Is this pattern solvable?

Question

The objective is to find what the next number is.

0, 1, 1, 0, 2, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 1, 0, 2, 1, 0, 2, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 1, 0, 2, 1, 0, 2, 0, 0, 1, 0, 2, 1, 0, 2, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 1, 0, 2, 1...

After every day, a number gets added onto the end of the list (it's either a 0, 1 or a 2). So the only way to know if we're right is to wait till the next day...

Anybody have any ideas as to how to approach this? It seems there's a pattern but it just hurts me.

Edit: Context is as follows...

In a city there are 2 houses. House 1 is north, and house 2 is south.
The owners of those houses might become forgetful and leave their house for the day with their door unlocked.
0 represents that both doors are locked (they are not accessible).
1 represents the northern house is unlocked. 2 represents the southern house is unlocked.

Only 1 house can be unlocked at a time. Either one house is unlocked, or both are not.

Edit2: 23/02/2018: 0
Edit3: 24/02/2018: 2
Edit4: 25/02/2018: 1

No matter how many terms you have, there's no way of being sure what the next one is. It's always possible that they're just random. If you have some deterministic process that produces the values, you may be able to guess what the pattern is, and prove it, but there's really nothing to do with a sequence like this. Once can always pass a polynomial of sufficiently high degree through the data points, for example. — saulspatz, Feb 22 '18 at 03:44
@saulspatz: There are 26 $1$'s, 26 $2$'s, and 74 $0$'s. If it were randomly generated, the chances that there is an equal number of $1$'s and $2$'s are pretty slim. — Tito Piezas III, Feb 22 '18 at 03:54
@Qmi: It may or may not be important that the "0"s occur in groups of $1$, $2$, or $4$ (ie, powers of $2$). Hard to say. Can you provide more context about the problem? If it's a classroom exercise, what's the grade level? What's the primary language at the school? In any case, since there are zillions of ways to continue the pattern mathematically, it isn't really a math question (which is probably why it's getting close votes). This question seems more appropriate to Puzzling.SE. — Blue, Feb 22 '18 at 06:20
@TitoPiezasIII Those are all the entries I have... A new day, a new number. So, I had to wait a lot of days to get these numbers lol — Qmi, Feb 22 '18 at 12:18
@Blue Okay so basically, in a city there are 2 houses. House 1 is north, and house 2 is south. The owners of those houses might become forgetful and leave their house for the day with their door unlocked.
0 represents that both doors are locked (they are not accessible). 1 represents the northern house is unlocked. 2 represents the southern house is unlocked.

Only 1 house can be unlocked at a time. Either one house is unlocked, or both are not. — Qmi, Feb 22 '18 at 12:20
@Qmi: I don't see how the "two houses" stuff matters to any of this, unless perhaps the "might become forgetful" aspect speaks to some probabilistic underpinning of a basically-random sequence or something. What makes you believe that this sequence has a pattern? — Blue, Feb 22 '18 at 12:31
@blue Well, I don't know if there's a pattern or not. But if there is one, I would like to crack it. But that's the context behind it and the 'forgetful' aspect doesn't influence these numbers in any way. Looking at Tito Piezas' answer below, it seems that they're onto something which convinces me there might be one... yeah more entries are needed and that's also the pattern I came up with. — Qmi, Feb 22 '18 at 12:40
@Qmi: So, what's the broader context of this problem? Who's providing these numbers and why? Can you provide a source? (I trust you aren't staking-out two houses in your neighborhood hoping to predict when one will be vulnerable. :) — Blue, Feb 22 '18 at 12:45
@Blue Sorry no source, but a computer is providing these numbers, every day a new number, no reward for figuring out the pattern, this is just a spin-off fun thing for one to do. Except it's probably not as fun anymore lol — Qmi, Feb 22 '18 at 12:49
@Qmi: "a computer is providing these numbers". That's still not very helpful. Ultimately, I guess you're saying that you're observing these numbers (somewhere, somehow), and that you suspect there's a pattern, even though there's no objective reason to believe that there is one. (For instance, this isn't a challenge or puzzle posted somewhere.) Without being very, very clear on that last bit, you're just wasting people's time. In the future, please be more considerate to those you're asking for help. — Blue, Feb 22 '18 at 13:05
@Qmi: Come back in blocks of 8 days and perhaps we can eventually see a pattern. :) — Tito Piezas III, Feb 22 '18 at 13:23
@TitoPiezasIII Haha okay, yes it does seem to be every 8 days that a number might repeat. For example, the second number (1) crops up after every 8 days until it breaks at the 37th number... but thereafter it repeats until it breaks again — Qmi, Feb 22 '18 at 13:43
@Qmi: I'm assuming the next two days are $0,2$. Pls see edited answer, and kindly come back in two days. :) — Tito Piezas III, Feb 22 '18 at 13:49

Tito Piezas III · Accepted Answer · 2018-02-23T16:00:32.880

0

(This is a new tentative answer.)

As explained by the OP, the numbers are generated daily by computer. It seems the first $8\times9 = 72$ numbers, labeled by columns $1,2,3,\dots8$ are "copied" by the next $72$ numbers with "shifted" columns $4,\color{blueviolet}1,6,\color{blueviolet}3,8,\color{blueviolet}5,2,\color{blueviolet}7$.

