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 Pandigitally Powerful (Posted on 2010-12-18)
R is a rational number such that 2123456789 degrees Celsius is precisely equal to R degrees Fahrenheit.

(I) Determine the first two digits (reading left to right) in [R].

(II) Determine the last two digits (reading left to right) in [R].

(III) Determine the first digit (reading left to right) following the decimal point in R.

Note: [x] denotes the greatest integer ≤ x.

 See The Solution Submitted by K Sengupta Rating: 5.0000 (1 votes)

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 Getting started, with part (I) (spoiler for that part) | Comment 1 of 3

print 123456789*log(2)/log(10)+log(9/5)/log(10)
37164196.912631546420776442430889649972415415276

Indicates that the required [R] has 37164197 digits when written in decimal, and the first few are

10^.912631546420776442430889649972415415276,

but written without the decimal point. That comes to

81777070058448542420198114358090551277928154067512436....

Of course, we've neglected to add 32, but that comes way at the end, so at some point there's a lack of carry, and we can safely say the first two digits are 81, and in fact, up to the terminal 6 we are safe for however many we show (the 6 itself might really be a 5 if rounding was up to make it a 6, or 7 in the unlikely event that carries percolated up this far).

 Posted by Charlie on 2010-12-18 13:35:43

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