Gunther's average on ten quizzes is 7.8. Each score is a positive whole number less than or equal to 10. He remembers that he scored at least one 5, at least 3 7's, at least two 9's and at least one 10. What is the sum of all the distinct possible values for Gunther's median quiz score?
Ok, I had no idea how to do this so I just wrote a python script. 
It appears the answer is 7+7.5+8=22.5
Whats an alternative answer to brute forcing though?
+2440 If you can write a computerized algorithm to solve it, then you would have to at least have some idea how to solve it.
Your question: Whats an alternative answer to brute forcing though? (sic)
The answer: Use an optimized partitioning algorithm.
Determine the sum of the missing scores by solving (10 + 10 + 10 + 9 + 9 + 7+ 7 + x)/10 =7.8. (x=24).
Find the partitions in set of three summing to 24. There are 48 of them; consider only the partitions that have values of 10 or less.
(This is easy enough to do without a computer, for this problem.)
10+10+4
10+9+5
10+8+6
10+7+7
9+9+6
9+8+7
8+8+8
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Calculate the medians:
{10, 10, 10, 9, 9, 7, 7, 7, 5, 4} Mean= 7.8. Median = 8.0
{10, 10, 9, 9, 9, 7, 7, 7, 5, 5} Mean= 7.8. Median = 8.0
10, 10, 9, 9, 8, 7, 7, 7, 6, 5} Mean= 7.8. Median = 7.5
10, 10, 9, 9, 7, 7, 7, 7, 7, 5} Mean= 7.8. Median = 7.0
10, 9, 9, 9, 9, 7, 7, 7, 6, 5} Mean= 7.8. Median = 8.0
10, 9, 9, 9, 8, 7, 7, 7, 7, 5} Mean= 7.8. Median = 8.0
10, 9, 9, 8, 8, 8, 7, 7, 7, 5} Mean= 7.8. Median = 8.0
Extract unique medians and calculate the sum (8+7.5+7) = 22.5
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I do wonder if StudyDaddy is scraping this forum.
GA
Oh, I see it was just "intuitive" brute forcing :P. I guess I could've manually solved it that way but I was too lazy. I don't know what I was expecting, some ingenious algorithm of some sort, but I guess some things are simple enough to just calculate.
Thanks! 
+2440 You’re welcome. It’s great to have a student who actually participates in the Q&A. You may want to consider joining the forum. There are mathematicians here with advanced skills in most branches of mathematics. Some also have skills in writing computerized algorithms for “brute forcing,” sorting, and Monte Carlo simulations.
GA
+118608 Thanks Ginger,
My answer is similar to yours with a couple of extra observations.
The known scores are 5 7 7 7 9 9 10
The median of these is 7 which is in the 4th position.
When 3 more scores are added the median will be the average of the 5th and 6th scores.
So the 5th and 6th ordered scores are the only ones that will matter.
Yes the 3 missing scores have to add to 24
24/3 = 8
So the average of these three scores must be 8
If all of these scores are 8 then the 5th and 6th will be 5 777 8 8 8 99 10 median is 8
If one of them is 8 then one will be bigger than 8 and the other smaller
then the 5th and 6th will be 5*77 78 99*10 median is 7.5
If two are bigger than 8 and one is smaller, then the 5th and 6th will be 5*777 99**10 median is 8
If two are smaller than 8 and one is bigger, then the 5th and 6th will be 5**777 99*10 median is 7
So the possible medians are 8, 7.5, and 7 The sum is 22.5
