Lesson Find a sequence of transformations of a given number to get a desired number
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<H2>Find a sequence of transformations of a given number to get a desired number</H2> <H3>Problem 1</H3>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. Find a list which starts with 51 and ends in 129. <B>Solution</B> I solved this problem with the help of Artificial Intelligence. 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 Google AI 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 Google AI 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 I asked Google AI another question: <pre> Find the degree of number 2 which starts 129. </pre> Google AI answered: the degree is 110, and it is first degree of 2 with this property. Then I checked using MS 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 1298074214633706907132624082305024, whose three starting/leading digits are 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, the problem is solved and the desired sequence of operations is presented explicitly. ////////////////////////////////////////// Yes, in my solution I used help from Google AI Overview - I explicitly referred to it, so it is not stealing - it is normal work in contemporary environment. Thanks to Google AI for help ( ! ) And it suggests some art asking right questions and interpreting the AI 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. 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