2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								---
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								id: 5900f54c1000cf542c51005f
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								title: 'Problem 480: The Last Question'
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								challengeType: 5
							 | 
						
					
						
							
								
									
										
										
										
											2019-08-05 09:17:33 -07:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								forumTopicId: 302158
							 | 
						
					
						
							
								
									
										
										
										
											2021-01-13 03:31:00 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								dashedName: problem-480-the-last-question
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								---
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								# --description--
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								Consider all the words which can be formed by selecting letters, in any order, from the phrase:
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								thereisasyetinsufficientdataforameaningfulanswer
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								Suppose those with 15 letters or less are listed in alphabetical order and numbered sequentially starting at 1.
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								The list would include:
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								1 : a
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								2 : aa
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								3 : aaa
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								4 : aaaa
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								5 : aaaaa
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								6 : aaaaaa
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								7 : aaaaaac
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								8 : aaaaaacd
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								9 : aaaaaacde
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								10 : aaaaaacdee
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								11 : aaaaaacdeee
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								12 : aaaaaacdeeee
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								13 : aaaaaacdeeeee
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								14 : aaaaaacdeeeeee
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								15 : aaaaaacdeeeeeef
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								16 : aaaaaacdeeeeeeg
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								17 : aaaaaacdeeeeeeh
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								...
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								28 : aaaaaacdeeeeeey
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								29 : aaaaaacdeeeeef
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								30 : aaaaaacdeeeeefe
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								...
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								115246685191495242: euleoywuttttsss
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								115246685191495243: euler
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								115246685191495244: eulera
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								...
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								525069350231428029: ywuuttttssssrrrDefine P(w) as the position of the word w.
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								Define W(p) as the word in position p.
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								We can see that P(w) and W(p) are inverses: P(W(p)) = p and W(P(w)) = w.
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								Examples:
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								W(10) = aaaaaacdee
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								P(aaaaaacdee) = 10
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								W(115246685191495243) = euler
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								P(euler) = 115246685191495243Find W(P(legionary) + P(calorimeters) - P(annihilate) + P(orchestrated) - P(fluttering)).
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								Give your answer using lowercase characters (no punctuation or space).
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								# --hints--
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								`euler480()` should return turnthestarson.
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								```js
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								assert.strictEqual(euler480(), turnthestarson);
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								```
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								# --seed--
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								## --seed-contents--
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								```js
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								function euler480() {
							 | 
						
					
						
							
								
									
										
										
										
											2020-09-15 09:57:40 -07:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								  return true;
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								}
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								euler480();
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								```
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							
								
									
										
										
										
											2020-11-27 19:02:05 +01:00
										 
									 
								 
							 | 
							
								
									
										
									
								
							 | 
							
								
							 | 
							
							
								# --solutions--
							 | 
						
					
						
							
								
									
										
										
										
											2018-09-30 23:01:58 +01:00
										 
									 
								 
							 | 
							
								
							 | 
							
								
							 | 
							
							
								
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								```js
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								// solution required
							 | 
						
					
						
							| 
								
							 | 
							
								
							 | 
							
								
							 | 
							
							
								```
							 |