Dynamic Programming?

This not an elegant solution but it will work for this particular problem (I'm guessing we have the same professor).

The logic is that for each V[n][c] we want to find the highest value possible for each unit of currency, and in order to do this we must calculate the maximum value out of 6 vales.

There are 6 values because there are 3 currencies, and each of those currencies has two possible ways that it can be converted into the target currency.

In this case since there are only 2 exchanges I simply do two statements rather than another loop. This is represented by the 0 in the array: rates[0][i][c]

I hope this helps!

    for (int n = 1; n <= numberOfTransactions; n++) {
        for (int c = 0; c < numberOfcurrencies; c++) {
            double max = Double.NEGATIVE_INFINITY;
            double temp;
            for (int i = 0; i < numberOfcurrencies;i++) {
                temp = rates[0][i][c]*V[n-1][i];
                if (temp > max)
                    max = temp;
                temp = rates[1][i][c]*V[n-1][i];
                if (temp > max)
                    max = temp;
            }
            V[n][c] = max;
        }
    }