For those confused why the two day solution doesn't work please see the below: Day 1 Mark's perspective Mark 12 Rose 6 8 Rose knows Mark has one of 10 12 14 If I had 14 I would say 20. If I had 10 I would say 18. I have 12 so pass
Rose perspective Mark 10 12 Rose 8 Mark knows Rose has one of 6 8 10 If Mark had 14 he would have said 20 so therefore Mark knows if I had 6 I would say 18. By passing I can eliminate 6 from the options he thinks I have. If I had 10 I would say 20. I have 8 so pass.
Day 2 Mark perspective Mark 12 Rose 8 As Rose passed Mark can eliminate 6 as a possibility and now knows that Rose has 8 so calls 20
If Rose actually had 6 bars then it plays out slightly differently but you seemingly reach the answer in the same amount of time:
Day 1 Mark's perspective Mark 12 Rose B 6 8 Rose knows Mark has one of 10 12 14 If I had 14 I would say 20. If I had 10 I would say 18. I have 12 so pass
Rose's perspective Mark 12 14 Rose 6 Mark knows Rose has one of 4 6 8 If Mark had 10 he would have said 18 so therefore if I had 8 I would say 20. By passing I can eliminate 8 from the options he thinks I have. If I had 4 I would say 18. I have 6 so pass
Day 2 Mark's perspective Mark 12 Rose 6 As Rose passed Mark can eliminate 8 as a possibility and now knows that Rose has 6 so calls 18
However, in both instances the actions taken by both parties are the same yet the conclusions reached are different.