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eight*two = sixteen (Posted on 2010-05-27) Difficulty: 3 of 5
Solve this alphametic multiplication problem, where each of the small letters denotes a different base 14 digit from 0 to D, and each of the asterisks denotes a base 14 digit from 0 to D whether same or different. None of the numbers can contain any leading zero.

e i g h t
t w o
---------------------
* * * * *
* * * * *
* * * * *
---------------------
s i x t e e n

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution computer solution | Comment 1 of 2

000100        DIM used(13)
000200
000300        FOR t = 1 TO 13
000400         IF used(t) = 0 THEN
000500     :    used(t) = 1
000600     :  FOR o = 1 TO 13
000700     :   IF used(o) = 0 THEN
000800     :    used(o) = 1
000900     :    n = (t * o) @ 14
001000     :    IF used(n) = 0 THEN
001100     :      used(n) = 1
001200     :
001300     :  FOR h = 0 TO 13
001400     :   IF used(h) = 0 THEN
001500     :    used(h) = 1
001600     :  FOR w = 0 TO 13
001700     :   IF used(w) = 0 THEN
001800     :    used(w) = 1
001900     :    en = ((h * 14 + t) * (w * 14 + o)) @ (14 * 14)
002000     :    e = en \ 14
002100     :    IF used(e) = 0 AND e <> 0 THEN
002200     :      used(e) = 1
002300     :                     :
002400     :  FOR g = 0 TO 13
002500     :   IF used(g) = 0 THEN
002600     :    used(g) = 1
002700     :    een = ((g * 14 * 14 + h * 14 + t) * (t * 14 * 14 + w * 14 + o)) @ (14 * 14 * 14)
002800     :    IF een \ (14 * 14) = e THEN
002900     :
003000     :  FOR i = 0 TO 13
003100     :   IF used(i) = 0 THEN
003200     :    used(i) = 1
003300     :    teen = ((i * 14 * 14 * 14 + g * 14 * 14 + h * 14 + t) * (t * 14 * 14 + w * 14 + o)) @ (14 * 14 * 14*14)
003400     :    IF teen \ (14 * 14 * 14) = t THEN
003500     :
003505     :    sixteen = ((e*14*14*14*14 + i * 14 * 14 * 14 + g * 14 * 14 + h * 14 + t) * (t * 14 * 14 + w * 14 + o))
003510     :    s=sixteen \ (14^6):i2=(sixteen\(14^5))@14:x=(sixteen\(14^4))@14
003515     :    if s<14 then
003520     :    if s>0  and used(s)=0 and i2=i and x<14 and used(x)=0 and x<>s then
003530     :
003535     :        eight=e*14^4+i*14^3+g*14^2+h*14+t:eightT=eight*t:eightW=eight*w:eightO=eight*o
003540     :       
003541     :
003600     :        PRINT e;i; g; h; t: PRINT t; w; o
003610     :        print 1+int(log(eightO)/log(14));
003611     :        print 1+int(log(eightW)/log(14));
003612     :        print 1+int(log(eightT)/log(14));:print
003613     :        PRINT s;i;x;t; e; e; n: print
003620     :
003710     :    end if
003715     :    end if
003800     :    END IF
003900     :    used(i) = 0
004000     :   END IF
004100     :  NEXT
004200     :
004300     :
004400     :    END IF
004500     :    used(g) = 0
004600     :   END IF
004700     :  NEXT
004800     :
004900     :
005000     :      used(e) = 0
005100     :    END IF
005200     :    used(w) = 0
005300     :   END IF
005400     :  NEXT
005500     :    used(h) = 0
005600     :   END IF
005700     :  NEXT
005800     :
005900     :
006000     :      used(n) = 0
006100     :    END IF
006200     :    used(o) = 0
006300     :   END IF
006400     :  NEXT
006500     :    used(t) = 0
006600     :   END IF
006700     :  NEXT

 

finds

 1  5  2  9  7
 7  8  6
 5  5  5
 10  5  12  7  1  1  0

where the 1st, 2nd and 4th lines represent the digits of eight, two and sixteen respectively, and the 5 5 5 confirms that the partial products all contain 5 base-14 digits. Decimal equivalents of the digits A - D are used rather than the letters.

The 15297 * 786 = A5C7110 solution is the only one found, even disregarding the lengths of the partial products.


  Posted by Charlie on 2010-05-28 12:55:30
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