Henry Ernest Dudeney/Modern Puzzles/49 - Exploring the Desert/Solution

Modern Puzzles by Henry Ernest Dudeney: $49$

Nine travellers, each possessing a motor-car, meet on the eastern edge of a desert.

They wish to explore the interior, always going due west.

Each car can travel $40$ miles on the contents of the engine tank,

which holds a gallon of petrol,

and each can carry $9$ extra gallon tins of petrol and no more.

Unopened tins can alone be transferred from car to car.

What is the greatest distance at which they can enter the desert without making any depots of petrol for the return journey?

$360$ miles out and home.

Let the $9$ men be denoted $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ and $J$.

All $9$ men travel $40$ miles into the desert, carrying $9$ gallons each, using the gallon they have in the tank.

$A$ transfers $1$ gallon to each of the other $8$ drivers, and uses his other gallon to get home.

Those $8$ then travel $40$ miles further into the desert on the gallon they have in the tank.

$B$ transfers $1$ gallon to each of the remaining $7$ drivers, and uses his other $2$ gallons to get home.

Those $7$ then travel $40$ miles further into the desert on the gallon they have in the tank.

$C$ transfers $1$ gallon to each of the remaining $6$ drivers, and uses his other $3$ gallons to get home.

Those $6$ then travel $40$ miles further into the desert on the gallon they have in the tank.

$D$ transfers $1$ gallon to each of the remaining $5$ drivers, and uses his other $4$ gallons to get home.

Those $5$ then travel $40$ miles further into the desert on the gallon they have in the tank.

$E$ transfers $1$ gallon to each of the remaining $4$ drivers, and uses his other $5$ gallons to get home.

Those $4$ then travel $40$ miles further into the desert on the gallon they have in the tank.

$F$ transfers $1$ gallon to each of the remaining $3$ drivers, and uses his other $6$ gallons to get home.

Those $3$ then travel $40$ miles further into the desert on the gallon they have in the tank.

$G$ transfers $1$ gallon to each of the remaining $2$ drivers, and uses his other $7$ gallons to get home.

Those $2$ then travel $40$ miles further into the desert on the gallon they have in the tank.

$H$ transfers $1$ gallon to the remaining driver $J$, and uses his other $8$ gallons to get home.

$J$ then travels $40$ miles further into the desert on the gallon he has in the tank.

Then using the $9$ gallons he is carrying, $J$ travels the $9 \times 40 = 360$ miles home.

$\blacksquare$