Brain EaserApplications of algebra
Two ferry boats serve the same route on a river, but travel at different speeds. They depart from opposite ends of the river at the same time, meeting at a point 720 yards from the nearest shore.
When each boat reaches the other side, it takes 10 minutes to unload and load passengers, then begins the return trip. This time, the boats meet at a point 400 yards from the other shore.
How wide is the river?
The ferry boat problem is created by well-known puzzle author Sam Loyd.
Given the iterative formula:
xn+1 = sqrt(xn) + a
There are some values of a for which a positive starting value for x results in this formula converging to an integer.
For example, if a = 2, you would observe that xn+1 = sqrt(xn) + 2 eventually converges to 4 for any positive starting value.
What form must a take in order for this formula to converge to an integer?
Find a combination of three different positive integers x, y, and z such that:
x3 + y3 = z4
Hint: there’s a better way than brute force / trial & error.
If x is an even integer that can be written as the sum of two perfect squares, prove that x / 2 can also be written as the sum of two perfect squares.