ags@pucc-k (Seaman) (11/23/83)
First, the corrected statement of the problem (as I gave it earlier):
The referee tells S and P that he has chosen two integers, each greater
than 1 but not greater than 100. He gives the sum to S and the product
to P.
(1) S announces that P does not know the sum.
(2) P says he now knows the sum.
(3) S says he now knows the product.
As I have already pointed out, the original instructions from the referee
(i.e. the stated bounds on the two integers) have a profound influence on
the solution. The upper bound can be as small as 62 without affecting the
outcome. If the upper bound is 61, there is no solution. If the upper
bound is on the SUM of the two integers, rather than on the integers
themselves, there is no unique solution.
Here is the solution for the problem as I have stated it. When S
makes statement (1), we can conclude that the sum is one of {11, 17, 23,
27, 29, 35, 37, 41, 47, 53}. Each of these numbers has the property that
every corresponding product can be factored in more than one way AS A
PRODUCT OF TWO INTEGERS WHICH SATISFY THE REFEREE'S CONDITIONS. The contents
of this list of possible sums is highly sensitive to the original conditions
stated by the referee.
When P makes statement (2), we know he has a product that can correspond
to exactly one of the sums in the list above. There are many such products:
{18, 24, 28, 50, 52, 54, 76, 90, ... 700, 702}. Note that 70 is not on the
list -- 70 can be factored as 7 * 10 or as 2 * 35, yielding sums of 17 or 37,
both of which are on the list of possible sums. Therefore the claimed
solution which was posted earlier is incorrect.
Finally, when S makes statement (3), we know that his sum can correspond to
only one of the possible products. There is exactly one sum with this
property: 17, corresponding to the product 52. The original numbers are
4 and 13.
Dave Seaman
..!pur-ee!pucc-k!agsags%pucc-k@phs.UUCP (Seaman) (11/23/83)
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Date:Wed, 23-Nov-83 10:54:18 EST
Organization:Purdue University Computing Center
First, the corrected statement of the problem (as I gave it earlier):
The referee tells S and P that he has chosen two integers, each greater
than 1 but not greater than 100. He gives the sum to S and the product
to P.
(1) S announces that P does not know the sum.
(2) P says he now knows the sum.
(3) S says he now knows the product.
As I have already pointed out, the original instructions from the referee
(i.e. the stated bounds on the two integers) have a profound influence on
the solution. The upper bound can be as small as 62 without affecting the
outcome. If the upper bound is 61, there is no solution. If the upper
bound is on the SUM of the two integers, rather than on the integers
themselves, there is no unique solution.
Here is the solution for the problem as I have stated it. When S
makes statement (1), we can conclude that the sum is one of {11, 17, 23,
27, 29, 35, 37, 41, 47, 53}. Each of these numbers has the property that
every corresponding product can be factored in more than one way AS A
PRODUCT OF TWO INTEGERS WHICH SATISFY THE REFEREE'S CONDITIONS. The contents
of this list of possible sums is highly sensitive to the original conditions
stated by the referee.
When P makes statement (2), we know he has a product that can correspond
to exactly one of the sums in the list above. There are many such products:
{18, 24, 28, 50, 52, 54, 76, 90, ... 700, 702}. Note that 70 is not on the
list -- 70 can be factored as 7 * 10 or as 2 * 35, yielding sums of 17 or 37,
both of which are on the list of possible sums. Therefore the claimed
solution which was posted earlier is incorrect.
Finally, when S makes statement (3), we know that his sum can correspond to
only one of the possible products. There is exactly one sum with this
property: 17, corresponding to the product 52. The original numbers are
4 and 13.
Dave Seaman
..!pur-ee!pucc-k!ags