I had seen this brainteaser back in 2003 and was able to solve it with a little deductive reasoning. The reason I am posting this now is because this teaser had recently come back up again with a few friends.
My first deduction came when I looked at the prime roots of 711 , which were 3,3,and 79. I realized that one of the solution numbers had to be a whole multiple of .79 because any non-whole multiple of .79 could not satisfy the multiplicative side of the answer (eg leaving a long tail decimal). My second deduction came when thinking of combinations of numbers that would add to a number ending with 1, while not trying to spin the multiplicative side of the answer out of control. Well, 5 and 6 add to 11 and multiple to 30, ending in ZERO, which is just what I wanted to see.
At this point, I was all-in that one number ended in 6, another number ended in 5, and the other two numbers ending in zero. After that it was just a little plug and chugging. The whole multiples of .79 that end in the 6 or 5 are 3.95 or 3.16, as there are no whole number multiples of .79 ending in zero. After testing 3.95 for possible solutions starting with 1.20 (as a possible solution number and testing for the other 2), it became quite evident 3.95 was too big of a number and that I was not going to find three numbers that added to 3.21 (7.11-3.95) and multipled to 1.8 (7.11/3.95).
At this point, I turned my attention to 3.16 and started goal testing at 1.75, lowering my test number by increments of .05 each time. To my much delight, when I got down and tested 1.50, boom a hit and answer. 3.16, 1.50, 1.25, 1.20
Charlie O
