Earlier today I set these puzzles, suggested by the Department of Physics at Oxford University.
1. Cuppa conundrum
You are in a rush to work. You pour yourself a cup of black coffee, but it is too hot. You intend to add a fixed amount of cold milk to it, but you know that even after that, the coffee will need to cool down for a few minutes before you can drink it.
In which case does the coffee cool down more:
1) Add milk right away, then wait a few minutes before drinking.
2) Wait a few minutes, then add milk just before drinking.
Solution 2) Wait a few minutes, then add milk just before drinking.
In case 2) the liquid in the cup is hotter at the start than in case 1) when the milk has been added. Hotter things cool faster than less hot things because there is a greater difference in temperature with the surrounding environment. Thus the rate of cooling will be greater in case 2). Since in both cases, the liquid is cooling for the same time at different rates, the liquid that spends most time cooling with the faster rate will be cooler at the end, once the milk is added.
2. Fly away
A fly has landed at the bottom of a glass that is sitting on a very sensitive digital scale. All of a sudden the fly takes off. What happens to the readout on the scale?
1) it goes up
2) it goes down
Solution: Both! First it goes up, and then it goes down.
Imagine you are standing on your bathroom scales. In order to jump off the scales you will push down on the scales, which will register a heavier weight. The situation with the fly is similar. Its wings push the air down, which, in turn, pushes the bottom of the glass. But when it is high enough so the air movement does not reach the bottom of the glass, the measured weight will be that of the empty glass.
3. Bounding balls
A weightless string connects two identical metal balls. The middle of the string is over the edge of a table. You let go of both balls at the same time. What will happen sooner: ball 1 flying off the table or ball 2 hitting its side?
Solution Ball 1 flies off the table
The tension force of the string is the same on both sides. But the horizontal component of that force is larger for ball 1 than for ball 2. Hence ball 1 will accelerate in the horizontal direction faster and take less time to travel the same horizontal distance.
4. Fun fraction
Prove (1/2) x (3/4) x …x (99/100) < 1/10
It looks complicated, but there is a two line solution.
Let (1/2) x (3/4) x …x (99/100) = A.
Then 101 A = (3/2) x (5/4) x … x (101/100), and 1/A = (2/1) x (4/3) x … x (100/99). Both 101 A and 1/A have 99 fractions in parentheses, and every fraction in 101A is less than the corresponding fraction in 1/A. Thus 101 A < 1/A. The rest follows:
or2 < 1/101 < 1/100. Thus A < 1/10
I have fleshed out the solution to more than two lines. In terms of actual mathematical detection, however, two lines is all you need.
Thanks to Professor Alexander Lvovsky at Oxford University for these puzzles. Prof Lvovsky runs the Comprehensive Oxford Mathematics and Physics Online School (COMPOS), a new initiative that teaches maths and physics to sixth formers from state schools in the UK at a deeper level than the A-level curriculum.
The course is free and anyone in years 12 or 13 is eligible. There are still spaces for this year’s intake, so please pass the word around!
I’ll be back with new puzzles in two weeks.
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.
I’m the author of several books of maths and puzzles, and also the co-author with Ben Lyttleton of the children’s book series Football School. The latest in the Football School series is The Greatest Ever Quiz Book, out now!
I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.