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
A 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?
Choose four consecutive whole numbers.
Multiply the first and last numbers together.
Multiply the middle pair together.
Choose several different sets of four consecutive whole numbers and do the same.
What do you notice?
Can you explain what you have noticed? Will it always happen?
Click below to see how Charlie and Alison explained what they noticed.
I can explain this by labelling the four consecutive numbers $n, n+1, n+2, n+3$.
Outer pair: $n(n+3) = n^2 + 3n$
Inner pair: $(n+1)(n+2) = n^2 + 3n + 2$
I drew a diagram, in which the product of each pair is represented by the area of a rectangle:
The outer pair is represented by the red rectangle.
The inner pair is represented by the blue rectangle.
The purple area is common to both.
The area of the red strip will always be two units less than the area of the blue strip.
Therefore, the product of the outer pair is always two less than the product of the inner pair.
Instead of doing lots of calculations, can you use these representations to compare the product of the first and last numbers with the product of the second and penultimate numbers, when you have:
Make up a few similar questions of your own. Impress your friends by giving them a calculator and 'predicting' what will happen!
Click here for a poster of this problem.