Puzzles
2 December
Holly adds up the first six even numbers, then adds on half of the next even number. Her total is 49.
Next, Holly adds up the first \(n\) even numbers then adds on half of the next even number. This time, her total is 465124. What is \(n\)?
Show answer & extension
Hide answer & extension
If we add up the first \(n\) even numbers then add on half of the next even number, we get \((n+1)^2\). This means that Holly added up the first \(\sqrt{465124}-1\) or 681 even numbers.
Extension
Can you show why adding up the first \(n\) even numbers and half of the next even number gives \((n+1)^2\)?
15 December
There are 3 even numbers between 3 and 9.
What is the only odd number \(n\) such that there are \(n\) even numbers
between \(n\) and 729?
Show answer & extension
Hide answer & extension
There are \((729-n)/2\) even numbers between \(n\) and 729, and so we want to solve \((729-n)/2=n\). The solution of this is 243.
Extension
For which odd numbers \(N\) does there exist an odd number \(n\) such that there are \(n\) even numbers between \(n\) and \(N\)?
