Return to my Puzzle pages
Go to
my home page
© Copyright 1999, Jim Loy
You were asked to find the only solution to ABCD x E=DCBA. Two people have sent me 1089x9=9801, which is not a valid solution. The actual solution is 2178 x 4=8712.
This is how I solve it:
Let's list the permutations:
E = 22 33 4 7 9 A = 14 12 2 1 1
Not very many. We can deduce a minimum and a maximum D from A??? x E=D???:
E = 22 33 4 7 9
A = 14 12 2 1 1
min D = 38 46 8 8 x
max D = 39 58 9 9
Or:
E = 2 22 33 333 44 77 A = 1 44 11 222 22 11 D = 3 89 45 678 89 89
Well, E x ...D=...A. So:
E = 2 22 33 333 44 77 A = 1 44 11 222 22 11 D = 3 89 45 678 89 89 A = 6 68 25 814 26 63
Only one of these columns works: 2??8 x 4=8??2. Since there is no carry to the last digit (8), then B is either 0 or 1: 20?8 x 4=8?02 or 21?8 x 4=8?12. 8 x 4 carries 3. So the second digit (from the right) is odd. So, 21?8 x 4=8?12 is the only possibility. ? x 4=...8 (so that a carry of 3 makes it a 1). So, the ? is either 2 or 7. 2 is already taken. So the only possible solution is 2178 x 4=8712. Multiplying it out (we have to do this), we find that it is indeed a solution.
Return to the original puzzle.