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# Road Maker 2

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### Good Approximations

### Rational Roots

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Age 16 to 18

Challenge Level

*This problem follows on from* Road Maker*, where the rules of making roads are detailed in full.*

The Munchkin road making authority have commissioned you to work out the possible destinations for their roads. Use Cartesian coordinates where the first tile is placed with opposite corners on $(0,0)$ and $(1,1)$.

Investigate ways in which you can reach your destination. You may like to consider these questions:

- Can you make roads with rational values for the $x$ coordinate of the destination?
- Can you make roads with rational values for the $y$ coordinate of the destination?
- Can you create a road with the $x$ coordinate equal to any integer multiple of one half?
- Can you make roads for which the coordinates of the destination are both rational? Both irrational?
- Can multiple roads lead to the same destination? For which destinations is the road unique?

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.