In turn 4 people throw away three nuts from a pile and hide a
quarter of the remainder finally leaving a multiple of 4 nuts. How
many nuts were at the start?
Three semi-circles have a common diameter, each touches the other
two and two lie inside the biggest one. What is the radius of the
circle that touches all three semi-circles?
Label this plum tree graph to make it totally magic!
You can choose any square numbers to start with but you'll find
that only certain combinations of odd and even numbers will work.
You'll need to solve three simultaneous linear equations in three
unknowns to find formulae for the three integers in terms of the
three square numbers that you started with.
For the second part you'll need a number which can be written as
the sum of two squares in at least three different ways. You could
use this example:
$1^2 + 132^2 = 28^2 + 129^2 = 36^2 + 127^2 = 55^2 + 120^2 = 63^2
+ 116^2 = 80^2 + 105^2 = 17425$