A diagonal through a box, step by step

Push a straight rod from one corner of a box of unit cubes to the opposite corner.

The question

How many of the little cubes does it pass through?

Count the walls, then the ones crossed together

naive:11

11−6=5

5+1=6

A rod goes corner to corner through an 2 × 3 × 6 block of unit cubes. It enters a new cube every time it crosses a wall — so a first guess is 2 + 3 + 6 = 11.
But where two walls meet on one line the rod crosses both at once, and that split was counted twice. It happens gcd(2,3) = 1, gcd(3,6) = 3 and gcd(6,2) = 2 times — 6 in all — so take them off: 11 − 6 = 5.
Where all three walls meet at a corner point it was subtracted once too often, so add gcd(2,3,6) = 1 back: 5 + 1 = 6.

How it works.

01
A rod goes corner to corner through an 2 × 3 × 6 block of unit cubes. It enters a new cube every time it crosses a wall — so a first guess is 2 + 3 + 6 = 11.
02
But where two walls meet on one line the rod crosses both at once, and that split was counted twice. It happens gcd(2,3) = 1, gcd(3,6) = 3 and gcd(6,2) = 2 times — 6 in all — so take them off: 11 − 6 = 5.
03
Where all three walls meet at a corner point it was subtracted once too often, so add gcd(2,3,6) = 1 back: 5 + 1 = 6.