[seqfan] Re: 32/25 - a Northern Summer puzzle

[wnmyers wnmyers]

Clearly the asymptote must be more than 5 because you have to divide by 5 at some point and get a number greater than one. The 6-step route gives an upper bound of 6.4, so we can ignore any multiplication or division by 7 or more. That means all bounds after that would be of the form 2^a*3^b*5^c.

steps  bound
4      8=2^3
6      6.4=2^5*5^-1
12     5.689=2^8*3^-2*5^-1
26     5.619=2^12*3^-6
76     5.605=2^-12*3^15*5^-4

I'm pretty sure the bound can't go any lower than 5.346=2^28*3^-22*5^4.

> On June 27, 2022 at 10:47 AM Peter Munn <techsubs at pearceneptune.co.uk> wrote:
> 
> 
> Amongst the data for a little research I have been doing, I saw the
> potential for a little puzzle.
> 
> How many operations, multiplying or dividing by an integer, does it take
> to get from 1 to 32/25? Clearly 2, but what if we restrict the range of
> the intermediate values? Let's keep the lower bound as 1. If we are not
> allowed as high as 32, we can do it in 4 steps: 1 -> 8 -> 8/5 -> 32/5 ->
> 32/25. If we are not allowed as high as 8, we can do it in 6 (simple
> puzzle, to work out the steps).
> 
> At each stage, we can find the length of the new shortest route, 2k, and
> the lowest upper bound that permits a route of 2k, and forbid the next
> stage to go as high. So how does the sequence 2, 4, 6, ... (the values of
> 2k) continue? I am confident it is infinite, which means there is an
> asymptote for the upper bound of the restriction range. What is this
> asymptote?
> 
> Best Regards,
> 
> Peter Munn
> 
> 
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