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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.

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 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.

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

Please note this solution only works because the numerical gap between the two possibilities of total bars is smaller than the numerical gap between the two sets of bars that Mark and Rose can see and Mark knows this to be true. If the gap was the same or smaller then the longer method would work but this would not.

If Rose actually had 6 bars then it plays out slightly differently but you 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

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.

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 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

Please note this solution only works because the numerical gap between the two possibilities of total bars is smaller than the numerical gap between the two sets of bars that Mark and Rose can see and Mark knows this to be true. If the gap was the same or smaller then the longer method would work but this would not.

If Rose actually had 6 bars then it plays out slightly differently but you 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

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

Please note this solution only works because the numerical gap between the two possibilities of total bars is smaller than the numerical gap between the two sets of bars that Mark and Rose can see and Mark knows this to be true. If the gap was the same or smaller then the longer method would work but this would not.

If Rose actually had 6 bars then it plays out slightly differently but you 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