Recently, I had occasion to re-read George

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1 DELVING deeper Charles F. Mario The Coffee-Milk Miture Problem Revisited Edited b Theodore Hodgso Recetl, I had occasio to re-read George Póla s (57) classic work o problem solvig i mathematics, How to Solve It. Earl i the book, Póla describes the differet phases of problem solvig, icludig the fourth ad fial phase lookig back. Describig that phase, he writes: Eve fairl good studets, whe the have obtaied the solutio of the problem ad writte dow eatl the argumet, shut their books ad look for somethig else. Doig so, the miss a importat ad istructive phase of the work. B lookig back at the completed solutio, b recosiderig ad reeamiig the path that led to it, the could cosolidate their kowledge ad develop their abilit to solve problems. A good teacher should uderstad ad impress o his studets the view that o problem whatever is completel ehausted. There remais alwas somethig to do; we could improve a solutio ad, i a case, we ca alwas improve our uderstadig of the solutio. (Póla 57, pp. 4 5) Póla s words seem to suggest that lookig back at the solutio of a problem should occur shortl after the problem has bee solved. I took Póla s advice Delvig Deeper offers a forum for ew isights ito mathematics egagig secodar school teachers to eted their ow cotet kowledge; it appears i ever issue of Mathematics Teacher. Mauscripts for the departmet should be submitted via For more iformatio o submittig mauscripts, visit Departmet editors Bria M. Dea, bdea@pasco.k.fl.us, District School Board of Pasco Cout, FL; Daiel Ness, essd@stjohs.edu, St. Joh s Uiversit, Jamaica, New York; ad Nick Wasserma, wasserma@tc.columbia.edu, Teachers College, Columbia Uiversit, New York some ears after first ecouterig a problem that is frequetl posed at professioal developmet workshops, i prit (Gamow ad Ster 5; Garder 5), ad o the Web (QED Ifiit 04; Wikipedia). Although m look back was log overdue accordig to Póla s guidelies, I was surprised ad delighted to fid just how far m revisit took me. THE PROBLEM AND INITIAL SOLUTIONS The problem that I revisited starts with two cups, each cotaiig eactl the same amout of differet liquids. Some versios of the problem mi water ad wie, others cream ad coffee. The versio I used mies milk ad coffee: Suppose that ou have a white cup of milk ad a black cup of coffee, either filled to the brim ad each cotaiig ouces of liquid. Suppose further that ou remove a ouce of milk from the white cup, place it i the black cup, ad thoroughl mi the resultig cotets. The remove a ouce of the coffee-milk miture ad place it i the white cup. What ca ou sa about the amout of milk i the black cup i relatio to the amout of coffee i the white cup? Are the two amouts equal, or is oe amout greater tha the other? After the iitial trasfer of a ouce of milk to the black cup, the white cup cotais 7 ouces of milk ad the black cup cotais ouce of milk ad ouces of coffee. Assumig a thorough miig of the cotets of the coffee-milk miture, the black cup ow cotais ouces of liquid, part milk ad parts coffee. Thus, / of the ouce trasferred back to the white cup will be milk ad / will be coffee. Table presets the results of these two trasfers; as the table illustrates, the amout of coffee i the white cup is the same as the amout of milk i the black cup. 5 MATHEMATICS TEACHER Vol. 0, No. September 05 Copright 05 The Natioal Coucil of Teachers of Mathematics, Ic. All rights reserved. This material ma ot be copied or distributed electroicall or i a other format without writte permissio from NCTM.

2 Table The Results of Si Trasfers betwee the White ad Black Cups Ouces White Cup Ouces Ouces Black Cup Iitial Coditio 0 0 After Oe Trasfer After Two Trasfers 7 0 Ouces Percetage of milk i the white cup is After Three Trasfers After Four Trasfers Percetage of milk i the white cup is 7.% % After Five Trasfers After Si Trasfers Percetage of milk i the white cup is % With this result i had ad Póla s admoitio that there remais alwas somethig to do i mid, I cosidered how I might build o m iitial work o this problem. I particular, I wodered what would be the result of cotiuig the process of trasferrig a ouce of the cotets of the two cups from oe to the other? Ituitivel, it seems evidet that the cotets of each cup would tred toward some stead state perhaps a eve distributio of milk ad coffee i each cup sice each trasfer from the white cup to the black cup would icrease the percetage of milk i that cup. Likewise, each trasfer from the black to the white cup would icrease the percetage of coffee i the milk, at least util the cotets of the two cups were idetical. However, questios about the umber of trasfers eeded to reach a stead state or a particular miture sa, a 75%-5% miture of the two liquids i each cup still remaied. I opted to eplore the latter of these questios. To determie the umber of trasfers required to reach a 75%-5% mi i each cup, I first used a arithmetic strateg. I particular, after two trasfers, the umber of ouces of milk i the white cup, 7 /, divided b the umber of ouces of liquid i the cup,, rouded to the earest iteger percetage poit is %. To fid the umber of trasfers eeded for that percetage to be 75, we ca eted the work of table, retaiig eact percetages after each trasfer to avoid roud-off errors. The percetage of milk i the black cup ad of coffee i the white cup first eceeds 5% after si trasfers of the liquids (or three complete back-ad-forth trasfers). I ow had the aswer to the questio, How ma trasfers of oe ouce of liquid from oe cup to the other would it take to reach a distributio of at most 75% milk i the white cup? Aside from havig a idea of the calculatios that I eeded to perform, however, I did ot have a efficiet wa to aswer more geeral ad comple questios, such as these: What percetage of the liquid i the white cup would be milk after 0 trasfers if, oce agai, each cup iitiall cotaied ouces of liquid ad each trasfer moved ouce? Vol. 0, No. September 05 MATHEMATICS TEACHER 5

