Discrete Mathematics Mathematics Department Phillips Exeter Academy Exeter, NH August 2012 Discrete Mathematics 1.Agraph is a network of dots and lines. What •.. ... • ... ... ........•.......................................•....... •....................•.. is the meaning of the graph shown at right? ........... ..... •. ... ....... .. ...... •......................•...... ..... ... ....... .. ..... ........ .... ... ..•...... .. ....... ..... .... .... ... ........ ............................•.. ... .. .... ... ... ... .... .•......... ..... What do the dots represent? Why are some . ... .. .... ... ... ... .... ... ......... .... ... ... ....... ... .... .. ... ......... .... ... ....... ... .... ... .. • ...... ... .... ....•........ ... •...... ... ... • dots joined by segments and others are not? . .... ..... ....... ....... ... ...... .... ... ... ......... ........ ..... .........•................... ... ..... .•...... .•........ ............ ....... .....•.. .... ....... ..... ..... ....•..... ....... ... ... ... ... .. ..... ..... ..... ... ..... •... .. .... ... .. ...... ...... .... .. .... ... ... ... .. .. ................................. .... ... ...... .... ... ... ... .•...... •... ..... ... ... .. ... ....... .. .. ..... ... •....... .. ..•. .. ... ....... .. .. ..... ... ............ ... ... ... .... ..... .. .. ..... ... ... ... ............ 2. Students at Pascal High School have a 25- . .. ... .....•....... .. ..... ... ... ... ...... .. ..•..... ..... ...... .. ... ......... ... ... •. ... ...... ..... ..... ...... .... ............ ... .. ..... ...... ..... ..................................................... ... ... ..... .......... ......•....... ..•................ .... member student council, which represents the . ........ •...... .......... ..... ... .............. •... ......... ...... ... ...... ... .........•. .... ... ............. ... ..... ... ......... .. .. .. ... ....... ... ..... ... ......... .. .. .. ... •. ... ..... ... ......... .. 2000 members of the student body. The class- . .. ... ... ... .......................... .. .. .. ... ... .............•...... ..........................•... .. ... .. ... ........... ....... .... .. .. .. ... ....•..... .... ... ... .. by-class sizes are: 581 Seniors, 506 Juniors, 486 . .. .. .... ... ............... ... ... .. .. .. ....... ....... ...... .. ... .... ... ... ... ............ ... .. .. ... ... ... ...•.. ... ... .•..... ... .. .. ... .... ... ..... .. ... ...... ... .. .. ... ... ....... Sophomores, and 427 Freshmen. How do you •................................... ..... ... .. ...... .......... •......... ...... ... .. ...••.. ....... •..... .......... .... .. .... ....... ..... •..... ... .. ..... ........ ..... ...... ..... ....... ....... ..... think that the seats on the council should be .......... ...... ........ ..... • ••.... .. ............. ............ distributed to the classes? .......•.. 3. Twelve business associates meet for lunch. As they leave to return to their offices a couple of hours later, one of them conducts a small mathematical experiment, asking each one in the group how many times he or she shook hands with someone else in the group. Thetwelvereportedvalueswere3,5,6,4,7,5,4,6,5,8,4,and6.Whatdoyouthinkof this data? Is it believable? 4. If you were President Washington, how would State 1790 Population you distribute the 120 seats in the House of Repre- Connecticut 236841 sentatives to the fifteen states listed at right? The Delaware 55540 table shows the results of the 1790 census. Georgia 70835 Kentucky 68705 5. Suppose that consumers prefer Brand X to Maryland 278514 Brand Y by a 2-to-1 margin, and that consumers Massachusetts 475327 141822 also prefer Brand Y to Brand Z by a 2-to-1 mar- New Hampshire New Jersey 179570 gin. Does it follow that consumers prefer Brand X New York 331589 to Brand Z? North Carolina 353523 Pennsylvania 432879 6. Three investors control all the stock of Amal- Rhode Island 68446 gamated Consolidated, Inc. One investor owns South Carolina 206236 46% of the stock, another owns 37%, and the third Vermont 85533 owns the remaining 17%. Which stockholder is the Virginia 630560 most powerful? Total 3615920 7. If you are a voter in a state that has one million other active voters, how likely is it that your vote will be pivotal (in other words, the deciding vote) in the next gubernatorial election? 