YOUR + YOU = HEART (Assume O = 4). Find the value of Y + U + R + E.

In a cryptarithmetic (alphametic) addition, each distinct letter stands for a unique digit from 0-9, and the sum must hold true column by column exactly as in ordinary vertical addition, with carries propagating from one column into the next. The strategy is to use the given digit and each column's carry constraint to pin down one letter at a time, checking that no two letters are ever assigned the same digit.

Align YOUR (a 4-digit number) and YOU (a 3-digit number) under HEART (a 5-digit number) by place value, with O = 4 given:

Y O U R

+ Y O U

-----------

H E A R T

1. Since adding a 4-digit number and a 3-digit number produces a 5-digit sum, there must be a carry into the leading position, so H = 1.

2. In the thousands column, Y (from YOUR) plus the carry-in from the hundreds column equals E, with a carry of 1 passing into H. Since Y is a single digit, the only way this column can generate that leading carry is Y = 9 with a carry-in of 1, which gives Y = 9 and E = 0.

3. In the hundreds column, O + Y plus the carry-in from the tens column equals A, and this column must itself carry a 1 into the thousands column (matching the carry-in just used above). With O = 4 and Y = 9, the column total is 13 plus the tens carry-in; if that tens carry-in were 1, A would equal 4 and clash with O, so the tens carry-in must be 0, which fixes A = 3.

4. In the tens column, U + O plus the units-column carry-in equals R, with no carry passed into the hundreds column (established above), so U + 4 + (units carry-in) = R. In the units column, R + U equals T plus 10 times the same units-column carry-in. Testing each digit not yet used for U (from the digits not already assigned to O, Y, E, H, A) against both columns together, only U = 2 keeps every column consistent with no units-carry: this gives R = U + 4 = 6 in the tens column and T = R + U = 8 in the units column.

5. This gives the complete digit assignment for every letter, with each digit used exactly once: O = 4, Y = 9, E = 0, H = 1, A = 3, U = 2, R = 6, T = 8.

6. Adding the four requested letters' digits together: Y + U + R + E = 9 + 2 + 6 + 0 = 17.

Substituting the assigned digits back: YOUR = 9426 and YOU = 942, and 9426 + 942 = 10368, which reads as HEART with H = 1, E = 0, A = 3, R = 6, T = 8 — exactly the digits assigned above, confirming the derivation is consistent.