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Nostra culpa (Posted on 2010-09-03) Difficulty: 2 of 5
WE + ARE +VERY = SORRY
For what value(s) of Y there is only one solution?

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (2 votes)

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solution | Comment 2 of 5 |
To begin with, let's see which letter values we can determine based on the values.

First off, we can see that S = 1, because there is only one 4 digit number on the left. Due to VERY being the only 4-digit number on the left, we also know that O = 0. Again because VERY is the only 4-digit number on the left, V = 9

Because WE and ARE both end in E, and VERY and SORRY both end in Y: E + E + Y = (Y or 10 + Y); E = 0 or 5, and since O = 0, E = 5.

So, we currently have S = 1, O = 0, V = 9, and E = 5; leaving W, A, R, and Y unsolved for, and 2, 3, 4, 6, 7, and 8 left to choose from.

Because VERY and SORRY end in the same 2 letters, let's subtract RY from both sides to get:

W(5) + AR(5) + (9500) = (10)R(00)

So, W(5) + R(5) must add to 100, that leaves W + R = 9. Replacing W(5) + R(5) on the left side of our equation with 100:

100 + A(00) + (9500) = (10)R(00) => A(00) + (9600) = (10)R(00)

So, A + 6 = 10 + R, leaving A = 6, R = 2 or A = 7, R = 3. Going back to our W + R = 9:

If A = 6, R = 2, W = 7 ... if A = 7, R = 3, W = 6. Since 6 and 7 are present in both solution types, Y can be 2, 3, 4, or 8. Which leaves 3 solutions for each combination of A, R, W. For both sets, Y = 4 and Y = 8 are both going to be solutions, but Y = 2 only works for the second set, and Y = 3 only works for the first set.

Thus, there are two values of Y such that there is only one solution: Y = 2 and Y = 3.

  Posted by Justin on 2010-09-03 15:00:53
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