A Puzzle by Midnight Math: February

midnightmathOn a circle are placed an equal number of “x”s and “o”s.

Starting with a value of zero, you chose a starting place on the circle and begin moving clockwise around the circle.

Every time you pass an “x”, you add 1 to your value, and every time you pass an “o” you subtract 1. Once you have returned to your starting location you stop.

Show that no matter how the “x”s and “o”s are placed (as long as there are an equal total number of each) there will always be a starting location such that your value is never less than zero.

Send your solutions (with proof) to midnight.math AT outlook.com. If you are correct, you will be given the highest of accolades: your name mentioned here, next issue.

Correct answers to last month’s puzzle:
Arash Ushani
Berit Johnson

(See last month’s puzzle and solutions at http://franklyspeakingnews.com/node/143)

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Solution:
Consider the graph of one’s value as one traverses the circle. Perhaps it dips below the axis, perhaps not. If it doesn’t, we are done. If it does, locate the minimum value achieved and start one’s traversal at the point corresponding to that minimum. The values achieved starting there will never be negative.