In the cryptarithmetic puzzle YOUR + YOU = HEART, each letter stands for a…

In a cryptarithmetic puzzle every letter stands for exactly one fixed digit (0-9), no two letters share a digit, and the addition must satisfy ordinary column-by-column arithmetic with carries. When a smaller number of digits added to another produces a result with one extra leading digit, that extra leading digit is always equal to the carry generated out of the highest column added — it is fixed by the carry, not solved independently.

Applying this to the puzzle:

1. YOUR has 4 digits and YOU has 3 digits, but their sum HEART has 5 digits, so the leading letter H must equal the carry out of the thousands column: H = 1.

2. In the thousands column, Y plus any carry coming from the hundreds column must total E plus 10 times the carry going into the ten-thousands place; since that outgoing carry is fixed at 1, the only digit that makes this balance is Y = 9, which forces the hundreds-column carry to 1 and gives E = 0.

3. In the hundreds column, O + Y + (carry from tens) must total A plus 10 times the carry passed into the thousands column; that carry into the thousands column is already fixed at 1 (from the previous step), so O + Y + (carry from tens) = A + 10. With O = 4 and Y = 9, this gives A = 3 + (carry from tens): a carry of 1 would force A = 4, which is already taken by O, so the carry from tens must be 0 and A = 3.

4. In the tens and ones columns, U + O (+ a possible carry from the ones column) must total R, and R + U must total T (+ a carry). Checking each remaining unused digit for U, the only assignment that keeps every letter's digit unique and every column valid is U = 2, R = 6, T = 8.

5. Substituting all eight digits back into YOUR + YOU confirms every column balances with no contradiction.

Cross-check: YOUR = 9426 and YOU = 942, and 9426 + 942 = 10368, which matches HEART with H = 1, E = 0, A = 3, R = 6, T = 8 — every column balances with no leftover carry.

So Y + U + R + E = 9 + 2 + 6 + 0 = 17.