For example, column $4$ has $0,0,2,2,2,2,2$ :

$$\begin{array}{|c|c|} \hline {\color{red}{1,2,3,4,5,6,7,8,}\\ \color{blue}{0, 1, 1, 0, 2, 0, 0, 0,\\ 0, 1, 1, 0, 2, 0, 0, 0,}\\ 0, 1, 1, 2, 2, 0, 0, 0,\\ 0, 1, 0, 2, 2, 0, 0, 0,\\ 0, 1, 0, 2, 1, 0, 2, 0,\\ \color{blue}{0, 0, 0, 2, 1, 0, 2, 0,\\ 0, 0, 0, 2, 1, 0, 2, 0,}} &{\color{red}{4,1,6,3,8,5,2,7,}\\ \color{blue}{0, 0, 0, 1, 0, 2, 1, 0,\\ 0, 0, 0, 1, 0, 2, 1, 0,}\\ 2, 0, 0, 1, 0, 2, 1, 0,\\ 2, 0, 0, 0, 0, 2, 1, 0,\\ 2, 0, 0, 0, 0, 1, 1, 2,\\ \color{blue}{2, 0, 0, 0, 0, 1, 0, 2,\\ 2, 0, 0, 0, 0, 1,}\color{brown}{0,2,}}\\ \hline 0, 0, 0, 1, 1, 0, 2, 0,\\ 0, 0, 0, 1, 1, 2, 2, 0,\\ \hline \end{array}$$

Thus, my guess is for the next two days, the next two numbers will be $0,2$ (in brown). This is a testable hypothesis, assuming the OP will come back with more data.

(Old answer.)

It's not completely random, but if it does have a pattern, it's tricky. Using the repeating $0,0,0,0$ as a place-holder, we have,

$$?,?,?,?,?,?,?,0,\\ 1, 1, 0, 2, 0, 0, 0, 0,\\ 1, 1, 0, 2, 0, 0, 0, 0,\\ \color{blue}{1, 1, 2, 2, 0, 0, 0, 0,\\ 1, 0, 2, 2, 0, 0, 0, 0,}\\ \color{brown}{1, 0, 2, 1, 0, 2, 0, 0, 0, 0,}\\ 2, 1, 0, 2, 0, 0, 0, 0,\\ 2, 1, 0, 2, 0, 0, 0, 0,\\ 1, 1, 0, 2, 0, 0, 0, 0,\\ \color{blue}{1, 1, 2, 2, 0, 0, 0, 0,\\ 1, 0, 2, 1, 0, 0, 0, 0,}\\ \color{brown}{1, 0, 2, 1, 0, 2, 0, 0, 1, 0,}\\ 2, 1, 0, 2, 0, 0, 0, 0,\\ 2, 1, 0, 2, 0, 0, 0, 0,\\ \color{blue}{1, 1, 2, 2, 0, 0, 0, 0,\\ 1, 0, 2, 2, 0, 0, 0, 0,}\\ 1,?,?,?,?,?,?,?,$$

edited Feb 23 '18 at 16:00

answered Feb 22 '18 at 05:03

Tito Piezas III

54,565

Hey, I added the new data for 23/02/2018 and it was a 0. I also predicted it was going to be 0 and then a 2, no doubt from the similar data. By the way, that new tentative answer was definitely something I didn't come around, and it looks good! If it works, then we just need to figure out how the numbers changes across lol – Qmi Feb 23 '18 at 09:49
@Qmi: We'll see if the third day yields a $1$. – Tito Piezas III Feb 23 '18 at 13:48
It did! Now I just hope that it follows the pattern you have found. (: – Qmi Feb 25 '18 at 01:10
@Qmi: Great! Assuming it follows the pattern, the next 7 days should be: $0, 0, 0, 0, 1, 0, 2$. Maybe don't edit your post anymore, as some of the guys may downvote it. Just add the numbers for the next few days as a comment here. – Tito Piezas III Feb 25 '18 at 02:14
@Qmi: What are the new numbers so far? – Tito Piezas III Feb 28 '18 at 02:24
0, 0, 0, 0, 1. Such is the fate of a waiting game. – Qmi Mar 02 '18 at 02:43
And 0, 2. Those 7 days were right. Onto the next 8 days... – Qmi Mar 04 '18 at 06:41
@Qmi: Hopefully the next 8 days will be $1, 0, 2, 0, 0, 1, 0, 2$. – Tito Piezas III Mar 04 '18 at 14:34
Turned out to be $1, 0, 0, 0$. ): – Qmi Mar 08 '18 at 01:31
Columns 6 and 8 could be swapped, no? Speculation, but it would temporarily fix the pattern... In any case, I'd rather wait for the next 4 days now and see what's up – Qmi Mar 08 '18 at 01:46
@Qmi: Yes, maybe they should be swapped. Let's see what happens in the new few days. – Tito Piezas III Mar 08 '18 at 02:29
It was $1, 0, 0, 0, 0, 1, 0, 2$. Everything was right except that one odd number... Also, the new number today is 1.
Thinking maybe I threw a human error and wrote down the old number wrong (since there's no way to check past numbers only by writing them down) or so so I'm gonna keep the layout you have and see what turns up
– Qmi Mar 13 '18 at 21:19
@Qmi: The pattern correctly predicted $17$ out of the last $18$ days, so you could be right. Assuming it really holds, the problem now is to determine the new column "shifts". – Tito Piezas III Mar 14 '18 at 02:36
The first row turned out to be $1, 0, 2, 0, 0, 0, 0, 1$. Same number of 0s, 1s and 2s, just in a different order. Will have to wait another eight days! – Qmi Mar 20 '18 at 23:10
@Qmi: Hopefully, it will also be $1,0,2,0,0,0,0,1$. – Tito Piezas III Mar 21 '18 at 02:09
Sure was. Waiting on next 8 days... – Qmi Mar 28 '18 at 00:08
@Qmi: I have a feeling the next 8 days will be $1, 2, 2, 0, 0, 0, 0, 1$. – Tito Piezas III Mar 28 '18 at 04:20

Is this pattern solvable?

1 Answers1