In each of the three additions below, each letter represents a different digit; however, each letter does not necessarily represent the same digit in one addition as it does in another addition.
G A L E E L S A N E A L
+ N E A L + G A L E + E L S A
------------- ------------- -------------
E L S A N E A L G A L E
Which addition has the smallest sum?
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Submitted by pcbouhid
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Rating: 3.0000 (1 votes)
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Solution:
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The first addition.
The "E" in addition I, the "A" in addition II, and the "L" in addition III, all behave in the same way.
In I, either:
(E + L) = A and (E + A) = 10 + L
or
(E + A) = L and (E + L) = 10 + A
In II, either:
(A + E) = L and (A + L) = 10 + E
or
(A + L) = E and (A + E) = 10 + L
In III, either:
(L + A) = E and (L + E) = 10 + A
or
(L + E) = A and (L + A) = 10 + E
Only the digit 5 behaves in this way (5 + 3 = 8... 5 + 8 = 13; 5 + 4 = 9... 5 + 9 = 14;...)
So one gets:
I II III
G A L 5 E L S 5 N E A 5
N 5 A L G 5 L E E 5 S A
------- ------- -------
5 L S A N E 5 L G A 5 EA substitution for L in (I), for E in (II) and for A in (III) yields a value for A in (I), for L in (II), and for E in (III). Trial and error produces (in the eliminated cases, 1 is carried from the second column to the third column, making the substitution invalid):
I (a) G 6 1 5 (b) G 7 2 5 (c) G 1 6 5
N 5 6 1 N 5 7 2 N 5 1 6
------- ------- -------
5 1 7 6 5 2 9 7 5 6 8 1
II (a) 6 1 S 5 (b) 7 2 S 5 (c) 8 3 S 5 (d) 9 4 S 5
G 5 1 6 G 5 2 7 G 5 3 8 G 5 4 9
------- ------- ------- -------
N 6 5 1 N 7 5 2 N 8 5 3 N 9 5 4
III (a) N 6 1 5 (b) N 7 2 5 (c) N 8 3 5 (d) N 9 4 5
6 5 S 1 7 5 S 2 8 5 S 3 9 5 S 4
------- ------- ------- -------
G 1 5 6 G 7 S 2 G 3 5 8 G 4 5 9
Inspection of the partial additions reveals that (I) has the smallest sum.
The additions can be completed further, and we can found 4 different solutions for (I), 1 solution for (II), and 2 solutions for (III), but in all of them, (I) has always the smallest sum, which is what was asked.
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