Quote from: albert0bsd on August 07, 2023, 10:39:20 PM

Quote from: james5000 on August 07, 2023, 08:30:20 PM

even though I'm skipping intervals.

We know that you are skipping those ranges that are some kind of unlikely to appper in random numbers... that is clear.

Let me show you this simulation for 64 bits, with 100 Million of iterations..

Code:

3 times 10 bits in 1, this is 0.000000
3 times 11 bits in 1, this is 0.000000
15 times 12 bits in 1, this is 0.000000
69 times 13 bits in 1, this is 0.000001
241 times 14 bits in 1, this is 0.000002
841 times 15 bits in 1, this is 0.000008
2677 times 16 bits in 1, this is 0.000027
7459 times 17 bits in 1, this is 0.000075
19405 times 18 bits in 1, this is 0.000194
47167 times 19 bits in 1, this is 0.000472
106263 times 20 bits in 1, this is 0.001063
222505 times 21 bits in 1, this is 0.002225
436048 times 22 bits in 1, this is 0.004360
795798 times 23 bits in 1, this is 0.007958
1357354 times 24 bits in 1, this is 0.013574
2178202 times 25 bits in 1, this is 0.021782
3259829 times 26 bits in 1, this is 0.032598
4588566 times 27 bits in 1, this is 0.045886
6064588 times 28 bits in 1, this is 0.060646
7533079 times 29 bits in 1, this is 0.075331
8784231 times 30 bits in 1, this is 0.087842
9628995 times 31 bits in 1, this is 0.096290
9933958 times 32 bits in 1, this is 0.099340
9637041 times 33 bits in 1, this is 0.096370
8782369 times 34 bits in 1, this is 0.087824
7530503 times 35 bits in 1, this is 0.075305
6064379 times 36 bits in 1, this is 0.060644
4589388 times 37 bits in 1, this is 0.045894
3260455 times 38 bits in 1, this is 0.032605
2175369 times 39 bits in 1, this is 0.021754
1356442 times 40 bits in 1, this is 0.013564
794205 times 41 bits in 1, this is 0.007942
435134 times 42 bits in 1, this is 0.004351
222946 times 43 bits in 1, this is 0.002229
106671 times 44 bits in 1, this is 0.001067
46833 times 45 bits in 1, this is 0.000468
19545 times 46 bits in 1, this is 0.000195
7516 times 47 bits in 1, this is 0.000075
2705 times 48 bits in 1, this is 0.000027
842 times 49 bits in 1, this is 0.000008
251 times 50 bits in 1, this is 0.000003
80 times 51 bits in 1, this is 0.000001
23 times 52 bits in 1, this is 0.000000
5 times 53 bits in 1, this is 0.000000
1 times 54 bits in 1, this is 0.000000
1 times 55 bits in 1, this is 0.000000

What that mean? for example if you want a ratio of the 50% of the bits are in 1

Code:

9933958 times 32 bits in 1, this is 0.099340

That is is near of ~10% of the TIME the random numbers had 32 of 64 bits in 1

Now you want to test numbers only with some ratio between 40% to 60% right?

Code:

>>> 64*0.40
25.6
>>> 64*0.60
38.4

For this example of 64 bits you need to test those numbers who have 26 to 38 bits in 1 that is from 32 +/- 6

In this example the values have the next probabilities:

Code:

3259829 times 26 bits in 1, this is 0.032598
4588566 times 27 bits in 1, this is 0.045886
6064588 times 28 bits in 1, this is 0.060646
7533079 times 29 bits in 1, this is 0.075331
8784231 times 30 bits in 1, this is 0.087842
9628995 times 31 bits in 1, this is 0.096290
9933958 times 32 bits in 1, this is 0.099340
9637041 times 33 bits in 1, this is 0.096370
8782369 times 34 bits in 1, this is 0.087824
7530503 times 35 bits in 1, this is 0.075305
6064379 times 36 bits in 1, this is 0.060644
4589388 times 37 bits in 1, this is 0.045894
3260455 times 38 bits in 1, this is 0.032605

That is 0.896575 or near of ~90% of all the KEYS in the whole range FIT in those % of Ratios of 1.

That is the % for a 64 bits range, but i bet that the % is very similar for a 66 bit range and others..

Quote from: james5000 on August 03, 2023, 02:23:39 PM

I guarantee that with a rate of 300 Mk/s we break the 66, 67 and 68 in one week

From you previous quote I going to asume a puzzle per week, not 3 in a week.

Code:

>>>2**65/300000000/60/60/24/52 * 0.0000365333
1.0000006458749602

That is one week and you need 0.0000365333 of the range that is 0.00365333%

Let me ask it again, because the last time you completely ignore my question.

How do you plan to reduce that 90% of the whole range to only 0.00365333%?

For many ranges that you skip, i don't believe any reduction near to the 1% of the whole range...

BTW if some one spot some miscalculation or some incoherent formula please let me know i would be happy of getting wrong.

Repeat, your Idea have a lot of logic, but the calcualtios about your expected time doesn't match...

First of all, thank you very much for the time you've dedicated and the painstaking calculations!

I will conduct some tests. None of the formulas I used to calculate the total intervals provided me with the correct result. I resorted to brute forcing puzzle #30 just to confirm.

Initially, I attempted using the binomial coefficient. It worked well for #10, but for #30, it didn't yield the correct result, which is around 77 million reduced from about 500 million.

If you are aware of a formula to calculate the number of combinations for a fixed-length string with 14 bits set to 0 and 16 bits set to 1, I would be grateful if you could correct my calculations.

Code:

The binomial coefficient, also known as "n choose k" or "combination," is used to calculate the number of different ways to choose k unique elements from a set of n elements, without considering the order in which they are chosen. The formula for the binomial coefficient is given by:

C(n, k) = n! / (k! * (n - k)!)

Where:

n is the size of the total set (in this case, the length of the string)
k is the number of elements you want to choose (in this case, the number of zeros or ones)
To calculate the binomial coefficient for a fixed-length string with 14 zeros and 16 ones, we have n = 30 (14 zeros + 16 ones) and k = 14 (number of zeros). Substituting these values into the formula, we get:

C(30, 14) = 30! / (14! * (30 - 14)!)

Here, 30! represents the factorial of 30 (30 * 29 * 28 * ... * 1), 14! represents the factorial of 14, and (30 - 14)! represents the factorial of 16.

Calculating these factorials and substituting them into the formula, we get the value of the binomial coefficient:

C(30, 14) = 30! / (14! * 16!) ≈ 145,422,675

Therefore, there are approximately 145,422,675 different ways to combine a fixed-length string with 14 zeros and 16 ones.