Tom, Dick and Harry each chose two different 2-digit semiprime numbers (a semiprime is the product of exactly two different prime numbers). Any given semiprime may have been used by more than one of these characters, but any one person chose two different ones. In each of the three cases, the sum of the two semiprimes was a 3-digit semiprime.

Also, in each instance, if the units digits of the two semiprimes were swapped, to be paired with the other tens digit, the result was two semiprimes that were different from the original pair. Of course they added up to the same 3-digit semiprime as the first pair. In each of Tom's, Dick's and Harry's three semiprimes, none of the six primes from which they were made were the same: for example, if Tom's semiprimes were 93 and 94 adding up to 187, all six primes (3 and 31 for 93, 2 and 47 for 94 and 11 and 17 for 187) would all be different, satisfying this rule. The same was also true for each of the three people after the swap of units digits between the two 2-digit numbers.

Tom's, Dick's and Harry's 3-digit sums were all different, but Tom's and Harry's 3-digit sums did share a common prime factor, while Dick's did not.

What were Dick's numbers?
What were the other two sets of numbers?

My first comment in this thread contains the thoughts around which this reply is built.


Within the spreadsheet numerous things were happening.  Each Px pairing is multiplied to produce an original 2-digit semiprime.  After the shuffle of units the recreated Number was tested for semiprimacy.

The pairings were added and then tested for semiprimacy as well as checking for a common factor.

                        2-digit     2-digit        3-digit
         Px        Original   Shuffled      semi-prime
                        T U           t  u                H T U
Tom
P1       5           9 5           9 1                   146        2 * 73
P2     19
P3       3           5 1           5 5
P4     17

Dick
P5        2          3 4           3 5                   129        3 * 43
P6      17
P7      19          9 5           9 4
P8        5

Harry
P8        3           5 7          5 5                    142        2 * 71
P10    19
P11    17           8 5          8 7
P12      5

                     
Note, of the original semiprimes selected Tom and Dick chose 95 which was allowed in the body text.

Sets of Primes:
Tom:     2, 3, 5, 17, 19 and 73
Dick:     2, 3, 5, 17, 19 and 43
Harry:   2, 3, 5, 17, 19 and 71.

Tom and Harry's 3-digit semiprime shared a 2.

Posted by brianjn on 2011-01-20 20:38:57