If ABC+DEF+GHI=JJJ, each letter stands for a different digit, and no number starts with zero, what is J?

Since A, D, and G represent three different digits, none of which can be zero, J must be 6, 7, 8, or 9.

Since C, F, and I represent three diferent digits, their sum cannot exceed 24; then, in order that J be 6, 7, 8, or 9, their sum cannot exceed 19.

If any two columns sum to 6, 7, 8, or 9, the third column must sum to 6, 7, 8, or 9. but since A through I represent nine different digits, this situation is impossible. So at most one column sums to 6, 7, 8, or 9.

Thus, from the above:

a) If A+D+G sums to 6, then C+F+I must sum to 16, 7, or 17.
b) If A+D+G sums to 7, then C+F+I must sum to 17, 8, or 18.
c) If A+D+G sums to 8, then C+F+i must sum to 18, 9, or 19.
d) If A+D+G sums to 9, then C+F+I must sum to 19.

The sum for B+E+H can be deduced from (a), (b), (c), and (d), and we have the following:

1) A+D+G = 6, B+E+H = 5,   C+F+I = 16, J = 6.
2) A+D+G = 6, B+E+H = 17, C+F+I =  7,  J = 7.
3) A+D+G = 6, B+E+H = 16, C+F+I = 17, J = 7.
4) A+D+G = 7, B+E+H =  6,  C+F+I = 17, J = 7.
5) A+D+G = 7, B+E+H = 18, C+F+I =  8,  J = 8.
6) A+D+G = 7, B+E+H = 17, C+F+I = 18, J = 8.
7) A+D+G = 8, B+E+H =  7,  C+F+I = 18, J = 8.
8) A+D+G = 8, B+E+H = 19, C+F+I =  9,  J = 9.
9) A+D+G = 8, B+E+H = 18, C+F+I = 19, J = 9.
0) A+D+G = 9, B+E+H =  8,  C+F+I = 19, J = 9.

Only for the cases (8) and (0) does the sum of the terms in the four columns total 45, as they should. So J must represent 9.

Continuing:

(8) A+D+G = 1+3+4, B+E+H = 5+6+8, C+F+I = 0+2+7.
                 = 1+2+5,             = 4+7+8,           = 0+3+6.

(0) A+D+G = 2+3+4, B+E+H = 0+1+7, C+F+I = 5+6+8.
                 = 1+3+5,             = 0+2+6,               4+7+8.
                 = 1+2+6,             = 0+3+5,            = 4+7+8.

150......140....205....104.....104
362......273....316....327.....237
487......586....478....568.....658
------------------------------------
999.....999.....999....999.....999

Basic solutions. All the others can be obtained the permuting digits in the corresponding column.

 

              

 

 

Edited on November 10, 2005, 9:38 am

Edited on November 10, 2005, 9:43 am

Edited on November 10, 2005, 9:51 am

Edited on November 10, 2005, 9:55 am

Edited on November 10, 2005, 10:46 am

Posted by pcbouhid on 2005-11-10 09:33:19