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billproj - MINI PROJECTS IN THEORY 1 Dishwasher Game 2...

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MINI PROJECTS IN THEORY 1. Dishwasher Game 2. Reciprocals 3. Divisibility problem
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DISHWASHER GAME Given: Numbers a , b , a set of tiles, and a threshold t . 1. Players I and II take turns placing tiles on the a × b grid. 2. Stop when a player cannot move. 3. If the number of open spaces is t then II wins, else I wins.
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WHAT TO ASK 1. Given a , b tiles and t , who wins? 2. Can approach mathematically and via computer program. 3. For a = 1 and b = n can probably be worked out mathematically. 4. Write a program to play this well (likely not perfect) using AI techniques.
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RECIPOCAL PROBLEM From MD Math Competition: (a) The equality 1 2 + 1 3 + 1 6 = 1 and 1 2 + 1 3 + 1 7 + 1 42 = 1 express 1 as the sum of three (respectively four) distinct positive integers. Find five positive integers a < b < c < d < e such that 1 a + 1 b + 1 c + 1 d + 1 e = 1 . (b) Prove that for any integer m 3 there exists m positive integers d 1 < d 2 < · · · d m such that 1 d 1 + · · · + 1 d m = 1 .
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