To get the Next Number, then, all you have to do is reverse the process. In this case you start with the common difference, 6, and add that to the last entry in the row below, 41, then add the sum, 47, to the last entry in the row below that, 159, and so forth until you get to the bottom row and the solution, 666.
 

The sophisticated puzzle solver, may be curious to know if that method will always work in puzzles that ask for the Next Number.

uppose the puzzle gave only the first four integers in the series instead of seven. If we apply the same method, we will run out of given numbers to subtract from one another before we find differences that agree with each other on top row.
 

The method still works, and we can calculate the Next Number, 183, just fine thank you.

The previous sentence does not have an exclamation point but the next one does. The sophisticated puzzle solver did not need all that extra information provided in the statement of the puzzle after all! He or she did need something, though...

By the way, in solving the puzzle with all of that extra information, we had to make an assumption just the same. We observed four 6s in a row and assumed that the fifth entry in that row would be a 6 also.

f the series above had as its fifth entry the number 184 instead of 183, then we need to create another row for the difference between 7 and 6. Here again, since we have run out of information, we deliberately assume that 1 is the common difference and produce 306 for the sixth entry instead of the number 301 in the original puzzle.
 

You might run this series out to discover what number replaces the 666 in our original solution. Also, try using a different number in place of the 184, like 182, which is smaller than the original 183. What happens when the fifth number is 99, which is smaller than the previous number, 100? Wild stuff, there.

King for a Day explores the matter of assumptions and predictions further. In the meantime, consider a slightly different test of our method for finding the Next Number. Let the fourth entry be the number 101 instead of the 100 in the original puzzle, but put the fifth entry back at the original number 183.
 

Whereas the original puzzle produces 666 for the eighth entry, we now get only 563. With -4 as the common difference, try running your predictions out into the 'future' and watch what happens.

he method we have been using still works. It is easy to show that the series will reach a peak and then the entries decrease and eventually become negative. Which is fine. Given any number of entries, apparently, we can make our assumption and then nonchalantly produce the Next Number.

We can do something else, too. All Next Number puzzles have a tacit requirement:

That the puzzle solver must provide a rationale for the solution.

Having made our assumption explicit, we always have a rationale for predicting the Next Number for any series of numbers.

onsider the series that starts out 0,  2,  24,  252,  3,120, ... We build a difference table as set forth below, and when we run out of differences to take, we make our explicit assumption of constant differences to form a rationale for projecting the series ... 13,310,  37,752,  63,624, ...
 

Thing is, each entry in this series was generated by the simple formula nⁿ - n. Accordingly, here is what the puzzle-master must have been expecting as a solution to the puzzle...