Friday Puzzler -- Sequence

A couple of hints

Alex Y

One hint is explicit and one is a word used in this post.

The sequence ends with the term you add to the list. An argument could be made for one additional term beyond that, but definitely no further.

That's it. Let us know if you base your solution on one of the hints here, or if you get it independent of these hints..

Re: A couple of hints

Larry Barrett

Seems like there could be a handful of additional terms. A couple could be 102, 103.

Not in the sequence I have in mind

Alex Y

I don't doubt that there is a rule that gives you the additional terms you mentioned, but the rule I am thinking of would generate only one additional term.

Hint 3

Alex Y

The rule that generates this sequence is not iterative--the nth term can be determined without regard to the previous or succeeding terms.

Hint 2

Alex Y

Was hidden in the original hint post, but is now revealed:

One hint is explicit and one is a word used in this post.

The sequence ends with the term you add to the list. An argument could be made for one additional term beyond that, but definitely no further.

That's it. Let us know if you base your solution on one of the hints here, or if you get it independent of these hints.

Re: Hint 2

Larry Barrett

Since the only digits that appear in the sequence, I thought they were quinary (base 5) numbers, which translate to 5,6,7,8,9,10,12,26 base 10. No discernible pattern, which is why I said there could be a handful of next numbers.

So right idea, but no cigar so far.

Hint 4

Alex Y

Kind of at a loss about how to put out hints that don't just give it away.

Here is another sequence, again with the last term missing. It uses the same formula for determining the terms, with one changed constant:

10

11

12

13

14

15

20

22

30

110

?

Next and final hint will be the missing last terms from these two sequences, with the challenge changed to "find the rule"

Hint 5 -- Last term

Alex Y

Here is the original sequence with the last term filled in:

10

11

12

13

14

20

22

101

1010

The second sequence I posted can be completed like this:

10

11

12

13

14

15

20

22

30

110

1100

Now, what is the pattern/rule?

The pattern

Alex Y

This sequence totally stumped me, but I thought that the last term might be a give-away.

The original sequence was ten, expressed in decreasing bases:

10 base 10 = 10

11₉ =10

12₈ =10

13₇ =10

14₆ =10

20₅ =10

22₄ =10

101₃ =10, and finally

1010₂ =10

The other sequence I posted in a hint is made up of 12 in different bases

10₁₂ =12

11₁₁ =12

12₁₀ =12

13₉ =12

14₈ =12

15₇ =12

20₆ =12

22₅ =12

30₄ =12

110₃ =12

1100₂ =12

I'm curious, Alex

Henry Higginbotham

You said of the first sequence, " An argument could be made for one additional term beyond that, but definitely no further."

Could you share the argument?

Re: I'm curious, Alex *LINK*

Alex Y

I tend to think of base two as the lowest base. But in some sense simply tallying is base one representation of number. 5 is represented at 11111 base 1.

1*1^4 +1*1^3 + 1*1^2 + 1*1^1 + 1*1^0

But this lacks features that make other bases "work". The main one that comes to mind is the lack of using the digit 0 as a "placeholder". Or if you do allow 0 within a base 1 number, then the representation of a number is not unique -- 3 is 111 or 100101, etc. in base one.

And I'm not the only one not quite sure whether to consider this a legitimate base--this Wikipedia article says "A base can be any whole number bigger than 1.", then a few lines later has a table of examples that include base 1.


Wikipedia article