000

# Puzzle solutions, Issue 01

These solutions relate to puzzles found in Issue 01 of the magazine.

### #2 – Circles

Call the blue area $B$ and the red area $R$. Then

\begin{align*}
R & = \mbox{Area of quarter circle} – \mbox{Area of semicircle} – (\mbox{Area of semicircle} – B)\\
& = \frac{1}{4}\pi(2r)^2 – \frac{1}{2}\pi r^2 – \left(\frac{1}{2}\pi r^2 -B\right)\\
& = \pi r^2 – \frac{1}{2}\pi r^2 – \frac{1}{2}\pi r^2 +B\\
& =0+B\\
&=B
\end{align*}

### #104 – Two Triangles

Draw on the following lines parallel to those which were added in the question.

Then a grid of copies of the smaller blue triangle has been created.

Now consider the three triangles which are coloured green, purple and orange in the following diagram:

Each of these traingles covers half a parallelogram made from four blue triangles. Therefore the area of each of these triangles is twice the area of the small blue triangle.

And so the blue triangle covers one seventh of the large triangle.

These solutions relate to puzzles found in Issue 01 of the magazine.

### #2 – Circles

Call the blue area $B$ and the red area $R$. Then

\begin{align*}
R & = \mbox{Area of quarter circle} – \mbox{Area of semicircle} – (\mbox{Area of semicircle} – B)\\
& = \frac{1}{4}\pi(2r)^2 – \frac{1}{2}\pi r^2 – \left(\frac{1}{2}\pi r^2 -B\right)\\
& = \pi r^2 – \frac{1}{2}\pi r^2 – \frac{1}{2}\pi r^2 +B\\
& =0+B\\
&=B
\end{align*}

### #104 – Two Triangles

Draw on the following lines parallel to those which were added in the question.

Then a grid of copies of the smaller blue triangle has been created.

Now consider the three triangles which are coloured green, purple and orange in the following diagram:

Each of these traingles covers half a parallelogram made from four blue triangles. Therefore the area of each of these triangles is twice the area of the small blue triangle.

And so the blue triangle covers one seventh of the large triangle.

• ### Issue 05

New issue out now! Graphical linear algebra, slide rules and game theory in nature. Plus all your favourite fun pages.
• ### Prize crossnumber, Issue 05

Win £100 of Maths Gear goodies by solving our famously fiendish crossnumber
• ### What’s hot and what’s not, Issue 05

Mathematical fashion advice for an ever-changing world
• ### Dear Dirichlet, Issue 05

Prof. Dirichlet tackles political arguments and werewolves in answering your personal problems
• ### Puzzles, Issue 05

Solve the puzzles that appeared in Issue 05.