Challenge question #35 (31/10/2016)
How many triangles are contained in this figure?
There are $48$ triangles.Worked Solution
We first notice that all of the triangles in the figure are equilateral.There are:
- $25$ triangles of side length $1$ unit
- $13$ triangles of side length $2$ units
- $6$ triangles of side length $3$ units
- $3$ triangles of side length $4$ units
- $1$ triangle of side length $5$ units.
So, there are $25+13+6+3+1=48$ triangles in total.
Suppose a group of numbers are written in a grid of rows and columns.
Amanda finds the maximum value of each row, and then finds the minimum of these values, which she calls $A$.
Bill finds the minimum value of each column, and then finds the maximum of these values, which he calls $B$.
Show that $A$ must be greater than or equal to $B$.
Consider the various possibilities for where $A$ and $B$ may lie on the grid relative to each other.Worked Solution
Since $A$ is the maximum value in the row, $A \geqslant B$.
Since $B$ is the minimum value in the column, $A \geqslant B$.
Consider the value $C$ which lies in the same row as $A$, and the same column as $B$.
Using the same reasoning as Cases 2 and 3, $A \geqslant C$ and $C \geqslant B$.
$\therefore\enspace A \geqslant B$.
In each of the cases, $A \geqslant B$.