The works of the sculptor Alexander Calder have been classified into two types, one of which is called a stabile (pronounced STAY-beel). Which four of the works above are stabiles, which one is the other type, and which one is a hybrid?
The works of the sculptor Alexander Calder have been classified into two types, one of which is called a stabile (pronounced STAY-beel). Which four of the works above are stabiles, which one is the other type, and which one is a hybrid?
At breakfast each person in Tanya's family had a full cup of coffee with milk, mixed to his or her favorite proportions. Everyone's cup is the same size. Given that Tanya drank one-quarter of all the milk and one-sixth of all the coffee, what was the ratio of milk to coffee in Tanya's cup?
It's tempting to use algebra: let m be the number of full cups' worth of milk consumed by the family, and c the amount of coffee. Then m/4 + c/6 = 1, and… ???
We do not seem to have enough information to go further! The key question is how many people are in Tanya's family. But this can actually be answered without any algebra.
Tanya had less than one-quarter of the total liquids. (We know this because she had exactly one-quarter of the milk, but less than one-quarter of the coffee.) Similarly, Tanya had more than one-sixth of the total liquids. Since everyone had the same size cup, Tanya must have had 1/N of the total liquids, where N is the number of people. So the only possibility for N is 5.
That gives us our second equation, m + c = 5, and now we are able to solve: m = 2, c = 3, and so Tanya's cup was exactly half milk and half coffee.
Solution
This puzzle is from a Russian math contest (I read it on Tanya Khovanova's math blog).
Congratulations to yesterday's solvers Anna K., Inca, Jacob C., Mr. Gregg, Graham, and the students of Number Theory Lab. Thanks to everybody who made a guess!
Ten years later, Dr. Shapiro revived Puzzle of the Day at Proof School, writing each day's puzzle on a name tag. After 600 puzzles or so, he was just starting to feel normal about students reading his chest all the time when campus closed and the puzzle, like the rest of our lives, moved online. New puzzles are posted daily on school days.
Want to catch up on old PotDs? There's an archive currently containing puzzles from March 2020 to March 2021.