Sunday, October 08, 2006

October 2006: The locker room (solution)

The state of locker n is changed when the kth student passes through, for every divisor k of n. Since factors usually come in pairs {j, k} where j x k = n, the net effect of students j and k on this locker is nil. The exception is when n is a perfect square, in which case there is no other divisor to cancel the effect of the sqrt(n) student. Therefore, the lockers which are open at the end are exactly the perfect squares, 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

Correct solutions: Evan Templeton, Matt Bachmann, Kaloyan Todorov, Theresa Sparacio ('04), Kristin Jekielek, Hiro Arai, Robert Pehlman, Jenny Witzeling, Dulguun Bayasgalan, Ritoban Basu-Thakur, Ritwik Niyogi, Sunil Baidar, Wade Robertson, Mike Scanish ('01), Lisa Dubbs, Amanda Janiec, Ben Raffeto

Thursday, October 05, 2006

October 2006: The locker room

Lockers numbered 1 to 100 stand in a row at the school gym. When the first student arrives, she opens all the lockers. The second student then goes through and recloses all the even-numbered lockers; the third student changes the state of every locker whose number is a multiple of 3.

This continues until 100 students have passed through. Which lockers are now open?

Dickinson College students can submit answers to Barry Tesman or Dave Richeson. The list of solvers will be posted at the end of the month.