Q7
How many surfers distributed their stars among exactly 2 bloggers?
Answer: 2
From the chart: A got M 10, N 25, P 5, Y 5, giving a total of 45. B got P 25, Y 20, giving a total of 45.
The total number of stars is 6 × 30 = 180. All four bloggers got the same total (fact 2), so each got 45, and C + D = 90.
What is left for C and D from each surfer: M 20, N 5, O 30, P 0, X 30, Y 5.
- Y: D > C (fact 5), so D gets 5 and C gets 0.
- M: A got 10 and B got 0. C and D must both get amounts different from 0 and 10 (fact 3), and they add to 20. So {C, D} = {5, 15}.
- Fact 4: M, N, P and Y each gave stars to at least two bloggers. So the two surfers who gave everything to one blogger are O and X. Each gives 30 to C or to D.
- C must total 45. If C got both 30s, C would have at least 60. If C got neither, C would have at most 15 + 5 = 20. So C gets exactly one 30 and needs 15 more from M and N. Only M = 15, N = 0 works.
| M | N | O | P | X | Y | Total | |
|---|---|---|---|---|---|---|---|
| A | 10 | 25 | 0 | 5 | 0 | 5 | 45 |
| B | 0 | 0 | 0 | 25 | 0 | 20 | 45 |
| C | 15 | 0 | 30 or 0 | 0 | 0 or 30 | 0 | 45 |
| D | 5 | 5 | 0 or 30 | 0 | 30 or 0 | 5 | 45 |
Which of O and X went to C (and which to D) cannot be fixed.
Number of bloggers each surfer gave stars to:
- M: A, C, D (3)
- N: A, D (2)
- O: one blogger (1)
- P: A, B (2)
- X: one blogger (1)
- Y: A, B, D (3)
Only N and P split their stars between exactly two bloggers.
Answer: 2