3 How ma trasfers would it take to reach a distributio that left at most 60% milk i the white cup if there were iitiall 0 ouces of the two liquids i each cup ad ouces of liquid were moved with each trasfer? To avoid the oerous arithmetic calculatios eeded to aswer these ad other more comple questios, I kew that I had to adopt a more comprehesive methodolog to geeralized coffee-milk miture problems. As is ofte the case whe cosiderig several variables ad seekig geeralized descriptios of umeric patters, the laguage of algebra is particularl useful. A GENERALIZATION OF THE COFFEE- MILK MIXTURE PROBLEM The geeralized problem ma be stated this wa: Suppose that we have a white cup of milk ad a black cup of coffee, either filled to the brim ad each cotaiig ouces of liquid. Suppose further that ouces of milk are removed from the first white cup ad placed i the secod black cup ad the cotets are thoroughl mied. The ouces of the miture are trasferred from the secod black cup to the first white cup. How much milk will be i the white cup after such trasfers? To solve this geeralized problem, I first assumed that each cup had a capacit of at least ouces; otherwise, the first trasfer would eceed the capacit of the cup, a situatio practicall impossible to model. Guided b the calculatios i table, I developed epressios to represet the results for the first si trasfers (see table ). As I reviewed the epressios i table after a eve umber of trasfers had bee completed, a patter seemed to emerge. I a attempt to make that patter more discerible, I rewrote those epressios with the variable factored from each, as i table. I kew ow that I was dealig with alteratig terms of biomial epasios. I particular, the umerators of the fractioal epressios were the sums of alteratig terms of the deomiators after the had bee epaded. For eample, cosider the epasio of ( ). The sum of the first ad third terms, those with odd powers of, forms the umerator of the fractioal epressio ( )/( ), used to represet the amout of coffee i the black cup. Similarl, the sum of the secod ad the fourth epressios forms the umerator of ( )/( ), used to represet the amout of coffee i the white milk cup. Whe we compare the epasios of ( ) ad ( ), it becomes apparet that the sum of the epasios has ol odd powers of, whereas the differece has ol eve powers of. Specificall, ad ) ( ) ) ( ). With these two idetities at m disposal, I cojectured that the amouts of liquid i the two cups after trasfers could be represeted i terms of the biomial epressios ( ) ad ( ), as i the last row of table. This cojecture ca be verified usig mathematical iductio. To start, whe, Similarl,.. What remais to be show is that if the amout of coffee i the black cup after trasfers (or complete back-ad-forth trasfers) is represeted as the B B would be that amout after such trasfers. After trasfers, the amout of coffee i the white cup is W ) ( ). 54 MATHEMATICS TEACHER Vol. 0, No. September 05

4 Table Amouts of Liquid for the Geeralized Problem Iitial Coditios After Oe Trasfer After Two Trasfers Amout White Cup Amout Amout Black Cup Amout After Three Trasfers After Four Trasfers ) ) ) ) After Five Trasfers ) ) ) ) ) ) ) ) ) ) ) ) After Si Trasfers ) ) 4 ) ) ) ) ) ) 4 Durig trasfer, a fractio, /, of the liquid is removed from the white cup ad added to the black cup, which ow cotais coffee i the amout of B B (/)W. The, durig trasfer, a fractio, /( ), of the liquid i the black cup is trasferred back to the white cup. Thus, it remais ol to prove that B (/) W /( ) B B. A equivalet equatio, after we multipl both sides b ( ) /, is this: ) ) ( ) ( ) ( ) ( ) ) ( ) ( ) ( ) ) ( ) The first ad third lies of the previous equatio combie to give. The secod ad fourth lies ma be combied as. Epadig these, addig, ad regroupig ields our desired outcome: ) ( ) Returig to our origial cotet b dividig this result b ( ) /, we recover the epressio we sought for the amout of coffee i the black Vol. 0, No. September 05 MATHEMATICS TEACHER 55

