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Showing posts from July, 2010

### Hats and Rooms

Source: Tanya Khovanova’s Math Blog

Problem: The Sultan decided to test the wisdom of his wizards. He collected them together and gave them a task. Tomorrow at noon he will put hats of different colors on each of the wizard’s heads.
The wizards have a list of the available colors. There are enough hats of each color for every wizard. The wizards also have a list of rooms. There are enough rooms to assign a different room for every color.
Tomorrow as the Sultan puts hats on the wizards, they will be able to see the colors of the hats of the other wizards, but not the color of their own. Without communicating with each other, each wizard has to choose a room. The challenge comes when two wizards have the same hat color, for they must choose the same room. On the other hand, if they have different hat colors, they must choose different rooms. Wizards have one day to decide on their strategy. If they do not all complete their task, then all of their heads will be chopped off. What strategy…

### Coin conundrum

Source: Australian Mathematical Society Gazette Puzzle Corner

Problem: There are coins of various sizes on a table, with some touching others. As often as you wish, you may choose a coin, then turn it over, along with every other coin that it touches. If all coins start out showing heads, is it always possible to change them to all tails using these moves?

Update (Nov 15, 2010):
Solution: Solution from the gazette author posted by me in comments! Interesting linear algebra solution posted by Siddhant Agarwal (EE, Senior Undergraduate, IITB) in comments!

### Differing views

Source: Australian Mathematical Society Gazette Puzzle Corner

Problem: An optimist and a pessimist are examining a sequence of numbers. The optimist remarks, ‘Oh jolly! The sum of any eight consecutive terms is positive!’ But the pessimist interjects, ‘Not so fast, the sum of any five consecutive terms is negative.’ Can they both be right? How long can this sequence of numbers be?

Incentive: Treat at H8 Canteen/Sodexho Cafeteria for the first person to solve it :P

Update (27/07/10): Solution: Posted in comments by Varun Jog (Berkeley Grad Student, EE IITB Alumnus) and Siddhant Agarwal (EE Senior Undergraduate, IITB)