### 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?

# Black and Gold Storeys

##### Age 11 to 14 Short Challenge Level:

Using reasoning

Using a table

$25-a=25-b\therefore a=b$

$\therefore a=14$

Using symbols for some of the numbers
Gold storeys in top half: $x$
Black storeys in bottom half: $y$
Given that $x+y = 28$

Gold storeys in the bottom half: $25-x$
Black storey in the bottom half: $y$
$25-x+y=25 \Rightarrow x=y$

So $x+y=28$ means $x$ and $y$ are both $14$, since $x=y$

Using symbols for all of the numbers
$a$ gold storeys in the top half of the building.
$b$ gold storeys in the bottom half of the building.
$c$ black storeys in the top half of the building.
$d$ black storeys in the bottom half of the building.

Then:
$a+b$ = 25   (1)
$c+d$ = 25   (2)
$a+c$ = 25   (3)
$b+d$ = 25   (4)
$a+d$ = 28   (5)

(1) $-$ (4): $a+b-b-d$ = 0 $\Rightarrow a-d$ = 0 $\Rightarrow a$ = $d$

(5) becomes $a+a$ = 28, so $a$ = 14

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