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September 2018 Puzzle Periodical - The F Sequence

By Wendell W., NSA Mathematician

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Creator Challenge Difficulty Rating: Easy

Problem

What comes next in this sequence?

1, 1, 2, 720!, (((120!)!)!)!, ?

Here n! = n x (n-1) x (n-2) x … x 3 x 2 x 1

Click to see the answer!

Solution

Notice that 720 = 6!, and 120 = 5!. Thus we can write this as:

1, 1, 2, (6!)!,((((5!)!)!)!)!.

Note further than 6 = 3!, so this can be written:

1, 1, 2, ((3!)!)!, ((((5!)!)!)!)!.

Finally, note that 1 = 1! = (1!)! = ((1!)!)! = … and likewise 2 = 2! = (2!)! = ((2!)!)! = … so we can rewrite the sequence as:

1!, 1!, (2!)!, ((3!)!)!, ((((5!)!)!)!)!.

This is the start of the Fibonacci Sequence (1,1,2,3,5,8,13,….) in which we apply the number of factorials (!) to each element equal to that number. So 1 has one factorial, 2 has 2 factorials, 3 has 3 factorials, and so forth. The next entry is thus 8 with 8 factorials:

(((((((8!)!)!)!)!)!)!)! = ((((((40320)!)!)!)!)!)!)!.