The Liars' Rail
Math brain teasers require computations to solve.
The people of Olde Mathville had unique ways of punishing wayward citizens. For example, those convicted of crimes of dishonesty were chained to the Liars' Rail until they solved a number of puzzles.
One such puzzle has been recently discovered!
In the multiplication below, each letter  L, I, A, R, and S  takes the place of a different digit. Find the digits to make the multiplication true.
L I A R
x S

R A I L
or: L I A R x S = R A I L
HintThe product of S x L must be less than 10. Combining that with the need for S x R to end in L limits the possibilities quickly.
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Answer
L=2, I=1, A=7, R=8, and S=4
2178 x 4 = 8712
One possible solution guide:
* SxL<10, as the product is a 4digit number; S can not = 1
... so S,L can be 2,3; 2,4; 3,1; 3,2; 4,1; 4,2
* but SxR ends in L  and if S or R is even, so is L
... so S,L,R can be 2,4,7; 3,1,7; 3,2,4; 4,2,8
* the first three are quickly rejected by checking the multiplication
... S=4, L=2, R=8
* Since 2000 x 4 = 8000, nothing has been carried from multiplying the hundreds digit
... so I=1 or 0
* I=0 is rejected as '3' tens have been carried forward from multiplying 4x8; the 3 must be added to 7 to give 0 in the bottom line, and we cannot have 4 x __ = 7.
... so I=1
... so A=7 (the only way to get a 1 in the tens column of the product)
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Comments
stil
Jul 18, 2010
 Nicely done, but pseudounique; too much like 50 others. 
coinjar876
Jul 21, 2010
 elf+elf=fool
This comment is not meant to be insulting, if you know what I mean you know what i mean. 
c5johnson
Aug 16, 2010
 Excellent puzzle! 
RRAMMOHAN
Oct 23, 2012
 A really tough one and luckily I was able to solve it, with reasoning similar to that of the creator of this problem. Obviously S cannot be 1, and L cannot be 5,6,7,8 or 9. Hence L can only be 0,1,2,3 or 4. By trying out each of these possible values for L and eliminating them one by one, taking different values for S, I got the answer LIAR=2178 and S=4.
I started with L=4 and so took time to arrive at the solution. Still, I had to eliminate any other possible solution by trying out all values for L.
A great puzzle. Thanks for making me think logically. 
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