### Hallway Borders

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?

### Not a Polite Question

When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

# Sum and Differences

##### Age 11 to 14 ShortChallenge Level

Answer: $28, 30, 42$

Using algebra
Middle number is $n$

\begin{align}n-2\ \ +\ \ n\ \ +\ \ n+12 &= 100\\ \Rightarrow 3n+10&=100\\ \Rightarrow 3n&=90\\ \Rightarrow n&=30\end{align}

So $n=30$, $n+12=42$ and $n-2=28$.

Guess and check - differences
Try some numbers which are the right distances apart

\begin{align}30 + 32 + 44 &= 106\\ &=100 + 6\\ &= 100+3\times2\end{align}
$\therefore (30-2)+(32-2)+(44-2)$ should work

$28+30+42=100$

Guess and check - sum
Try some numbers which add up to $100$

$33+33+34=100$

Need to make largest number larger, always balancing out changes:
$(33-2)+\ \quad33\ \quad+(34+2)=31+33+36=100$

$(31-1)+(33-1)+(36+2)=30+32+38=100$

$(30-2)+(32-2)+(38+4)=28+30+42=100$

You can find more short problems, arranged by curriculum topic, in our short problems collection.