Easy Sequences

[Coyote]

There have been countless 'number sequence' puzzles posted in VSP, and I'll confess that most of them have been too difficult for me to solve. A few of them, IIRC, have never been solved by anyone.

So just as a change of pace, here's eight very easy number sequences -- and by 'very easy', I mean they should be instantly recognized by most of the people who frequent this site.

Of course, if I just wrote down the sequences there wouldn't be much of a puzzle, so I've taken the first ten positive integers of each sequence and arranged the digits in numerical order. For example, the digits in the first sequence can be rearranged to form
(spoiler) 1 3 5 7 9 11 13 15 17 19, which are the first ten positive odd integers.

You've probably noticed that while I spoke of eight easy sequences, there are actually nine puzzles shown. That's because the final sequence is something a little different. I'd be very surprised if anyone instantly recognized this one, since I made it up as a bit of a meta-puzzle. It has a very specific connection with the other puzzles, and in fact you won't be able to solve it until you've cracked the first eight. Once you do, the final answer should be clear.

-----------------------------------------------------------------

#1
11111133557799

#2
11112233345558

#3
1111122233357799

#4
001111234445666899

#5
01111223345555668

#6
1111222223445566688

#7
0001111122222334455667789

#8
000000000112222222333444566678888

#9
00011122222233444688

[ralphmerridew]

#4: 00111234445666899 = 1 4 9 16 25 36 49 64 81 100
Except it would need one more 1.

[ralphmerridew]

#3 1111122233357799 = 2 3 5 7 11 13 17 19 23 29

[ralphmerridew]

#2 11112233345558 = 1 1 2 3 5 8 13 21 34 55

[ralphmerridew]

#5 01111223345555668 = 1 3 6 10 15 21 28 36 45 55

[Coyote]

Crud. I thought I double checked everything, but you're right of course. I'll go edit #4 and then triple check the rest.

Also, correct on #2 and 3.

[edit] ...and #5 and probably whatever others you post while I type this.

[ralphmerridew]

#8 000000000112222222333444566678888 1 2 6 24 120 720 5040 40320 362880 3628800

[Coyote]

Found a second typo--#9 should have three 4's instead of two.

And correct on #8.

[Trojan Horse]

#6 1111222223445566688 = 1 2 4 8 16 32 64 128 256 512

[Trojan Horse]

#7 0001111122222334455667789 = 1 8 27 64 125 216 343 512 729 1000

And that leaves just the final one.

[Trojan Horse]

Looks like we're stuck on the last one.

[ralphmerridew]

Putting everything in one place:

#1: 1 3 5 7 9 11 13 15 17 19
#2: 1 1 2 3 5 8 13 21 34 55
#3: 2 3 5 7 11 13 17 19 23 29
#4: 1 4 9 16 25 36 49 64 81 100
#5: 1 3 6 10 15 21 28 36 45 55
#6: 1 2 4 8 16 32 64 128 256 512
#7: 1 8 27 64 125 216 343 512 729 1000
#8: 1 2 6 24 120 720 5040 40320 362880 3628800
#9: 00011122222233444688

[Coyote]

Well, I'd planned on posting a few hints on the one week mark, but since there's still some interest being shown here, I'll put a couple in spoiler tags rather than posting them outright.

If I'd included another sequence-code (which we'll call puzzle #0, because the order in which the sequences are listed has no relevance) that read as follows:
#0
000011112222344555668

Then the final sequence would have been changed so that the last puzzle read:
#9
00111222223344444688

This should lay enough extra groundwork for you to figure out the final puzzle.

[Trojan Horse]

Well, I got puzzle #0, at least:

000011112222344555668 = 1 4 10 20 35 56 84 120 165 220 (pyramid numbers)

Now back to work on the last one.

[Trojan Horse]

Looks like we all give up.

[Coyote]

I had one extra hint hidden in the final sentence. The key words are "groundwork" and "figure". Referencing the list ralphmerridew posted might help here.

[Trojan Horse]

There are 35 definitions of the word "ground" on dictionary.com. None seem applicable.


Both the old #9 and the new #9 have exactly 20 digits. All the other sequences have 10 numbers. Perhaps #9 also has 10 numbers, each of which have two digits.


Not much difference between the old #9 and the new #9; a 0 and a 2 both turned into 4s. Perhaps with #0 added in, there is now 2 more of something and 4 more of something else.


I thought of counting how many times each digit appears within sequences #1 through #8. No good; there are 15 0s, and there's no 5 available in sequence #9.


Counting the amount of numbers that END in each digit doesn't work either; 9 numbers end with 0, and there's no 9 available in sequence #9.


I'm just rambling now.


This must be how my math students feel when they come to office hours for help. "Enough hints! Why can't you just tell me the answer?"

[Coyote]

Okay, Trojan Horse--here you go (and thanks for sticking with this puzzle right up to the end).

There are indeed many definitions of 'ground', but 'figure' was also given as a keyword, and 'figure & ground' has a rather specific meaning.

In the case of this puzzle, the 'ground' is supplied by the numbers in the first eight sequences. The 'figure' of course is supplied by whatever other numbers are left.

Or, to put it more succinctly, the 9th sequence is the list of all numbers that don't appear in any of the other sequences. Checking ralphmerridews' list, it's clear the first ten numbers in that sequence would be:

12 14 18 20 22 26 30 38 40 42

Which, after rearranging the digits, gives:

00011122222233444688

And with that list of figures, I think we have sufficient grounds to lay this puzzle to rest.

[Trojan Horse]