Number patterns

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Mona Pereth
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10 Jul 2019, 9:04 pm

Here, in another thread:

Mona Pereth wrote:

hurley4456 wrote:

synchromystic wrote:

hurley4456 wrote:

Can you complete the following pattern (next row in the sequence)....?

1
11
21
1211
111221
.................

312211
13112221
1113213211
31131211131221
13211311123213112211
1113122113311213121113212221
311311222123211211131112311312113211

Very good!

For those interested in the recurcive pattern, you use the following rules:
- The successive row is derived from preceeding row (a given)
- By progessing from left to right, consider each number set as an individual element ( i.e. Row 5: 111, 22, and 1 are individual sets
- Count the number of elements per set and arrange left to right numerically ( i.e. Row 5: There exists three (3) 1's, two (2) 2's, and one (1) 1.
(312211) = row 6

I might have been able to see this pattern had I been shown maybe three or four more rows.

hurley4456 wrote:

I have another one, but this is derived from vedic mathematics, and I spent my some time working on it....Note: This is a visual pattern based on the shifting on numerical elements. Also, row N is supposed to be shifted to the right N-1 times (hint-space bar). So, if you write it out again by shifting row 2 one time, row 3 two times and so forth, it may help. This is is how you convert a square into an inverted triangle... You can explain verbally or visually based on preferree method..Consider the following term written as ABCD^(2)

3568^(2) =

09253664
306096
3680
48

The way you can preserve visual patterns in text, here on this message board, is by using the "code" tag, which among other things uses a monospaced font and preserves indentation. For example:

Code:

(Quote this message to see how to use the "code" tag.)

Anyhow, could you provide a few more examples of the pattern you are trying to show here?

Then hurley4456 posted the following additional examples of the pattern:

hurley4456 wrote:

Code:

For N = 1
6^(2) =

36

Code:

For N = 2
26^(2) =

0436
  36

Code:

For N = 3
894^(2) =

648104
 14472
    64

Code:

For N = 5
12345^(2) =

0104091625
  04122440
    061630
      0820
        10

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Last edited by Mona Pereth on 10 Jul 2019, 9:13 pm, edited 1 time in total.

Mona Pereth
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10 Jul 2019, 9:12 pm

Here, I see that hurley4456 deleted the above-quoted additional examples and posted the following instead:

hurley4456 wrote:

Code:

 N = 1 
5^(2) = 

25

Code:

 N = 2 
13^(2) = 

0109
 06

Code:

Code:

 N = 5 
12345^(2) = 

0103091625
 04122440
  061630
   0820
    10

Hurley, are the previously-posted examples incorrect, or do you just think the newer ones are clearer?

Anyhow, in all these examples, I see that the first row consists of a sequence of two digit squares of the digits of the original number. I still can't figure out how the subsequent rows are derived. I'm not asking you to tell us yet, but maybe a few more examples might be helpful.

hurley4456
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Joined: 6 Jul 2019
Gender: Male
Posts: 16
Location: Arlington, TX

10 Jul 2019, 10:18 pm

Mona Pereth wrote:

Here, I see that hurley4456 deleted the above-quoted additional examples and posted the following instead:

hurley4456 wrote:

Code:

 N = 1 
5^(2) = 

25

Code:

 N = 2 
13^(2) = 

0109
 06

Code:

Code:

 N = 5 
12345^(2) = 

0103091625
 04122440
  061630
   0820
    10

Try the following...It will be easier to start with N = 2 or 3 where N = # of rows.

Code:

N= 2
49^(2) = 

1681
 72
............
2,401

Code:

369^(2) = 

093681
 3608
  1
  54
...........
136,161

Code:

123^(2) = 

010409
 0412
  06
...........
15,129

synchromystic
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Joined: 3 Jul 2019
Age: 30
Gender: Female
Posts: 48

11 Jul 2019, 12:24 pm

I did it!!

I'll take a picture of my notes when I get home as that might be easier than trying to put it into words.
This was super fun, thank you hurley!

synchromystic
Tufted Titmouse

Joined: 3 Jul 2019
Age: 30
Gender: Female
Posts: 48

11 Jul 2019, 3:05 pm

synchromystic wrote:

I did it!!

I'll take a picture of my notes when I get home as that might be easier than trying to put it into words.
This was super fun, thank you hurley!

I just realised I haven't actually solved it completely lol :lol:

I have worked out how the numbers in the triangle represent/can be used to calculate the square number, BUT I haven't worked out how the numbers in the triangle are calculated beyond the first row.

Agh I got ahead of myself lol.

My notes showing how the triangle can be used to calculate the square.

hurley4456
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Joined: 6 Jul 2019
Gender: Male
Posts: 16
Location: Arlington, TX

12 Jul 2019, 10:05 am

synchromystic wrote:

I did it!!

I'll take a picture of my notes when I get home as that might be easier than trying to put it into words.
This was super fun, thank you hurley!

I just realised I haven't actually solved it completely lol :lol:

Very interesting solution! I liked the way you color coded it...I do the same when creating plots in excel. You are on the right track by figuring out that the solution is derived using the tenths placement (zeroes).

The key is to figure out how each integer (substituted by A,B,C, and D) is shifted via two pair couples. So, I suggest you write out ABCD^(2) and determine the possible ways a pair (i.e AD, BD) can be shifted and then extract an ordered pattern. Also, find the mathematical operations (multiplucation, addition, exc) performed on the pairs. I hope this clarifies everything.

hurley4456
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Joined: 6 Jul 2019
Gender: Male
Posts: 16
Location: Arlington, TX

13 Jul 2019, 3:01 pm

Use this to complete the pattern.

I realized it's for N = 5, but the same rule applies. Try to complete the orientation of all the remaining nodes. When they intersect, that means that the two digits are combined using a math operation (i.e. multiplication, exc). Once you figure out the operation, it should be no problem.

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