To a set of 4 numbers B adds a card easily recognized
as a 'guide" : the number that changes the average
of the other four more than any other card (out of final 5)
would do .
For instance : if 1,2,3,4 were selected - clearly any card
greater than 5 would be recognized as B's card.
Now out of 95 possible numbers a subset of 24
should be defined and used with a predefined
correspondence table: 1st ABCD,2nd ABDC ,3rd ACBD,
...24th DCBA. in our case 6==>ABCD ,7==>ABDC 8==>ACBD
For set (10,15,37,99) B should use a number between 75 and 98
75 as ABCD 98 as DCBA / etc
When A sees a set 10,15,37, 77, 99 he might hesitate
between 77, 99 as to decision which is the "guide" number
but finally(wise guy) he will understand that 77 is the right
decision , because for the set 10,15,37, 77 ther are no 24
consecutive choices between 78 and 98 , and B would choose a
number between 38 and 61 (10,15,37, 77 average 34.75).
It seems to me that there will be always a 24- number- long
uniquelly defined subset to enable coding and decoding with
It is very late here,and although I like this puzzle
and feel that I can perform the trick- I leave it to someone else
to finalize a formal proof.
Hope it can be done.