Problem 9. (Mary's Random Walk)

Mary lives in a town which has streets and avenues crossing each other at right angles. One day at exactly noon Mary decides to take a walk starting at the intersection of a street and avenue, whereupon she covers one block in every 15 minutes. When she reaches the end of a block, she turns either to the left or right. Show that Mary can return to the starting point only on the hour.

Solution

One thing is clear. If Mary makes a right turn at every block, then when she returns home she will have traveled an even number of blocks in the horizontal direction (either left of right) and the same even number of blocks in the vertical direction (either up or down). Hence, the total number of blocks she will have traveled is twice an even number, or a multiple of four. Therefore, if Mary started her walk on the hour, she will return to the starting point on the hour.

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Last modified on Wednesday, March 17, 1999