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## The Liars' Rail

Math brain teasers require computations to solve.

 Puzzle ID: #47469 Fun: (2.5) Difficulty: (2.77) Category: Math Submitted By: dalfamnest

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

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 4-digit 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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 stil Jul 18, 2010 Nicely done, but pseudo-unique; 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.