Hello, blame_canada!
The reasoning is simpler than you think . . .
If you multiply a 4-digit number ABCD by 4, the digits get reversed.
That is: ABCD X 4 = DCBA. .Find ABCD.
Number the columns for reference.
I will use ? for a possible "carry".
Also, → will mean "ends with" . . . as in: 4 x 7 → 8 Code:
1 2 3 4
A B C D
x 4
- - - -
D C B A
In column-1, we have: 4 x A + ? = D
. . Hence, A = 1 or 2.
But if A = 1, column-4 says: 4 x D → 1 . . . which is impossible.
. . Therefore: A = 2
So we have: Code:
1 2 3 4
2 B C D
x 4
- - - -
D C B 2
In column-1, we have: 4 x 2 + ? = D
. . Hence: D = 8 or 9.
In column-4, we have: 4 x D → 2
. . Therefore: D = 8 (and there is no "carry" from column-2)
So we have: Code:
1 2 3 4
2 B C 8
x 4
- - - -
8 C B 2
If there is no "carry" from column-2, then B = 0 or 1.
If B = 0, then column-3 has: 4 x C + 3 → 0 ... which is impossible.
. . Therefore: B = 1
So we have: Code:
1 2 3 4
2 1 C 8
x 4
- - - -
8 C 1 2
In column-3, we have: 4 x C + 3 → 1
. . Hence: C = 2 or 7
Since 2 is already used, C = 7 Code:
Solution
2 1 7 8
x 4
- - - -
8 7 1 2