MATH STINGER # 68

by 
Dr. Joel C. Fowler 
Assistant Professor of Mathematics 

A game between two players, A and B, is played with four sided dice marked with the numbers 1, 2, 3, and 4. 
In each round each player chooses to roll either one or two of these dice. Their score is the average of the numbers 
that come up on their dice (i.e. either a single number or the average of two numbers, depending on how many dice 
that player chose to throw in that round). The player with the highest average wins that round. After two rounds of 
play A and B had each thrown a total of three dice and A had lost both rounds. Looking at the record of t
he throws, however, A noticed the following surprising fact. Although A had lost each round individually, if all three rolls for 
both players were averaged over the two rounds A would have won that single "super" round. The puzzle for this issue 
s to determine from the above information exactly how many dice A and B tossed in each round and exactly what 
numbers came up on those tosses. You should find that there is only one possible answer apart from the order of 
the rounds. There are two parts to the puzzle. The first is finding any set of rolls that fit the conditions. The second 
is proving that your set is the only possible solution.

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