Numberplay: Student Groups - NYTimes.com Our puzzle this week was suggested by Numberplay contributor Marco Moriconi, who says he likes the puzzle because "it requires very little new math, but it is quite ingenious." The puzzle is called Student Groups. In a certain school there are 289 students. They have to go out in 17 groups of 17 students each day, with the condition that no two students can be in the same group more than once. For how many days can they do this? Recap: Denominations Last week we flipped over Denominations. The puzzle: If a currency system could have only two coins, what should the values of those coins be? The solution: Sumanth Ravipati was the first to propose the standard solution: Easy counting implied keeping it in base 10 as much as possible. Also, we would have to keep a coin with value 1 to account for any given value. Therefore, the two coins I would keep are 1 and 10. Between the values 1 and 100, the most number of coins you would carry is 18, for the value of 99. There were other proposals as well: Rich in Atlanta: a 9- and 11- cent coin (in addition to a penny) would also work. Aaron: twenty-five and fifty cents. Retailers would round up or down. Old Mister: a quarter and a dime. Change how change is done. Georgia: a quarter and a fifty cent piece, due to inevitable inflation Simon closed out the week by showing total how a 9- and 10-cent piece were best if you could trade coins. You could pay for a 1-cent item, for example, by trading a 10-cent piece for the item and a 9-cent piece. Simon summarized the process in the following chart. I thought it might be helpful to see a graphical comparison of the standard solution (the dime and penny) with Simon's trading solution of a dime and 9-cent piece. The following shows one way to compare the two methods. The y-axis on each chart indicated number of dimes. The x-axis on the left indicates pennies and on the right indicates 9-cent pieces. The zig-zag traces values from 1 to 100. The line on the left begins at [1 penny, 0 dimes] and ends up at [0 pennies, 10 dimes]. The line on the right begins at [-1 9-cent piece, 1 dime] and also ends up at [0, 10], but is much more efficient along the way. Let's hear it for everyone who made the conversation happen last week. In order of appearance: Gary, Dr W, Rich in Atlanta, unest, Golden Dragon, polymath, Aaron, Vaal, Old Mister, L. Arnaud, Sumanth Ravipati, charlie, David Smith, David, Giovanni Ciriani, Seth, Maya, Steve Kass, Hans, frobozzz, Andrew, JGR, Conuly, deo, Georgia, Paddy D, Jocelyn, Theodore Norvell and Neil Lehto. Student Groups was suggested by longtime Numberplay contributor and theoretical physicist Marco Moriconi, who currently lives in Rio de Janeiro. He first encountered the problem at the 6th Brazilian Mathematics Olympiad. Thank you, Dr. Moriconi. Have any ingenious puzzles?