Crossing the river

Twelve soldiers (four Romans, four Carthaginians and four Greeks) are on one bank of the Tiber River and need to go to the other bank, for which they must use a small boat that can accommodate a maximum of three people.

If all the soldiers know how to row, What movements must be made to cross the river, taking into account that, for obvious reasons, there should be no greater number of Romans than Carthaginians, or Carthaginians than Greeks, on any shore or in the boat during the trip?

Solution

There may be other similar solutions. Here we propose the following in which to abbreviate each group of soldiers is called by its initial and a number is indicated indicating the number of soldiers traveling in the boat or remaining on each shore:

In short, soldiers are named by initials (and a number that represents the amount, if there is more than one) that is on one side, in the boat or on the other side.

They cross in the boat 1R1C1G, leaving 3R3C3G on the shore.

They leave on the other shore 1R1C and return 1G in the boat, gathering 3R3C4G.

They cross 3C in the boat, remaining on one shore 3R4G and on the other (when they arrive) 1R4C.

Return 1R in the boat, leaving 4C on the other shore and in the starting 4R4G.

Now they cross 3R, leaving 1R4G and joining the other shore, where there are now 3R4C.

They return in the boat 1R2C, leaving on the other shore 2R2C. When they join those of the starting shore, they are 2R2C4G.

Now they cross 2G, leaving 2R2C2G and joining an identical amount on the other side.

Again, the full boat must return, with 1R1C1G, leaving on the other side 1R1C1G and on the initial side 3R3C3G.

Cross the boat again with 3G, reaching the final shore 1R1C4G and leaving in the starting 3R3C. The boat returns with a 1C. Now there is the starting 3R4C and the arrival 1R4G.

They cross 1R2C, leaving in the starting 2R2C and in the arrival 2R2C4G.

The boat returns with 1R1C, leaving on arrival 1R1C4G and on departure 3R3C.

They now cross 3C, leaving in the starting 3R and in the arrival R4C4G.

1R returns now, leaving on arrival 4C4G, and on departure 4R.

They cross 3R, leaving in the starting 1R and in the arriving 3R4C4G.

Return 1R in the boat, leaving on arrival 2R4C4G and on departure 2R.

They cross the last two Romans in the boat.