Each of the small letters in bold represents a different base x digit from 0 to x-1 to satisfy this alphametic equation. None of the numbers can contain any leading zero.

                              (x-1)*number = integer

Determine the minimum positive integer value of x such that the above equation has at least one solution. What is the next smallest value of x having this property?

There are 9 letters used: numberitg so x is at least 9.

Since x is the base we can distribute (x-1)*number = number0-number = integer or

integer
+number
number0

r+r ends in 0 so either r is zero or it is half the base and this base must be even.  If r is zero e+e ends in zero so e is half the base and the base must be even.  [either way the base is even]  If r is not zero, r+r has a carry so for e+e to end in r, r must be odd.

n=i+1

n+n gives a carry so n is at least half the base.  I can't be half the base (e or r is) so it is at least one more than half the base and n is at least two more than half the base.

I probably will not be able to derive all that much more, but if nothing else this speeds up a computer search.

Posted by Jer on 2011-06-14 02:03:15