Substitute each of the capital letters by a different base six digit from 0 to 5 to satisfy the following set of alphametic relationships. None of the numbers can contain any leading zero.
- The base six number ICUS is a perfect square, and:
- A is a perfect square, and:
- The base six number MI is a perfect fifth power, and:
- The base six number US is a perfect square.
Determine the value assumed by the base six number
AMICUS.
MI must be 32 (base 10) = 52 (base 6)
A must be 1 or 4
US must be 25 (base 10) = 41 (base 6)
or 16 (base 10) = 24 (base 6)
or 9 (base 10) = 13 (base 6)
But US (base 6) cannot be 41 (because A is 1 or 4)
and cannot be 24 (because MI = 52)
so US (base 6) = 13
So A = 4
So C = 0 (the only remaining digit)
AMICUS = 452013
And, as a check, 2013 (base 6) = 441 (base 10) = 21*21
Analytical solution (spoiler)
