An Upscale Weight Problem

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An Upscale Weight Problem PUZZLE CORNER ALLAN J. GOTTLIEB, '67 An Upscale Weight Problem This being the first issue of a calen JAN 2. Donald Savage asks: The present est legal go game on a 2x2 board with no passes. dar year, we again offer a "year U.S. flag has 50 stars arranged in alter The following solution was from the propos ly problem" in which you are to nate rows of 6 and 5. If Puerto Rico er himself: Note that by the "ko" rule, a legal game cannot repeat a game position with the express small integers in terms of the were to become a state, what would be same player to move. The longest legal game on digits of the new year (1,9,9, and 4) and an appropriate arrangement of the stars a 2 x 2 board without passes is 23 moves (24 the arithmetic operators. The problem is on the revised U.S. flag? positions); it is given by: formally stated in the "Problems" sec _ X_ XO XO 0 O 00 O_ 0_ 00 OO X 0X0 _0 X_ X_ XX _ 0_ tion, and the solution to the 1993 year Speed Department ly problem is in the "Solutions" section. _ O_ OX O_ OX 0 X _X XX X_ _0 X _X _X O_ O_ 00 O_ O_ CO CO 00 Speedy Jim Landau wants to know (There are a total of 57 legal positions on the Problems what is the matter with the function 2x2 board; 8 if we all take symmetries into account. The longest legal game with passes 1 Y1994. Form as many as possible of the fU) = I've been able to construct has over 50 moves, and traverses 44 of the legal positions!) integers from 1 to 100 using the digits 1, 9, 9, and 4 exactly once each and the where L is the latitude? A/S 2. Thurston Sydnor wonders where, in the operators +, -, x (multiplication), / (divi first quadrant, the curve xV = yx intersects itself. Al Cangahuala writes that one solution to the sion), and exponentiation. We desire Solutions equation is clearly x=y. One can find where the solutions containing the minimum num second solution intersects this line by perform Y1993. The following solution is from John ing the substitution y=kx. Then ***=(*£)>< and ber of operators; and, among solutions Drumhellcr: having a given number of operators, thus 3r=fe1/(«l). We get the x-coordinate of the 1 intersection by taking the limit as k approach those using the digits in the order 1,9,9, 30 (19-9)*3 59- 2 93-91 31 19+9+3 60 (1+9)* (9-3 es 1 (since the intersection occurs at the x=y and 4 are preferred. Parentheses may be 3 199*3 32 (9/9)+31 61 - line. But, letting »=l/(/Sr-l) we get that used for grouping; they do not count as 4 3 33 (l*99)/3 62- lim operators. A leading minus sign does 5 9-13+9 34 l+(99/3) 63 (9-3+l)"9 6 «l*9)+9)/3 35 (9»3)-l+9 64 91-(9*3) So the point of intersection is {e,e). count as an operator. 7 19-9+3 36 (13-9)"9 65- 8 (9°3)-19 37 l+9+(9*3) 66 9+(3*19) A/S 3. Dave Mohr has noticed that the tempera 93 ture sign in his bank alternates integer readings JAN 1. Theodore Hoffman is dismayed 9 38 39-1' 67- 10 193+9 39 1**39 expressing Fahrenheit and Celsius. Assuming that to realize that he gains weight just by 68 99-31 11 (9/3)1+9 40 19+39 69 ((9-l)*9)-3 the readings are perfect (and perfectly rounded), moving his scales. He writes: 12 13-(9/9) 41 - 70- for what temperature(s) is one's uncertainty of The puzzle surfaced when I moved 13 19-9+3 42 - 71 - the precise temperature at a minimum? The following solution is from Eric Lund: my bathroom "Detecro" scales from a 14 13+(9/9) 43 - 72 (9+9)"(3+l) 15 (l*9)+9-3 44- 73 - Let F be the thermometer reading in degrees section of bare wood floor to a rug. 16 19-(9/3) 45 ((3+l)*9)+9 74 93-19 Fahrenheit and C be the thermometer reading Imagine my surprise when I found that, 17 (9»3)-l-9 46 19+(9*3) 75 3-((l-9)*9) in degrees Celsius. Let Fnom and Cnom be the exact temperature, i.e Fnom=1.8 C +32. If according to the scales, I had gained 10 18 (9*3*l)-9 47 9+39-1 76- 19 l-9+(9*3) 48 (l*9)+39 77 (9*9)-3+l (r,C) is a solution to this problem, then for all pounds in the process of moving them. 20 39-19 49 31+9+9 78 (l*9*9)-3 integers n, (F+9«, C=5n) is also a solution. So, I made a few readings under varied 21 (l*9)+9+3 50 (9*9)-31 79 91-9+3 Therefore we need only consider the range 32 £ conditions. Here they are: 22 19+(9/3) 51 (9-l+9)*3 80 (9'9)-V F<41,0£C<5. Construct the following table: 81 (l/9)*93 '"mill 'max Object Scales Location/ReadingonScales 82 (9*9)+^ 0 32 31.5 32.5 83 93-1+9 0 33 32.5 bn'flobr On3/8" Qnf/8". On3/4" wood on' 84 (19+9)*3 32.9 1 33 32.9 33.5 or wood foam pad 'pile'rug top ofjrug or foam 85 1-9+93 1 34 33.5 34.5 Scales alone 0 0-0 86 99-13 0 1 35 34.5 34.7 87 ((l+9)*9)-3 Me 145 153 155 • 145 -, 2 35 34.7 88 91-(9/3) 35.5 2 weights I8V2 19V2 193/4 2 36 35.5 " I8V2 89- 36.5 3 37 36.5 90 9(3-">+9 37.5 (The Detecto scales register from 0 to 255. The overhang of the 3 38 37.5 91 - 38.3 weight-carrying top platform clears all surfaces by 3/8") 4 38 38.3 92 93-19 38.5 4 39 38.5 93 1»*93 39.5 My curiosity abounds as to the expla 4 40 39.5 23 - 52 91-39 94 l'+93 40. r nation. 5 40 40.1 40.5 24 ((9-l)»9)/3 53 ((9+9)"3)-l 95 99-1+3 5 41 40.5 41.5 SEND PROBLEMS, SOLUTIONS, AND 25 19+9-3 54 ((l»9)+9)»3 96 (l»99)-3 COMMENTS TO ALLAN]. GOTTLIEB. 26 ((9*9)/3)-l 55 l+((9+9)*3) 97 1+99-3 From the table we see that for any integer n, the '67, THE COURANT INSTITUTE, 27 (l'9*9)/3 56 - 98 99-P combinations F=9n+35, C=5n+1 and F=9n+38, NEW YORK UNIVERSITY, 251 MER 28 l+(9*9/3) 57 (19*9)/3 99 -13*99 C=5n+4 give the temperature to within 0.2 CER ST.. NEW YORK. N.Y. 10012, OR 29 39-1+9 58 19+39 100 degrees Fahrenheit. TO: [email protected] A/S 1. The late Bob High wanted to know the long Continued on Page MIT 40 MIT 50 JAMW1994 company's guru on lead paint in housing— . NUCLEAR Franz Neubachcr, SM '83, visited MIT this past BRIDGEGATE GROUP XXII ENGINEERING summer. I Ic has founded an environmental consulting company, UV&l'-Neiibacher and James K. Liming, SM '83, writes from Fountain Partner, doing innovative environmental man Valley, Calif.: "I am currently the Southern agement and incineration work in Austria and ONE OF Tilt OLDEST AND LARGEST RETAINED EffiCCTlVE SEARCH California branch office manager for Erin Engi Germany. He also reports that he miraculously Firms spkciauziw is teciinoiogv-based organhatioss. neering and Research, Inc., where I supervise survived an extroaordinary automobile crash and perform risk analysis and reliability engi and is lucky to be alive Eric Paillas, SM '84, neering services for complex facilities world is still with GTM participating in the construc wide." ... J. H. Goldberg, SM '60, reports tion of a major bridge between England and from Jupiter, Fla.: "I am president of Florida Wales. Eric now lives in Bristol and has two Frank I'oirier, 'K9 Power 8c Light's Nuclear Division, which owns daughters, Claire and Juliette Francois Vice President and operates four nuclear plants." ... From Jacques, SM '85, dropped by TPP to say he has Augusta, Maine, Jeffrey H. Musk, SM '88, been director of strategy and part of the man sends word: "I will be living in Seoul, South agement team of Cimcnts Lafarge, a Sl-billion- Korea, for the next two years serving with the per-year company, since early 1993—Bravo army." ... Vincent P. Manno, SM 78, NUE and Bonne Chance!... Louisa Koch, SM "88, is 600 Montgomery Street '79, ScD "83, writes to tell us that he has been currently with the U.S. Office of Mangagement 37ni Floor appointed chair of the Department of Mechani and Budget Lola Matysiak, '91, SM '92, San Francisco. CA 941H cal Engineering at Tufts University. Mass. has announced her engagement. A 1994 wed Manno lives in Sudbury, Mass Ken Brooks, ding is planned. Boston, MA 02109 SM '89, reports: "I have just finished a PhD this Steve Thomas, SM '92, has moved to (415)788-7800 summer in the Woodruff School of Mechanical Chicago to be an investment analyst with Euro Engineering's Radiological Engineering progra- American Arbitrage Congratulations are in mat the Georgia Institute of Technology.
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