If POINT + ZERO = ENERGY, then E + N + E + R + G + Y = ?

In a cryptarithmetic (alphametic) puzzle, every distinct letter stands for exactly one fixed digit (0-9), the same letter always represents the same digit, no two different letters share a digit, and the leading letter of any number cannot be 0. Such puzzles are solved column-by-column from the units place, carrying any overflow into the next column exactly as in ordinary addition.

Applying this to POINT + ZERO = ENERGY, a 5-digit number plus a 4-digit number giving a 6-digit result:

1. A 5-digit number added to a 4-digit number can produce at most a 6-digit result only if there is a carry all the way into a new leading place — so the new leading digit, E, must equal 1.

2. Units column: T + O gives Y, with some carry c1 into the tens column.

3. Tens column: N + R + c1 gives G, with carry c2 into the hundreds column.

4. Hundreds column: I + E + c2 gives R, with carry c3 into the thousands column.

5. Thousands column: O + Z + c3 gives E (= 1), with carry c4 into the ten-thousands column.

6. Ten-thousands column: P + c4 gives N, and the resulting carry (c5 = 1) becomes the new leading digit E, confirming E = 1 is self-consistent.

7. Testing digit assignments that satisfy all ten columns simultaneously (with all ten letters P, O, I, N, T, Z, E, R, G, Y distinct, and P, Z, E nonzero) yields exactly one consistent mapping: P = 9, O = 8, I = 5, N = 0, T = 4, Z = 3, E = 1, R = 6, G = 7, Y = 2.

8. Substituting these digits: POINT = 98504, ZERO = 3168, and ENERGY = 101672.

Cross-check: 98504 + 3168 = 101672, confirming the column-wise solution; and the ten digits used (9, 8, 5, 0, 4, 3, 1, 6, 7, 2) are all of 0-9 exactly once, confirming every letter maps to a distinct digit as required.

So E + N + E + R + G + Y = 1 + 0 + 1 + 6 + 7 + 2 = 17.