# How to solve this alphametic (verbal arithmetic)?

I know I can get the answer for this puzzle but I'm struggling to see how to solve it.

Every letter represents a different number (0-9):

    PLAYS
+   WELL
=======
BETTER


So far I know that:

• B = 1 (has to be)
• P = 9 (because P + 1 ≥ 10)
• E = 0 (9 + 1 = 10)
• L + W ≥ 10
• A + 1 = T

How do I continue from here? I can't find any more hints:

    1 1
9LAYS
+   W0LL
=======
10TT0R

-
Why couldn't $S+L \ge 10$ occur, so that $Y+L = 9$? –  Lord_Farin Oct 15 '12 at 17:22
Yes, I meant that Y + L (+ 1) ≥ 10. But I'm not sure if it's helpful because you can't know if Y + L ≥ 10 or S + L ≥ 10 and Y + L = 9. –  tempy Oct 15 '12 at 17:24
You mean $9 \le Y+L \lt 10$ You also know it is no greater. –  Ross Millikan Oct 15 '12 at 17:38
Note that A cannot be 9, because P is. So there can be no carry from A+0. So L+W = 10+T, and T has to be at least 3 (0,1 taken and T=A+1) so L+W $\ge$ 13. –  Mark Bennet Oct 15 '12 at 17:44
Since A cannot be 0 or 1 and since T cannot be more than 5 (8 + 7 = 15 is the maximal sum for L and W), $2 \le A \le 4$ and $3 \le T \le 5$. Subsequently, since W must be 8 or less and the sum of L and W must be greater than 12, $5 \le L \le 8$ and $5 \le W \le 8$. Thus $2 \le Y \le 4$. The combination of $L=5$ and $Y=4$ would force $R=3$ which precludes consecutive solutions for A and T. So $6 \le L \le 8$. –  cheepychappy Oct 15 '12 at 18:16

Following from the comments, including mine. There are only four possible combinations for the trio of $(A,T,Y)$: $(2,3,4), (3,4,2), (4,5,2), (4,5,3)$.

The first two are impossible because the possible values left for R and S keep the tens column from summing to 0. The third case gives both L and W as 7 or 8. That leaves 3 and 6 for R and S, the only combination of those four numbers that would work in the rightmost column is $L=7, S=6, R=3$. This gives you your final answer:

    97426
+   8077
======
105503

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Wow, it's pretty complicated. Thanks for the help! –  tempy Oct 15 '12 at 18:37

If you like Verbal Arithmetic, check out the free android application, AlphaMetic.

http://goo.gl/bTkNI

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Nice! Thanks for the link. –  tempy Oct 24 '12 at 21:47