I'm going to be in the UKMT Team Challenge in a few days and revising some questions used in the previous year. The questions are really bugging my mind. I know it may seem like a lot and quite easy but thanks if you help out, I will be grateful.
$Question$ $1)$
How many times does the digit $3$ appear when the whole
numbers between $1$ and $150$ are written down?
My thoughts:
I wrote down all the numbers from $1$ to $150$ with the digit $3$ in them and I only got $12$ digits. Also, is there any other way to count them that is faster without writing them down.
$Question$ $2)$
A car’s milometer shows that the number of miles it has
travelled is $15951$. This number is palindromic which means it
reads the same forwards as backwards. The next time a
palindromic number appears on the milometer is exactly two
hours later. Find the average speed of the car during these two hours.
My thoughts:
What I did was changed the number $15951$ to $16961$ and found out the difference which was $1010$. The answer is somehow $55mph$. How?
$Question$ $3$
A cube is divided into $64$ identical smaller cubes. Seven of these
smaller cubes are removed. The resulting shape has a volume of
$1539$cm3.
What is the surface area of the original cube?
My thoughts:
I have no idea how to do this.
Best Answer
The next palindrome is $16061$, not $16961$.
$7$ of $64$ cubes were removed, leaving $\frac{57.1539}{ 57} = 27$, so each small cube is $3×3×3\,\text{cm}$. The big cube was divided into $4×4×4$ cubes, so it was $12×12×12\text{ cm}$. The surface is six sides times $144\text{ cm}^2$, or $864\text{ cm}^2$.