Solve this alphametic, where each of the capital letters in bold represents a different decimal digit from 0 to 9. None of the numbers contains any leading zero.

(TO)! = (KNOW)*(TITANS)

I set up the equation:
(TO)! = (KNOW) * (TITANS) within a spreadsheet and then defined certain constraints which are discussed below.  At a certain point I could not further develop the logic of those constraints and resorted to 'trial and error'.


What is the extent of ([TO]!)?
Allow both T and K the vaues of 9.
  KNOW * TITANS
 = 9000 * 900000
 = 8100000000
or 10 digits.

Now allow T and K the values of 1.
  KNOW * TITANS
 = 1000 * 100000
 = 100000000
or 9 digits.

Therefore (TO)! must have either 9 or 10 digits.

12! = 9 digits while 13! = 10 digits so TO is either 12 or 13 and so 100000 < TITANS < 200000 and 1000 < KNOW < 9000.  The product of 9000 and 200000 is 1800000000 (10 digits) but way below the value of 13! so TO = 12.

Now S or W must have one of the values of 0 or 5.  If one is 5 the other is 0, 4, 6 or 8.

Since TO! begins with 47 and T = 1 then K must be either 3 or 4 (factor of division and 1 & 2 are discounted).  Also TO! ends with 600 so the TU digits of both numbers need to take into account that value whether that be 3 or all 4.

With that logic in place I resorted to 'trial and error' swapping values of decreasing placevalue in my spreadsheet while attempting to keep the product value under the TO! value and eventually achieving the follow results:  
A I K N O S T W
3 0 4 7  2 6  1  5

T O   K N O W  T I T A N S
1 2   4  7  2  5   1 0 1 3 7 6.

Posted by brianjn on 2009-06-14 21:51:15