### Tennis

A tennis ball is served from directly above the baseline (assume the ball travels in a straight line). What is the minimum height that the ball can be hit at to ensure it lands in the service area?

### Square Pegs

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

### The Old Goats

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?

# Pythagorean Triples

##### Stage: 3 Challenge Level:
This problem can be solved by a short computer program as follows:

10 $D=0$

20 FOR $C = 1$ TO $99$

30 FOR $B = 1$ TO $C$

40 FOR $A = 1$ TO $B$

50 IF $A^2 + B^2 = C^2$ PRINT $A$, $B$, $C$

60 IF $A^2 + B^2 = C^2$ $D = D + 1$

70 IF $A^2 + B^2 = C^2$ AND $A*B = A+B+C$ PRINT Area is half perimeter for '';$A$,$B$,$C$

80 NEXT $A$

90 NEXT $B$

100 NEXT $C$

110 PRINT Total number is ''; $D$

120 END

You can check for yourself that, when you put integer values of $P$ and $Q$ into the formulas $A=2PQ$, $B=P^2 - Q^2$ and $C=P^2 + Q^2$, you get $A^2 + B^2 = C^2$. The formulas give infinitely many Pythagorean triples.

Can you find $P$ and $Q$ such that $A=9$, $B=40$ and $C=41$? Play a game with a friend. Each choose integer values of $P$ and $Q$ and calculate $A$, $B$ and $C$. Then give your opponent just the values of $A$, $B$ and $C$. The winner is the first one to find $P$ and $Q$.

It needs to be proved that all Pythagorean triples come from values of $P$ and $Q$ in this way. Look out for future articles with some proofs and showing how to derive these formulas.