# Algebra

Applications of algebra

## Who Scored How Many Points

Lalo, Tyson, and Michael played a number of games of pick-up basketball. At the end, their combined points across all the games were:

• Lalo: 20
• Tyson: 10
• Michael: 9

They noticed that in every game, one of them scored x points, one of them scored y points, and one of them scored z points, where x > y > z and all three are distinct positive integers.

If Tyson got the highest score in the first game, who got the second highest score in the second game?

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## Ferry Boat Problem

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.

## Converge to an Integer

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?

Continue reading “Converge to an Integer”