document.write( "Question 1203324: Eighteen points are equally spaced on a circle, from which you will choose a certain number at random. How many do you need to choose to guarantee that you will have the four corners of at least one rectangle? \n" ); document.write( "

Algebra.Com's Answer #838787 by ikleyn(52847) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Eighteen points are equally spaced on a circle, from which you will choose a certain number at random.
\n" ); document.write( "How many do you need to choose to guarantee that you will have the four corners of at least one rectangle?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Let the points on the circle are numbered consequently from 1 to 18, inclusive.\r\n" );
document.write( "\r\n" );
document.write( "Let call them  , , , . . . , .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Notice that if a rectangle is inscribed in a circle, then its diagonals are DIAMETERS\r\n" );
document.write( "of the circle. So, the opposite verices of the rectangle are the opposite points on the circle.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus, diagonals of our rectangle connect opposite points on the circle.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Of 18 points , , , . . . , , we have 9 pairs with opposite points\r\n" );
document.write( "(,), (,), . . . , (,).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now my statement is that the ANSWER to the problem's question is 11 points of 18.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Indeed, among these 11 points, there is at least one pair of opposite points.\r\n" );
document.write( "Removing this pair, you have remaining 16 points and 11-2 = 9 selected points.\r\n" );
document.write( "Among these 9 selected points, there is at least one another pair of opposite points.\r\n" );
document.write( "\r\n" );
document.write( "So, 11 points are ENOUGH, even in the WORST CASE.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From the other side, taking 10 points , , , . . . , , it is clear, that\r\n" );
document.write( "among them, there is ONLY ONE pair (,) of opposite points to make a diameter/(=a diagonal),\r\n" );
document.write( "and there are NO points to make another pair/(=another diagonal).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  11 points are needed to choose to GUARANTEE \r\n" );
document.write( "         that you will have the four corners of at least one rectangle.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "--------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "I want to make three POST-solution notes.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "    (1)  This problem is a TWIN to another popular classic problem about 18 socks in a drawer. \r\n" );
document.write( "         These 18 socks are of 9 different colors, are separated and lie in disorder.\r\n" );
document.write( "         How many socks should a person take from a drawer without looking (without seeing) \r\n" );
document.write( "         to guarantee that among selected socks there are at least two proper pairs?\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    (2)  The condition that the points are uniformly distributed on the circle DOES NOT MATTER\r\n" );
document.write( "         in this problem, and I did not used it in my solution. What really does matter is that \r\n" );
document.write( "         these  18 points are, actually, 9 pair of opposite points.\r\n" );
document.write( "         The words \"uniformly distributed\" are used to disguise this fact and to distract \r\n" );
document.write( "         attention of a reader from using this conception of opposite points.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    (3)  Interesting, that in the Internet, there is a solution to this problem, \r\n" );
document.write( "         generated by the artificial intelligence chatGPT\r\n" );
document.write( "\r\n" );
document.write( "https://www.quora.com/Eighteen-points-are-equally-spaced-on-a-circle-from-which-a-number-are-chosen-at-random-How-many-must-you-choose-to-guarantee-that-you-will-have-to-the-four-corners-of-at-least-one-rectangle\r\n" );
document.write( "\r\n" );
document.write( "              +--------------------------------------------------+\r\n" );
document.write( "              |    AND THIS SOLUTION IS INCORRECT ( !  !  ! )    |\r\n" );
document.write( "              +--------------------------------------------------+\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "      It confirms my idea that so called AI in its contemporary status \r\n" );
document.write( "      is simply big (VERY BIG, huge) GARBAGE BIN.\r\n" );
document.write( "\r\n" );
document.write( "      Regarding this so called AI in Math education, it is simply a way \r\n" );
document.write( "      to make money without taking any responsibility for a correctness\r\n" );
document.write( "      of solutions and quality of final product.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "///////////////////\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To see many other similar (and different) solved problems, look into the lesson\r
\n" ); document.write( "\n" ); document.write( " - In the worst case \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Learn from good sources ( ! ) Then everything will be OK.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );