DAY 122
You have nine identical coins. One of them is just slightly lighter than the other eight. How many weighings on a balance scale would it take to determine the lighter coin?

Answer:

2..........Weigh 3 against 3. If they do not balance, take 2 from the lighter side and weigh them against each other. If they balance, the third coin is the lighter. If they don't balance, the lighter coin is obvious. If the 3 against 3 weighing balances, follow the same procedure with the remaining 3 coins.

DAY 123
Divide 59 into two portions such that the sum of 1/7 of the greater portion and 1/3 of the lesser equal 13.

Answer:

35 and 24............1/7X + 1/3(59 - X) = 13