## Infinity Pizzeria

Infinity Pizzeria offers an unlimited variety of toppings for its pizzas. One day Infinity receives an order for 1000 two-item pizzas. You notice that any three pizzas on the order have a common topping. Show that there is a topping common to all of the pizzas on the order.

## Solution

Suppose the first pizza is mushroom and pepperoni. If there is no topping common to all of the pizzas then there is some pizza, say A, that doesn't have mushrooms and some pizza, say B, that doesn't have pepperoni. But then the first pizza together with pizzas A and B are three pizzas without a common topping. (If pizzas A and B were the same pizza then this pizza, the first pizza, and any other pizza would be three pizzas without a common topping.)

Rules:

1. Answers must be written neatly on 8.5 by 11 inch paper.
2. Answers must be submitted to Dr. Buchanan either at his office (Wickersham 113) or placed in his department mailbox by the department secretary.
3. The contest will open on July 20, 1998.
4. The first complete and correct answer will be awarded the sum of \$5.00CND (five Canadian dollars).
5. All complete and correct answers will be listed on a "Mathematical Puzzle List of Distinction" to be posted outside Wickersham 113 as well as on a "Mathematical Puzzle List of Distinction" web page.

Page maintained by: Robert.Buchanan
Robert.Buchanan@millersville.edu

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