Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Label this plum tree graph to make it totally magic!
Find all the ways of placing the numbers 1 to 9 on a W shape, with
3 numbers on each leg, so that each set of 3 numbers has the same
The 2-digit number is either $65$ or $56$.
Any 2-digit number can be represented as $10a + b$. We need
$(10a+b)^2 - (10b+a)^2 = 99a^2 - 99b^2 =9 \times 11 \times (a^2 -
b^2)$ to be a square.
This means that $(a^2 - b^2)$ must be 11 and so $(a - b)(a + b)
= 11$ making, $a - b = 1$ and $a + b = 11$. This gives $a = 6$, and
$b = 5$.
If we find a solution with $a > b$ then, by reversing the
digits, we get a second solution.