The Orbit of Mercury

Date: 9/6/95 at 18:35:50
From: Anonymous
Subject: math word problem

The number of days it takes Mercury to revolve around the sun is a 
palindrome.  Each digit is a cube greater than one. How many days 
does it take Mercury to complete its orbit?

This is a question from my niece who is in grade 7 in a school in 
Saint John.

I would like an answer as it has taken 3 adults from 4 until now (7)
to give in and use the computer to find a correct answer.  

Thank you very much for your time.

Date: 9/13/95 at 14:20:3
From: Doctor Ethan
Subject: Re: Planets and Palindromes

Hey,

Well, the problem is that there are a lot of numbers that satisfy the
criteria given.  But let us get started and then see where we get stopped.

First, what digits are perfect cubes?  Well 1 is, but none of the rest 
except for 8 are perfect cubes. 

So we know that this number must only have ones and eights in it, so let us 
start listing palindromes with only ones and eights.

So we have

1,  8,  11,  88,  111,  181, 818,  1111, 1881, 8118, 8888,  ........

This gives us a lot to choose from, so we can only guess from here.

Knowing that it takes earth 365 days to revolve around the sun, and that 
Mercury is a lot closer to the Sun than the Earth, reasonable guesses seem 
to be 88 and 111.

According to 

   The Nine Planets
   http://www.seds.org/billa/tnp/ 

the orbit of Mercury is 87.97 days, which is very close to 88 days.  So 
that must be it.   Hope that helps.

- Doctor Ethan,  The Geometry Forum