Particle Moves to Target

A particle starts at integer coordinate (startX, startY) and must reach (targetX, targetY) on an unbounded grid. You may use these operations any nonnegative number of times:
PROTON: move one cell right, from (x, y) to (x + 1, y).
NEUTRON: move one cell down, from (x, y) to (x, y - 1).
ALPHA: move two cells up and two cells left, from (x, y) to (x - 2, y + 2).
Return a three-element integer array [protons, neutrons, alphas] whose operations reach the target using the minimum possible total number of operations. The result is ordered as PROTON, NEUTRON, ALPHA.

Function
particleMoves(startX: int, startY: int, targetX: int, targetY: int) → int[]

Examples
Example 1
startX = 0
startY = 0
targetX = 3
targetY = -2
return = [3,2,0]
Three PROTON operations add 3 to x, and two NEUTRON operations subtract 2 from y. No ALPHA operation is needed.
Example 2
startX = 4
startY = -3
targetX = -2
targetY = 5
return = [2,0,4]
Four ALPHA operations change the position by (-8, 8), and two PROTON operations restore x by 2. The net change is (-6, 8).
Example 3
startX = -5
startY = 7
targetX = -8
targetY = 0
return = [1,11,2]
Two ALPHA operations contribute (-4, 4). One PROTON and eleven NEUTRON operations then produce the required net change (-3, -7).

Constraints
-10^6 <= startX, startY, targetX, targetY <= 10^6.
Movement occurs on an unbounded integer grid.
A minimum-operation solution always exists and every returned count fits a signed 32-bit integer.

Let dx = targetX - startX and dy = targetY - startY. With p protons, n neutrons and a alphas, you need p - 2a = dx and 2a - n = dy. So p = dx + 2a and n = 2a - dy. Total operations is dx - dy + 5a, which only grows with a. So pick the smallest a that keeps p, n and a nonnegative: a = max(0, ceil(-dx/2), ceil(dy/2)). Then compute p and n from it. Check example 3: dx = -3, dy = -7, so a = 2, p = 1, n = 11. Matches. The pitfall is ceiling division on negatives. Languages that truncate toward zero will give wrong answers, so use (v + 1) // 2 with floor division or guard the sign. Another trap is trying to fix parity with extra moves. You never need to. If the formula slips under pressure, StealthCoder is the hedge on the live OA.

Drill it cold or hedge it with StealthCoder. Either way, don't walk into the OA hoping you remember the trick.