Puzzles

This is a collection of some of my favourite puzzles. Many of these, I love because of the elegance and simplicity of their solution. Others have a cleverly crafted story around the puzzle. Still, others are intriguing because of their complexity. Whatever the reason, they’re all challenging. Enjoy!

How did I get around?

I decide to cycle around the local park in an anticlockwise direction, but never once do I turn left. How did I do it?

This solution has fewer loops.

Exactly half full

I have a cylindrical glass and an unlimited supply of water. How can I fill the glass exactly half full of water without the use of any measuring aids?

Fill the glass more than half full with water and then tip some water out until the water level touches the bottom and front lip of the glass as shown.

A cross-section of the centre reveals that we’re dealing with identical triangles.

When uprighted, the glass will be exactly half full of water.

Where is the father?

A mother is 21 years older than her child. In exactly six years from now, the mother will be exactly five times as old as the child. Where is the father?

A mother is 21 years older than her child: $M=21+C$ In exactly six years from now, the mother will be exactly five times as old as the child: $M+6=5(C+6)$ Substituting for $$M$$ and solving for $$C$$: \begin{align*} (21+C)+6&=5(C+6)\\ 27+C&=5C+30\\ -3&=4C\\ C&=-\frac{3}{4} \space years \end{align*} Any guesses what the father is doing?

A bottle and a cork

A bottle and cork cost $1.10. If the bottle costs a dollar more than the cork, how much does the cork cost? A bottle and cork cost$1.10: $B+C=1.1$ The bottle cost a dollar more than the cork: $B=1+C$ Substituting for $$B$$ in the first equation and solving for $$C$$: \begin{align*} (1+C)+C&=1.1\\ 2C&=0.1\\ C&=\0.05 \end{align*} So the cork costs 5¢ and the bottle costs1.05.

Exactly 4 litres

I have two cylindrical containers. One has a five-litre capacity; the other has a three-litre capacity. I have an unlimited supply of water. How can I get exactly four litres of water?

This solution is the least environmentally sustainable as it wastes 6L of water.

This solution, while longer, is slightly more environmentally sustainable as it only wastes 5L of water.

This is not only the quickest solution. It is also the most environmentally sustainable solution as it wastes very little water. This makes it a very efficient solution.

Shortest path

I went camping with the family. We had pitched the tent not far from a riverbank at the spot marked B. My daughter had gone to fetch some water in a billy, but on the way got distracted and wandered off picking wildflowers at A. “Where’s the water?” I called out. “I can’t make tea without it!”. If my daughter is to take the shortest path from A to B via the riverbank, where on the riverbank would she fill the billly?

The shortest distance between two points is a straight line.

Mirror point A on the opposite side of the riverbank so it appears at C. Draw a line between B and C. For the shortest path, where that intersects with the riverbank at D is where the water should be collected.