Collatz Series 2
If you don't know what the Collatz series is, see the previous post.
We learned lots of the initial numbers give a series that contain 40.
Give two initial numbers between 100 and 200 that are ODD and don't contain a 40 in their series.
We learned lots of the initial numbers give a series that contain 40.
Give two initial numbers between 100 and 200 that are ODD and don't contain a 40 in their series.
Labels: mathschallenge
posted by Ragknot at
4:44 PM
7 Comments:
113 and 151 satistfy these conditions
Cam
Collatz series 2
Once again, I followed an iterative approach, starting from 32. Left hand column= N*2, Right Hand column=(N-1)/3 ,"X" indicates (N-1) Mod 3 != 0 OR (N-1)/3 Mod 2 = 0
Why start from 32 ?
I start from 32, because the series 16,8,4,2,1 can be generated with either a 5 or 32. If it contains 5, then the series will also contain 40. In this manner, I "dodge" 40.
I terminated each series below, once the LH column was > 2500, as it was not likely to converge quickly to a number between 100-200. Smallest remaining RH number is used for next series, as it will be most likely candidate to yield a solution.
32,
64, X
128, 21
256, X
512, 85
1024, X
2048, 341
4096, X
21
42, X
84, X
168, X
336, X
672, X
1344, X
2688, X
85
170, X
340, X
680, 113
1360, X
2720, 453
***113 is a solution***
341
682, X
1364, 227
2728, X
227
454 X
908 151
1816 X
3632 605
***151 is a solution***
ANSWER:
113 and 151 are odd numbers between 100 and 200 that do not contain 40 in their collatz series.
Cam
Cam,
I don't quite understand your logic but it seems to work since you found the correct numbers.
I am still facinated with things I find in the Collatz series.
I notice something I think of as twins and triplets. When consective number yield almost identical series. For example 124,125, and 126. They each has 109 steps and all but 2 or 4 steps are identical.
Initial numbers 188 and 189 are almost idential twins of 107 steps.
There are several other twins and triplets less than 200 (I suppose millons above 200)
Another very odd thing is that every single initial number between 1 and 200 that has more than 100 steps has a common "high point". That's 39 initial numbers with steps varying from 101 to 125 with the same high. And of course they would identical after than high point.
Can you figure out the high point?
Had a quick look at the first 10 steps of #124,125,126,188,189. (only need to look at 1st ten since we are told # of steps for each #, and that the steps may vary from 101 to 125) hence series after high point is 100 more steps. (series starting with high point is shortest series with 101 steps)
124,125,188,189 start sharing numbers after 142.
Calculate series for 142, and count the number of steps. 103 steps after 142. So need to pick 3rd # after 142 to find high point (103-3=100 steps after high point).
142,71,214,107
107 is the high point.
Cam
Cam,
I think you missed something.
Taking 142 as the initial number, the series has 103 steps.
They are...
142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232,(the high point)
4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, 20, 10, 5, 16, 8, 4, 2, 1
and the high point is 9232.
In fact each series with more than 100 steps reaches 9232 before begins it's decent to one.
Oh I see, I interpreted "high point" differently, but I see now that by high point you, in fact, meant largest number in the series.
Anyhow, all series eith greater than 100 steps contain the series with 107 (for initial numbers between 1-200).
It's trivial to see that 9232 is the largest number in the series started by 107.
Cam
Oh I see, I interpreted "high point" differently, but I see now that by high point you, in fact, meant largest number in the series.
Anyhow, all series eith greater than 100 steps contain the series with 107 (for initial numbers between 1-200).
It's trivial to see that 9232 is the largest number in the series started by 107.
Cam
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