From MI to MU
This is an interesting little problem I ran into back when I was in high school:
Change the string of letters 'MI' to the string 'MU', using the following rules.
- If a string ends with 'I', 'U' can be added ('MI'
can be changed to 'MIU') - Three 'I's in succession can be changed to a 'U' ('MUIII' can be changed to 'MUU')
- The string 'Mx' (where x is any sequence of letters) can be changed to 'Mxx' ('MUIU' can be changed to 'MUIUUIU')
- Two 'U's in succession can be deleted ('MIUU' can be changed to 'MI')
Solution
The annoying thing about this problem is this: It can't be solved. I had a friend who spent several hours attempting to solve it before bringing it to me; he was not happy to find out
the problem was unsolvable. So how do I know it is unsolvable?
Only two of the rules alter the number of I's: the second rule decreases the number of of I's by 3, and the third rule doubles the number of I's.
If the number of I's is not a multiple of three, neither subtracting 3 or doubling will result in a multiple of three. Since 0 is a multiple of three, we can never get rid of all the I's.
Isn't that slick?
"Slick Math" is written by Douglas Twitchell, and hosted at The Problem Site.
Contents copyright 2008 by Douglas Twitchell. Contents of this page may not be reproduced without permission of the author.
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