English spelling of the numbers, so four belongs after five and nine comes after fourteen.