Question 1204042: There are 18 questions in a math competition. The scores of each question are allotted in the following ways: 1 mark will be given for a correct answer, 1 mark will be deducted from a wrong answer and 0 mark will be given for a blank answer. find the minimum number of candidate(s) to ensure that 3 candidates will have the same scores in the competition
Answer by ikleyn(52903)   (Show Source):
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There are 18 questions in a math competition. The scores of each question are allotted
in the following ways: 1 mark will be given for a correct answer, 1 mark will be deducted
 for a wrong answer and 0 mark will be given for a blank answer.
find the minimum number of candidate(s) to ensure that 3 candidates will have the same scores in the competition
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First, the set of possible scores is the set of all integer numbers from -18 to 18
{ -18, -17, -16, . . . , 0, 1, 2, 3, . . . 16, 17, 18 }.
It is so because for all 18 correct answer the score is 18;
for 17 correct and 1 blank the score is 17;
for 16 correct and 2 blank the score is 16, same as for 17 correct and 1 incorrect;
for 15 correct and 3 blank the score is 15;
and so on till 18 incorrect, which gives the score -18.
Thus we have 18 + 18 + 1 = 37 possible different scores.
In order to ensure that 3 candidates will have the same scores in the competition,
2*37 + 1 = 75 candidates should participate (according to the pigeon principle).
ANSWER. In order to ensure that 3 candidates will have the same scores in the competition,
minimum 75 candidates should participate.
Solved.
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