### Brain Teasers

# The Liars' Rail

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

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

### Hint

The 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.### Answer

L=2, I=1, A=7, R=8, and S=42178 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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