You are blindfolded before a table. On the table are a very large number of pennies. You are told that 2007 of the pennies are heads up and the rest are tails up. How can you divide the entire set of pennies into two subgroups, each with the same number of heads facing up?
Solution 1 by Jerrod Kempf , Charlotte High School
Simply count out 2007 pennies and flip them over. This works because the 2007 pennies you extracted will have n number of heads while the remaining pile (of coins you did not pick) has 2007- n number of heads. By flipping the 2007 coins, you create 2007- n number of heads in the first pile, which matches the number of heads in the other pile.
Solution 2 by Joan Gary , Forest Hills Eastern High School
There are two possible cases to consider:
Case 1: If there are only 2007 pennies, all will be heads up. Take the 2007 pennies and flip them over so now only tails are facing up. Then separate the coins into two groups. This will make two groups with an equal number of heads: zero .
Case 2: If there are more than 2007 pennies, split them into one group with 2007 pennies and another group with the rest. In the 2007 penny group you have x heads and 2007- x tails. In the other group you have 2007- x heads, which happens to be equal to the number of tails in the 2007 penny group. Now, flip all the pennies in the 2007 penny group. This will leave you with 2007- x heads in each group.
Also solved by: Ben Beck (Bullock Creek H.S.) ; Galina Gagin (East Grand Rapids H.S.); Clare Boals, Kristen Bova, Eric Graverson, Eric Hart, Melvin Joice, Sonya Kavalam, Ryan Klock, Kristin Kopec, Amanda Ly, Mary Maier, Caitlin McCarthy, Nikhil Nandigam, Olivia Ruiz, Rebecca Vogelsang, Christopher Yang, Marc Walker, Sheila Waslawski, Alex Zheutlin (Forest Hills Eastern H.S.); Murali Tej Bolla (Forest Hills Northern H.S.); Darren Brown, Robert Harrison, Alex Robinson (Lansing Catholic H.S.); Spencer Frost, Britt Gabel, Rachael Juengel, Katlynn Kish, Nate Weller (Linden H.S.) ; Hiu-Tung Wong (Salem H.S.); David Maniez (St. Louis H.S.); Chirag Pathre (Troy H.S.); Mallory Sparr (Alamo Heights H.S., San Antonio, TX).