Since the vehicle has four wheels the total mileage is 4*27,000=108,000. The tyre mileage of 12,000 divides into this 9 times, so the minimum number of tyres to provide the mileage is 9. It's just a question of how.
The journey can be split into 9 stages of 3,000 miles each. Label each tyre with numbers 1 to 9, so that each one is identified by its number. Label the journey stages A to I. Stage A uses tyre numbers 1-4; B 2-5; ... F 6-9; G 1,7,8,9; H 8,9,1,2; I 9,1,2,3. So every 3,000 miles, a tyre is swapped. Between stages A and B, 5 replaces 1; between B and C 6 replaces 2, and so on. Over the whole journey then, tyre #1 is used in stages A, G, H, I so its total mileage is 12,000; B is used in stages A, B, H, I, so its total mileage is also 12,000. Each tyre takes part in 4 stages only and is unused for 5 stages. This arrangement fully utilises all 9 tyres.
JOURNEY STAGES
A |
B |
C |
D |
E |
F |
G |
H |
I |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
2 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
2 |
3 |
The table above shows the tyre-switching arrangements. Each stage is 3,000 miles.