Question 1166184
.
Starting with a positive integer, apply the following operations any number
of times and in any order to produce a list of numbers:
1. double the current term, or
2. delete the last digit of the current term.
An example of such a list is
231, 23, 46, 92, 9, 18.
(a) Find a list which starts with 51 and ends in 129.
(b) Show that every starting number can produce a list ending in 1.
A cycle is a list which eventually returns to the starting number, such as
24, 48, 96, 9, 18, 36, 3, 6, 12, 24.
(c) Show that every number from 1 to 41 occurs in a cycle with at most
13 distinct terms.
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        In this my post,  I will answer question  (a),  ONLY.



To solve this problem,  I asked  Google  AI  OVERVIEW:


<pre>
    Is it true that for every k-digit positive integer n, 
    there is a power of 2 whose first k digits coincide with the number n.
</pre>

Below is the &nbsp;Google &nbsp;AI &nbsp;Overview answer.


<pre>
The statement is true. [1]  

For every k-digit positive integer n, there is a power of two whose first digits coincide with the number n. 
This is a consequence of the fact that log_{10}(2) is an irrational number, which implies that the fractional parts 
of m*log_{10}(2) are uniformly distributed in the interval [0, 1) (by the Kronecker's Approximation Theorem). [1, 2, 3]  


Understanding the condition 

For a positive integer P to be the leading digits of a power of two, say 2^m, it must satisfy the inequality: 
   P times 10^d <= 2^m < (P+1) times 10^d 

for some integer d >= 0. This means 2^m has the same leading digits as P, followed by d more digits. 

Answer: 

The statement is true because any finite sequence of digits can be the leading digits of a power of two. 
This relies on the irrationality of log_{10}(2) and the uniform distribution of the fractional parts of its multiples. [1, 5]  

References

[1] https://www.quora.com/Prove-that-for-any-natural-number-n-there-esists-a-power-of-2-lets-say-2-k-such-that-2-k-in-its-decimal-expression-is-1-followed-by-n-zeros-and-eventually-other-digits-For-example-for-n-2-2-196-works-How-to-prove-it
[2] https://www.antonellaperucca.net/didactics/Powers-of-2.pdf
[3] https://math.stackexchange.com/questions/1370645/using-kroneckers-theorem-can-we-prove-theres-some-power-of-two-yielding-a-numb
[4] https://math.stackexchange.com/questions/328655/proving-prime-p-divides-binompk-for-k-in-1-ldots-p-1
[5] https://www.reddit.com/r/askmath/comments/18o86pr/is_it_true_that_for_any_positive_integer_n_there/


The link to this &nbsp;Google &nbsp;AI &nbsp;response is

https://www.google.com/search?q=Is+this+statement+true%3F+For+every+k-digit+positive+integer+n%2C+there+is+a+power+of+two+whose+first+n+digits+coincide+with+the+number+n.&rlz=1C1CHBF_enUS1071US1071&oq=Is+this+statement+true%3F+For+every+k-digit+positive+integer+n%2C+there+is+a+power+of+two++whose+first+n+digits+coincide+with+the+number+n.&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIHCAEQIRiPAjIHCAIQIRiPAjIHCAMQIRiPAtIBCTE3OTZqMGoxNagCCLACAfEFNet2ZYDKsLI&sourceid=chrome&ie=UTF-8

of Dec.4, 2025.
</pre>

Then &nbsp;I &nbsp;asked &nbsp;Google &nbsp;AI &nbsp;another question:


<pre>
    Find the degree of number 2 which starts 129.
</pre>


Google &nbsp;AI &nbsp;answered:  &nbsp;the degree is &nbsp;110, &nbsp;and it is first degree of &nbsp;2 &nbsp;with this property.


Then I checked using &nbsp;MS &nbsp;Excel in my computer


<pre>
    {{{2^110}}} = 1.29807E+33.
</pre>

Now I can present the desired sequence of operations


    51 -> 102 -> 10 -> 1 -> 2 -> 4 -> 8 -> 16 -> 32 -> 64 -> 128 -> . . . . -> {{{2^110}}} = 1.29807E+33,


which is a 34-digit integer number &nbsp;1298074214633706907132624082305024, &nbsp;whose three starting/leading digits are &nbsp;129.


Notice that I do not state that this sequence is the shortest possible.


I only state that this explicit sequence produces a desired number.
</pre>

Thus part &nbsp;(a) &nbsp;is solved and the desired sequence of operations is presented explicitly.



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Yes, &nbsp;in my solution &nbsp;I &nbsp;used help from &nbsp;Google &nbsp;AI &nbsp;Overview - I explicitly referred to it, 
so it is not stealing - it is normal work in contemporary environment. &nbsp;&nbsp;Thanks to &nbsp;Google &nbsp;AI &nbsp;for help &nbsp;(&nbsp;!&nbsp;)


And it suggests some art asking right questions and interpreting the &nbsp;AI &nbsp;answers properly,
as well as organizing pieces and thoughts into a logically coherent text, 
which opens new knowledge and provides new material for teaching and learning.