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# MATH PROB SOLV MA 310

UK

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This 7 page Class Notes was uploaded by Kennith Herman on Friday October 23, 2015. The Class Notes belongs to MA 310 at University of Kentucky taught by C. Lee in Fall. Since its upload, it has received 9 views. For similar materials see /class/228139/ma-310-university-of-kentucky in Mathematics (M) at University of Kentucky.

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Date Created: 10/23/15

Tools MA310 Spring 2001 Tools 2 1 Winning Positions We have considered some two person games A typical game has a certain usually nite number of positions from each one of which either player has a certain usually nite number of moves to other positions If either player has the same choices of moves from each position then the game is called impartial So Tic Tac Toe is not impartial because di erent players must use di erent symbols but the game in which players take turns removing one or more stones out of one of three piles of stones is impartial One way to try to analyze an impartial game is to identify each position as either winning or losing The goal positions of the game is of course winning By working backwards77 from the goal positions we can identify other positions as winning or losing by the following requirements 1 If all moves from a given position P lead to losing positions then P is a winning position 2 If there is at least one move from a given position P that leads to a winning position then P is a losing position If the total number of possible positions in the game is nite and if there are no sequences of moves that allow you to return to a previous position in the game then the above method can in principle identify the winning positions But certain games may be quite complicated and we may instead want to gure out and prove some sort of criterion to identify the winning positions 2 Symmetry Some of our games have winning positions described by symmetry Points A and B are symmetrical with respect to a point P if P is the midpoint of the line segment joining A and B Two points A and B are symmetrical with respect to a line Z if the line Z is the perpendicular bisector of the line segment joining A and B 3 Remainders Given integer a and positive integer b there are integers q and r such that a qb r and 0 S r lt b q is called the quotient and r the remainder In this case we can say that a is congruent to r mod b It is not hard to prove that if a has remainder r when divided by b then so does a b and so does a plus any integer multiple of b Tools 4 Magic Squares An 71 gtlt 71 magic square is an arrangement of numbers7 usually the integers from 1 to 7127 in an n gtlt n array such that the sum of the numbers in each row7 column7 and diagonal is the same Except for rotating or re ecting it7 there is only one 3 gtlt 3 magic square 8 1 6 3 5 7 4 9 2 Connections 1 Music Tuning a Piano A geometric sequence is a sequence of the form aarar2ar3 That is to say the rst term is some number a and thereafter each term is obtained from the previous one by multiplying by a speci c number 7 For example 1545 135 405 is a geometric sequence The 88 keys of a piano are tuned according to a geometric sequence The note A desig nated A4 above middle C designated C4 has frequency 440 HZ The C note C5 above middle C has twice the frequency of middle C There are 13 notes between C4 and C5 inclusive Determine the frequencies of these 13 notes explaining your reasoning Round your answers to two decimal places 1 0 C4 2 on 3 D 4 D11 5 E 6 F 7 F11 8 G 9 on 10 A 440 Hz 11 A11 12 B 13 c 05 To the human ear two notes played together tend to sound more harmonious if the ratios of their frequencies is a ratio of two small integers For each of the fractions below nd a note above C4 so that the ration of the frequency of this note to the frequency of C4 is approximately equal to that fraction Connections 1 21 2 32 3 43 4 54 5 53 6 65 Here is a website with a Virtual keyboard to test notes and Chords WWWbg orgbg CustomresourcesftpClientjtpstmusiCpianoindexhtm Connections Mathematical Magic TwentySeven Card Trick This trick is described by Eric Shrader on Cut the Knot7 wwwcut the knotorgarithmeticrapidCardTrickshtml My grandfather taught me this simple trick when l was young Deal out 27 cards face up into a grid of 9 rows and 3 columns Do this by dealing out 3 cards horizontally in a row7 then 3 more cards just below the rst 37 then 3 more7 etc7 until you have 9 rows lt7s best to overlap cards in a column so that the columns aren7t so long just make sure that the values of all cards are visible Discard the remaining cards Only these 27 will be used to play Ask a spectator to mentally pick a card and remember it Ask him to tell you only which of the 3 columns it is in Collect the 27 cards into a deck Gather them vertically such that the column containing the spectators card is second Pick up cards from the top of the column to the bottom7 keeping them in the same order For example7 if the spectator tells you column 17 then rst gather column 2 or 37 then column 17 and then the remaining column When you7re done7 the top card of the deck should now be the top card of the column you collected rst7 followed by the rest of that column in order Then the 10th card of the deck will be the card at the top of the column containing the spectators card7 etc If you overlapped the cards as suggested7 gathering and keeping them in the correct order is easy Now deal the cards again Deal exactly as described aboveihorizontally across the rows rst Again7 ask him to tell you only which column contains the card Pick up the columns verticallyiexactly as aboveimaking sure that the column con taining the card is picked up second Finally deal them out a third time in exactly the same way7 ask which column contains the card7 and gather them up in the same manner The spectator7s card will now be the 14th one in the deck To add drama7 I usually deal out the cards one at a time face down I dont make it obvious that l7m counting and I dont look at any of the cards lnstead7 l7ll hesitate over certain cards7 pretending to get a vibe77 from them7 and then l7ll nally settle on the right one This seemed like pure magic when I rst learned it Only when I got older did I realize that its actually simple math Do you see how it works7 Connections 2 Eight Card Trick This is a variant of the previous trick This time use 8 cards though it can be modi ed for more cards and two columns The magician asks one spectator to name aloud a whole number N from 0 to 77 and a second spectator to select a card Cards are dealt and collected three times as before7 but now7 at the very end after the nal gathering7 the magician is able to count off precisely N cards7 revealing the chosen card as the next one By what method does he stack the columns to force the chosen card to the desired position

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