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Mi, Us and Amicus (Posted on 2009-09-13) Difficulty: 2 of 5
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.

See The Solution Submitted by K Sengupta    
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Solution Analytical solution (spoiler) | Comment 1 of 2
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

  Posted by Steve Herman on 2009-09-13 13:54:02
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