
Earlier today I set you the following puzzle:
A full whisky bottle has a height of 27cm and a diameter of 7cm, and contains 750 cubic centimetres of whisky. It has a dome-like indentation at the bottom like many bottles do.

Whisky was drunk. Now the bottle only has 14cm of whisky left in it. When you turn the bottle over, it has 19cm of whisky. How much whisky is still in the bottle, in cubic centimetres?

Solution
This is a beautiful puzzle. It’s nicest solution is brief and elegant, relying on a single insight. If you used simultaneous equations, like many who commented or shared the puzzle on social media, you were overcomplicating the matter.
Bernardo Recamán, the Colombian maths educator who suggested it to me, says that his students always find the solution a surprise.
“When they ask for a hint, I tell them to drink some of it, though not so much to blur their minds.”
How much should they drink?
Exactly 3cm. And here’s why.
When the upright bottle has 14cm of whisky, there is 13cm above it with no whisky.
If I drink 3cm of the whisky, leaving 11cm, there would now be 16cm above it with no whisky.
And if we upturned the bottle, it would now have 19 –3 = 16cm of whisky. In other words, the volume of whisky in the bottle is equal to the volume with no whisky, meaning the bottle is half full.
So, the whisky in the bottle is equal to half the bottle’s volume plus 3cm of whisky.
The full bottle is 750cc, so half the bottle is 375cc.
The volume of 3cm of whisky is πr2 x 3cm, where r is the radius of the bottle. Since the diameter is 7cm, the radius is 3.5, so the volume of 3cm is:
3.14 x (3.5)2 x 3 = 115 cc (approximately).
The whisky left in the bottle = 375 +115 = 490 cc approx.
Lovely. No equations.
Chin chin!

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
It is believed today’s puzzle, which has appeared on The Math Forum, was originally devised by the US mathematician Robin Hartshorne.
My most recent book Can You Solve My Problems? A Casebook of Ingenious, Perplexing and Totally Satisfying Puzzles is available from the Guardian Bookshop and other retailers. My children’s book Football School: Where Football Explains The World was recently shortlisted for the Blue Peter Book Award 2017.