It’s a Puzzlement
The Slippery Ring
frictionless ring contains N identical beads spaced evenly around it. Each second, every bead independently jumps one position clockwise or counterclockwise with equal probability. If two or more beads land on the same position, they instantly merge into a single bead at that position. The process continues forever.
Question:
Extra credit: How does the answer grow as N grows to infinity?
To Freeze Or Not To Freeze?
If the vending machine temperature were in the range of –1°C to 0°C, diet sodas would freeze and regular ones would remain liquid. This is due to the scientific phenomenon of freezing point depression, whereby the freezing temperature of a solution decreases as more solute is dissolved (the boiling point is likewise elevated). Ordinary water would have a freezing point of 0°C. In the U.S. market, a 12 oz. can of regular Mountain Dew has 46g of sugar (fructose, molar weight = 180g), or 0.255 mol. For ordinary water, this would translate to a molality of 0.255 mol / (12 oz. = 0.355 kg) = 0.72 molal. This in turn depresses the freezing point by 0.72 m * (–1.86°C / m) = –1.3°C. Other brands such as Sprite, Dr Pepper, Coke, and Pepsi have less sugar, but it is still in the upper 30s of grams. On the other hand, a 12-oz. can of Diet Coke would contain only about 200mg of aspartame, which furthermore has a much larger molar weight of 294g. The resulting freezing point depression is (0.2 g / 294 g/mol) / (0.355 kg) * (–1.86 °C / m) = –0.004 °C, basically unchanged from regular water.
Solutions were also submitted by Daniel Aarhus, Bob Conger, Daniel Heyer, Fresa Luo, Sean Porreca, John Noble, and Chris Shatto.