Optimising Konsistency

Probability Level 3

Konsistent Karl is taking a specific contest with the following rules:

  1. There are 15 problems.
  2. The n n th problem is worth n n points.
  3. The contest is 15 minutes long.

Karl can solve the first five problems in one minute each, the next five in two minutes each, and the last five in three minutes each. What is the maximum possible score Karl can earn?


The answer is 70.

Sharky Kesa by Sharky Kesa , 19, United Kingdom

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2 solutions

Marco Brezzi
Aug 20, 2017

Assign to each problem a value in the following way:

value ( problem n ) = n time taken \text{value}(\text{problem }n)=\dfrac{n}{\text{time taken}}

This table summarizes the values

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 3 3.5 4 4.5 5 11 3 \frac{11}{3} 4 13 3 \frac{13}{3} 14 3 \frac{14}{3} 5

Solving the first 6 6 most valuable problems, namely problems 5 , 10 , 15 , 14 , 13 , 9 5,10,15,14,13,9 takes 14 14 minutes, we now have 1 1 minute left to solve problem 4 4 . Hence the maximum we can obtain is

5 + 10 + 15 + 14 + 13 + 9 + 4 = 70 5+10+15+14+13+9+4=\boxed{70}

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Sharky Kesa
Aug 20, 2017

Note that the last 5 questions will take up all of Karl's time and give him 11 + 12 + 13 + 14 + 15 11+12+13+14+15 points. However, 11 + 12 11+12 is less than 8 + 9 + 10 8+9+10 , and they would take the same amount of time to solve. Furthermore, 8 8 is less than 4 + 5 4+5 , and they would take the same amount of time.

Thus, the maximum score Karl can attain is 4 + 5 + 9 + 10 + 13 + 14 + 15 = 70 4+5+9+10+13+14+15=70 .

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