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Hexadecimal Hinder (Posted on 2016-06-19) Difficulty: 3 of 5
x is a randomly chosen hexadecimal real number on the interval (0, (10)16)

Determine the probability of each of the following:

(i) x and 3x have the same first digit.

(ii) x and x3 have the same first digit.

(iii) x3 and 3x have the same first digit.

(iv) x, x3 and 3x have the same first digit.

*** “First digit” denotes the first nonzero digit of the number when expressed in hexadecimal form.

No Solution Yet Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Solution part (iv) solution | Comment 5 of 6 |
Might as well start off with the table produced in the part iii solution, in simplified form:

 x in hex         x^3 and 3^x leading digit       decimal length
.1 to .1428...            1                      .0162450656184296

.285... .32cb             1         no match with x

.6597f... .8              1         no match with x

2.148... 2.188d...        9         no match with x

...

3.a25d...  3.c91b...      3                      0.151337335764465

...


Ellipses indicate none of the ranges' x values matched the functions' leading digits.

The two stretches on the number line total  .1675824013828946.

That covers the numbers above .1 hex.

Below .1 hex, all 3^x begin with a 1, we need to know what fraction of that 1/16  length of number line has a leading hex digit 1 and has x^3 also begin with hex digit 1. This fraction, I'm sure is cuberoot(2)/16, so the portion of the number line (total length) is cuberoot(2)/256 ~=  .004921566601151848.

This brings the total length of number line occupied to  .1725039679840464.

As we're seeking the probability, this is divided by 16 to give  .0107814979990029.



  Posted by Charlie on 2016-06-20 10:37:02
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