document.write( "Question 1145247: Maria have NINE NUMBERS 1, 2,3,4,5,6,7,8 and 9 she adds 2 to some of them and 5 to the others.what is the smallest number of different results she can obtain? \n" ); document.write( "

Algebra.Com's Answer #766480 by ikleyn(52794) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Maria $\"highlight%28cross%28have%29%29\"$ has NINE NUMBERS 1, 2,3,4,5,6,7,8 and 9 she adds 2 to some of them and 5 to the others.
\n" ); document.write( "what is the smallest number of different results she can obtain?
\n" ); document.write( "~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Let \"a\" denotes any number of the set {1,2,3,4,5,6,7,8,9}, and let \"b\" denotes any number of the same set.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Maria forms new numbers a'= a+2 and b'= b+5.  She wants to select some subset A for \"a\"s \r\n" );
document.write( "(and automatically to get its complement B for \"b\"s)\r\n" );
document.write( "\r\n" );
document.write( "in a way that the number of distinct differences a'-b' would be minimal.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The difference a'-b' is \r\n" );
document.write( "\r\n" );
document.write( "    a' - b' = (a+2) - (b+5) = (a-b) - 3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, she actually wants to select the subset A and its complement B in a way to have MAXIMUM possible \r\n" );
document.write( "number of solutions of the equation\r\n" );
document.write( "\r\n" );
document.write( "    a - b = 3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In other terms, she wants to have maximum possible number of pairs (a,b), \"a\" belongs to A, \"b\" belongs to B =  with  a - b = 3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now, after this series of reformulations, the answer is OBVIOUS:\r\n" );
document.write( "\r\n" );
document.write( "    take the pairs (a,b), where \"a\" is 3 units greater than \"b\", and try to collect maximum possible number of such pairs.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "My answer is:  A = {4, 5, 6} U {7,8,9},\r\n" );
document.write( "\r\n" );
document.write( "               B = {1, 2, 3}. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, my A is the union of two subsets,  X = {4, 5, 6}  and Y = {7,8,9}.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In all, she will have 6*3 = 18 different pairs (a',b');  of them,  3 pairs will give zero difference (a'-b'). \r\n" );
document.write( "\r\n" );
document.write( "These three pairs are (4,1), (5,2) and (6,3).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At this selection, the number of pairs with NON-ZERO difference (a'-b') is 18-3 = 15, and, according \r\n" );
document.write( "to the solution, this number is maximal.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Notice that the solution, i.e. selection/choice of sets A and B is not unique.\r\n" );
document.write( "\r\n" );
document.write( "Maria can select them in different way, but can not make \"the number of different results\" smaller.\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "/\/\/\/\/\/\/\/\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Below is my addition after posting the above solution.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Still I think that the problem formulation is incomplete and missed something.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    For example, she can put A = {1,2,3,4,5,6,7,8,9} and B = {empty}.\r\n" );
document.write( "\r\n" );
document.write( "    Then the number of pairs and differences is 0 (zero, ZERO).   (!)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Another choice is to select A of 8 elements and B (as the complement) of only 1 element.\r\n" );
document.write( "    Then the number of different pairs is 8 --- very small, but the mathematical meaning of the problem totally disappears. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Third choice is to select A of 7 elements and B of remaining 2 elements.\r\n" );
document.write( "    Then the number of different pairs is 7*2 = 14 only (!), but the the mathematical meaning of the problem \r\n" );
document.write( "    disappears, again (at least, in half).\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );