Comments on: Freeman Dyson and 1/19 http://nealabq.com/blog/2009/04/11/freeman-dyson-and-119/ Probing dark corners while dodging the grues Thu, 22 Jul 2010 01:02:34 +0000 hourly 1 http://wordpress.org/?v=3.0 By: SoldAtTheTop http://nealabq.com/blog/2009/04/11/freeman-dyson-and-119/comment-page-1/#comment-78 SoldAtTheTop Tue, 14 Apr 2009 17:30:58 +0000 http://nealabq.com/blog/?p=1180#comment-78 WOW AGAIN!... Ill have to start taking a closer look at numbers... I never realized how interesting all these natural patterns are. WOW AGAIN!… Ill have to start taking a closer look at numbers… I never realized how interesting all these natural patterns are.

]]> By: Neal http://nealabq.com/blog/2009/04/11/freeman-dyson-and-119/comment-page-1/#comment-77 Neal Tue, 14 Apr 2009 05:50:41 +0000 http://nealabq.com/blog/?p=1180#comment-77 Hey SATT, nice to see you! My son likes calculating stuff out and looking for patterns. He's done it since he was 2. Squares, cubes, primes, Pascal triangle stuff, all kinds of integer sequences. And you know how N/7 works: 1/7 0.142857... 2/7 0.285714... 3/7 0.428571... 4/7 0.571428... 5/7 0.714285... 9/7 0.857142... Same 6 digits (repeated forever), just started at different places. And 14*2=28, 28*2=56, and you add to 56 to make 57 because 57*2=114 which is 3 digits so you have to take the 1 off and carry it back to the 56. Or you can start doubling at 42 instead of 14 -- the even/odd patterns zip together. So once my son saw how the 7ths worked he's looked for the same thing elsewhere. The 19ths are almost the same. Hey SATT, nice to see you!

My son likes calculating stuff out and looking for patterns. He’s done it since he was 2. Squares, cubes, primes, Pascal triangle stuff, all kinds of integer sequences.

And you know how N/7 works:

1/7 0.142857…
2/7 0.285714…
3/7 0.428571…
4/7 0.571428…
5/7 0.714285…
9/7 0.857142…

Same 6 digits (repeated forever), just started at different places. And 14*2=28, 28*2=56, and you add to 56 to make 57 because 57*2=114 which is 3 digits so you have to take the 1 off and carry it back to the 56. Or you can start doubling at 42 instead of 14 — the even/odd patterns zip together.

So once my son saw how the 7ths worked he’s looked for the same thing elsewhere. The 19ths are almost the same.

]]> By: SoldAtTheTop http://nealabq.com/blog/2009/04/11/freeman-dyson-and-119/comment-page-1/#comment-76 SoldAtTheTop Tue, 14 Apr 2009 03:17:39 +0000 http://nealabq.com/blog/?p=1180#comment-76 WOW! That is really cool... How do you think he figured that n/19 pattern? Did he already know the pattern or did he figure a clever way to solve the problem and then seek the pattern out? WOW! That is really cool… How do you think he figured that n/19 pattern?

Did he already know the pattern or did he figure a clever way to solve the problem and then seek the pattern out?

