## The Last Question

Published on Sunday, 14th September 2014, 01:00 am; Solved by 461;
Difficulty rating: 45%

### Problem 480

Consider all the words which can be formed by selecting letters, in any order, from the phrase:

Suppose those with 15 letters or less are listed in alphabetical order and numbered sequentially starting at 1.
The list would include:

• 1 : a
• 2 : aa
• 3 : aaa
• 4 : aaaa
• 5 : aaaaa
• 6 : aaaaaa
• 7 : aaaaaac
• 8 : aaaaaacd
• 9 : aaaaaacde
• 10 : aaaaaacdee
• 11 : aaaaaacdeee
• 12 : aaaaaacdeeee
• 13 : aaaaaacdeeeee
• 14 : aaaaaacdeeeeee
• 15 : aaaaaacdeeeeeef
• 16 : aaaaaacdeeeeeeg
• 17 : aaaaaacdeeeeeeh
• ...
• 28 : aaaaaacdeeeeeey
• 29 : aaaaaacdeeeeef
• 30 : aaaaaacdeeeeefe
• ...
• 115246685191495242: euleoywuttttsss
• 115246685191495243: euler
• 115246685191495244: eulera
• ...
• 525069350231428029: ywuuttttssssrrr

Define P(w) as the position of the word w.
Define W(p) as the word in position p.
We can see that P(w) and W(p) are inverses: P(W(p)) = p and W(P(w)) = w.

Examples:

• W(10) = aaaaaacdee
• P(aaaaaacdee) = 10
• W(115246685191495243) = euler
• P(euler) = 115246685191495243

Find W(P(legionary) + P(calorimeters) - P(annihilate) + P(orchestrated) - P(fluttering)).