perplexus.info :: Probability : Hexadecimal Hinder

Hexadecimal Hinder (Posted on 2016-06-19)

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 3^x have the same first digit.

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

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

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

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

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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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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