Counting, Probability & Logic: GK UKMT Intermediate GK 2020 Q19

GK 2020 Q19

GKCurrent spec6 marksCounting, Probability & Logic

19. In the final of a dancing competition, each of the three members of the jury gives each of the five competitors 0 points, 1 point, 2 points, 3 points or 4 points. No two competitors get the same mark from any individual judge. Adam knows all the sums of the marks and a few single marks, as shown. How many points does Adam get from judge III?

	Adam	Berta	Clara	David	Emil
I	2	0
II		2	0
III
Sum	7	5	3	4	11

Grey Kangaroo 2020, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk

Solution (UKMT)

The table can be partially completed as follows. Berta scored 5 points in total. Therefore her score from judge III is 3. Clara’s total is 3. Therefore she cannot have been given 4 by any of the judges. David’s total is 4. He cannot have been given a score of 4 by any judge, since, if so, both the other two judges must have given him 0. This is impossible, as judges I and II give 0 to Berta and Clara, respectively. Therefore judge I gives 4 to Emil, and judge II gives 4 to either Adam or Emil. Emil’s total is 11. So if he gets 4 from judges I and II, he gets 3 from judge III which is not possible as judge III gave Berta 3. Hence judge II gives 4 to Adam. Because Adam’s total is 7, it now follows that Adam gets a score of 1 from judge III.
(Note: the final four scores are not uniquely determined.)

	Adam	Berta	Clara	David	Emil
I	2	0			④
II	④	2	0	①	③
III	①	③			④
Sum	7	5	3	4	11