]]> By: Neal http://nealabq.com/blog/2009/04/11/freeman-dyson-and-119/comment-page-1/#comment-75 Neal Mon, 13 Apr 2009 01:35:18 +0000 http://nealabq.com/blog/?p=1180#comment-75 There is another way to generate this sequence, but left-to-right instead of right-to-left like we've done so far. It uses powers of 5 instead of powers of 2, and it shifts 2 by digits (not just 1) for each new term. Here's how it looks: <pre> 1 x05 xxx25 xxxx125 xxxxxx625 xxxxxxx3125 xxxxxxxx15625 xxxxxxxxxx78125 xxxxxxxxxxx390625 xxxxxxxxxxxx1953125 xxxxxxxxxxxxxx9765625 xxxxxxxxxxxxxxx48828125 xxxxxxxxxxxxxxxx244140625 xxxxxxxxxxxxxxxxx1220703125 xxxxxxxxxxxxxxxxxxx6103515625 xxxxxxxxxxxxxxxxxxxx30517578125 ------------------------------- 10526315789473684210xxxxxxxxxxx </pre> You can also do this 2 digits at a time but you have to predict how much carrying you're going to get. It's not hard though. And if you do generate this 2 digits at a time, you can start anywhere. You can start with 10 or 05 or 52 or 26. You don't have to worry about even or odd alignment -- it works both ways. Since you can generate the 1/19 sequence right-to-left with powers of 2, or left-to-right with powers of 5, what about 1/49? The post mentions that you can generate the 1/49 sequence (42 digits before it repeats) right-to-left with powers of 5. So can you also generate 1/49 left-to-right with powers of 2? Yes. 1/49 is 0.020408163265306122448979591836734693877551... You can travel left-to-right through this series 2 digits at a time by doubling the previous 2 digits mod 100. And if the result is 50 or above you add one (to carry from the next double). So from 02 you get 04, 08, 16, 32, and then 65 (you add 1 to 64 because it 64 >= 50). From 65 you get 130 which is 30 when you mod 100. What's really cool about this sequence is that you can start doubling from anywhere, even though this generates 2 digits at a time. You can start with 02 or with 20. Or with 08 or 81 or 16. It's like we have two sequences of 2-digit numbers that zip together. The even-aligned sequence (02 04 08 16 32 65 30 etc) and the odd-aligned sequence (20 40 81 63 26 53 06 12 24 etc). Both follow the same doubling rule, and both are the same when viewed as a sequence of 1-digit numbers. There is another way to generate this sequence, but left-to-right instead of right-to-left like we’ve done so far. It uses powers of 5 instead of powers of 2, and it shifts 2 by digits (not just 1) for each new term. Here’s how it looks:

1
x05
xxx25
xxxx125
xxxxxx625
xxxxxxx3125
xxxxxxxx15625
xxxxxxxxxx78125
xxxxxxxxxxx390625
xxxxxxxxxxxx1953125
xxxxxxxxxxxxxx9765625
xxxxxxxxxxxxxxx48828125
xxxxxxxxxxxxxxxx244140625
xxxxxxxxxxxxxxxxx1220703125
xxxxxxxxxxxxxxxxxxx6103515625
xxxxxxxxxxxxxxxxxxxx30517578125
-------------------------------
10526315789473684210xxxxxxxxxxx

You can also do this 2 digits at a time but you have to predict how much carrying you’re going to get. It’s not hard though.

And if you do generate this 2 digits at a time, you can start anywhere. You can start with 10 or 05 or 52 or 26. You don’t have to worry about even or odd alignment — it works both ways.

Since you can generate the 1/19 sequence right-to-left with powers of 2, or left-to-right with powers of 5, what about 1/49? The post mentions that you can generate the 1/49 sequence (42 digits before it repeats) right-to-left with powers of 5. So can you also generate 1/49 left-to-right with powers of 2?

Yes. 1/49 is 0.020408163265306122448979591836734693877551… You can travel left-to-right through this series 2 digits at a time by doubling the previous 2 digits mod 100. And if the result is 50 or above you add one (to carry from the next double). So from 02 you get 04, 08, 16, 32, and then 65 (you add 1 to 64 because it 64 >= 50). From 65 you get 130 which is 30 when you mod 100.

What’s really cool about this sequence is that you can start doubling from anywhere, even though this generates 2 digits at a time. You can start with 02 or with 20. Or with 08 or 81 or 16. It’s like we have two sequences of 2-digit numbers that zip together. The even-aligned sequence (02 04 08 16 32 65 30 etc) and the odd-aligned sequence (20 40 81 63 26 53 06 12 24 etc). Both follow the same doubling rule, and both are the same when viewed as a sequence of 1-digit numbers.

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