8. Avery and Cameron want to share a Snickers bar. Describe an algorithm (a set of rules that, when followed mechanically, are guaranteed to produce a result) that will create a fair division. August 2012 1 Phillips Exeter Academy Discrete Mathematics The Apportionment Problem In every method of apportioning the House of Representatives, the ideal quota for a state is now calculated by the formula 435·(state population/total population). Because this is not likely to be an integer, it is necessary to either round up to the upper quota or round down to the lower quota to obtain a meaningful result. A method of apportionment must specify exactly how this rounding is to be done. Another quantity of significance in apportionment is the ideal district size, which is the total population divided by the total number of representatives. The 2000 census puts this figure at 646952 = 281424177/435. This is how many constituents each representative should have (and would have, if Congressional districts were allowed to cross state boundaries). 1. What do you get if you divide a state’s population by the ideal district size? The simplest method of apportionment was proposed in 1790 by Alexander Hamilton, and it is so intuitively appealing that you may have thought of it yourself already: Calculate each state’s share of the total number of available seats, based on population proportions, and give each state as many seats as prescribed by the integer part of its ideal quota. The remaining fractional parts of the quotas add up to a whole number of uncommitted seats, which are awarded to those states that have the largest fractional parts. Apply the Hamilton method to the following small, three-state examples. (The names of the states are simply A, B, and C.) You should notice some interesting anomalies. 2. Suppose that the populations are A = 453000, B = 442000, and C = 105000, and that there are 100 delegates to be assigned to these states on the basis of their populations. 3. Suppose that the populations are A = 453000, B = 442000, and C = 105000, and that there are 101 delegates to be assigned to these states on the basis of their populations. 4. Suppose that the populations are A = 647000, B = 247000, and C = 106000, and that there are 100 delegates to be assigned to these states on the basis of their populations. 5. Suppose that the populations are A = 650000, B = 255000, and C = 105000, and that there are 100 delegates to be assigned to these states on the basis of their populations. ——————————— 6. A third-grade teacher is arranging a field trip for eight of the boys in his class. These boys do not always behave well together. Adam gets along with Ben, Frank, and Hugh; Ben gets along with Adam, Chris, David, and Hugh; Chris gets along with Ben and Frank; David gets along with Ben, Eric, and Frank; Eric gets along with David, Frank, and Guy; Frank gets along with everyone except Ben; Guy gets along only with Eric and Frank; Hugh gets along with Adam, Ben, and Frank. Draw a graph that models this situation; tell what the vertices and edges represent. 7. Avery, Cameron, and Denis want to divide a rectangular sheet cake fairly. Describe an algorithm for doing so. August 2012 2 Phillips Exeter Academy Discrete Mathematics •.. ... 1. Show that it is possible to color a map of • ... .... ........•.....................................•....... •..................•.. ........... .... •. ... ....... .. ...... •......................•...... ... ... ....... .. ..... the United States using only four colors, so that . ......... ... ... ..•...... .. ....... ... .. .... .... ... ........ ............................•... ... .. .... ... ... ... .... ..•........ ...... ... .. .... .... ... ... .... ... ......... .... ... ... ....... ... .... ... ... ......... adjacent states do not receive the same color. .... .... ....... ... .... .. .. • ...... ... .... ....•........ ... •..... ... ... • . .... ..... ....... ....... ... ...... .... ... ... ......... ........ ..... .........•.................... ... ..... .•..... .•........ ........... ........ ....•... The graph at right represents the fifty states and .... ....... ..... ..... ....•...... ....... ... ... .... ... .. ..... .... ..... .. ..... •... .. ... ... .. ...... ...... .... ... .... .... ... .... .. .. ................................ .... ... ...... their borders — two states (vertices) are joined .... ... ... ... .. ..•...... •... ..... .. ... .. ... .... .... .. ..... ... •...... .. ..•. ... ... ........ .. .. ..... ... ............ .. .. .. .... ..... .. .. ..... ... ... ... ............ .. ... .....•....... .. .. ..... ... ... ... ...... when they share a border. .. ..•..... ..... ...... .. .. ......... ... ... •. .. ........... ..... ...... ... ............ ... .. ..... ...... ..... ................................................... ... .... ..... .......... ......•....... .•................ .... ........ •....... .......... ..... ... .............. •.. ......... ...... ... ...... ... .........•. ..... .... ............. ... ..... ... ......... .. .... .. ....... ... ..... ... ......... .. .. .. .. •. ... ..... ... ......... ... 2. Is it possible to do the coloring