Guess the digits
A neutral player randomly extracts two balls (without replacement) from a ballot box. The box contains ten balls labeled from to , no two balls have the same number. He then reveals to player only the product of these two digits, and to player only their sum . The following discussion takes place in this order:
- : "I don't know the two digits";
- : "I don't know them either";
- : "Now I do know them";
- : "So do I".
What are the two digits?
If is the product and the sum, give the answer as .
The answer is 35.
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1 solution
There are 45 (10 x 9 / 2 ) possible combinations for the two balls.
A cannot solve the problem immediately. Therefore the Product which A has heard, can be factorised in more than one way from the set {0,1,2,3,4,5,6,7,8,9}. The product is one of {0, 6, 8, 12, 18, 24}. B knows this
[note a lot of composite numbers such as 16 and 21 can be factorised one way only, from the set. square factorisation is not possible as the balls are not the same].
If A has heard 0, B could have heard any one of 5,6,7, 8 or 9 or more. A can do nothing with the fact that B cannot solve the problem and statement 3 will fail. The product is not 0. Of course A knows this. B does not know this .
Therefore A was told that the product is one of {6, 8, 12, 18, 24}. A knows that the number pair is one of (2,3) (1,6) (2,4) (1,8) (3,4) (2,6) (3,6) (2,9) (4,6) or (3,8). Respectively, the sums of these pairs which would be told to B are 5, 7, 6, 9, 7, 8, 9, 11, 10, 11. A knows B has received one of these sums.
Notice that the only sum in this list which can be totalled from 2 numbers from the set {0,1,2,3,4,5,6,7,8,9} in exactly one way is the number 10. For example, 5 is 0+5 and 1+4. Therefore if B had heard "10", he would be able to solve the puzzle before his first statement. A knows this [note, from B's point of view, 0 is still a possible product].
Therefore the pair (4,6) has just been excluded. And yet we know this little gem is a massively useful piece of information for A. All the products except 24 remain ambiguous. Therefore the product is 24, and A now knows the balls are (3,8).
B can observe this in the same way we can, and can also now solve.
(3 x 8) + (3 + 8) = 35