5 Table A Modified Versio of Table Iitial Coditio After Trasfers Amout White Cup Amout Amout Black Cup Amout 0 0 After 4 Trasfers ) ) ) ) After 6 Trasfers ) ) ) ) After Trasfers ) ( ) ) ) ( ) ) ) ( ) ) ) ( ) ) cup after trasfers. This completes the iductio process ad proves that our cojecture was true. Further, we ca rewrite the formula as ( ) Similarl, ( ) B. W. I his descriptio of the lookig back compoet of problem solvig, Póla suggests that, oce a problem has bee solved, if there is some rapid ad ituitive procedure to test either the result or the argumet, it should ot be overlooked (Póla 57, p. 5). To test the reasoableess of equatio (), eamie specific cases for each variable. Whe ad, for istace, the epressios i both equatios () ad () reduce to /. The assumptios ad correspod to iitiall trasferrig all the milk i the white cup ito the black cup; accordigl, after the secod trasfer, each cup would cotai equal amouts, /, of milk ad coffee. A USEFUL SUBSTITUTION The fractio ( )/( ) is cetral to the mathematics i this problem. For coveiece, we will use X ( )/( ). Now, suppose that is less tha so that the fractio X is a positive umber less tha. I that case, if is greater tha m, the X < X m. This correspods to what we would epect to happe as the umber of trasfers is icreased; amel, the amout of milk i the white cup decreases. O the other had, o matter how large becomes, the value of X is alwas greater tha zero, idicatig that there will alwas be more milk tha coffee i the white cup. Eve though the limit as approaches ifiit of / ( X) is /, for fiite values of it is true that / ( X) is alwas greater tha / ( X). I keepig with Póla s suggestio that we eamie the solutio i multiple was, I decided to use equatio () to solve three additioal problems that start with the coditios assumed i m geeralizatio of the coffee-milk miture problem. Problem : Suppose that we start with 0 ouces of liquid i each cup ad ouces are shifted with each trasfer. After trasfers, what percetage of liquid i the white milk cup will the be milk? Solutio: Rememberig that the cup cotais ouces of liquid ad usig the variable X ( )/( ), we kow that the fractio of milk i the white cup after trasfers is / ( X ) divided b, or ( X )/. With 0,, ad 4, we have X (0 )/(0 ) 7/ ad ( X )/ ( (7/) 4 )/ 54%. Problem : Suppose that we start with 0 ouces of 56 MATHEMATICS TEACHER Vol. 0, No. September 05

6 the two liquids i each cup ad that ouces of liquid are shifted with each trasfer. After how ma trasfers will the percetage of milk i the white cup be at most 5%? Solutio: Defiig p to be the percetage of milk i the white cup after trasfers, we have p ( X )/. Our iitial coditios correspod to 0,, ad X (0 )/(0 ) 7/. Thus, we solve 0.5 ( (7/) )/. Equivaletl, we ca write (7/) 0.0. Takig the logarithm of both sides gives us log(7/) log(0.0), or 6.. Sice we caot have a fractioal umber of trasfers, this value ca be iterpreted as tellig us that it will take 4 trasfers (or 7 complete back-adforth trasfers) before the white cup cotais less tha 5% milk. Problem : Suppose that we start with 0 ouces of liquid i each cup ad wish to have at most 60% milk i the white cup after 0 trasfers. How much liquid should be trasferred at each step to obtai these coditios? Solutio: The coditios i this problem are reflected i the equatio p ( X )/ with p 0.6, 0/ 5, ad 0, leadig to X (0 )/(0 ). Thus, we start with 0.6 ( X 5 )/ or, equivaletl, 5log(X) log(0.). From this, we have X 0 log(0.)/ Returig to the variable, we fid that equall well. We ca ol imagie the mathematical eploratios ad discoveries that will come to light whe, as Póla recommeds, other problems previousl solved ad put aside are revisited. REFERENCES Gamow, George, ad Marvi Ster. 5. Puzzle- Math. New York: Vikig Press. Garder, Marti. 5. The Scietific America Book of Mathematical Puzzles ad Diversios. New York: Simo ad Schuster. Póla, George. 45 (57). How to Solve It: A New Aspect of Mathematical Method. Priceto, NJ: Priceto Uiversit Press. QED Ifiit. Problem : Miig Milk ad Coffee. Problem of the Da: Jauar, 0. ad-coffee/ Wikipedia: The Free Ecclopedia. Wie/water miig problem. /water_miig_problem CHARLES F. MARION, charliemath@ optolie.et, taught mathematics at Lakelad High School i Shrub Oak, New York, for thirt-two ears. I retiremet, he has ejoed searchig for ad writig about udiscovered patters i elemetar umber theor ad sharig his ethusiasm for mathematics with his five gradchildre , 5 5 so 0( 0.)/( 0.).. This value tells us that if we eecute 0 trasfers of about. ouces, the white cup will cotai about 60% milk. TIMELESS ADVICE We have come a log wa from solvig the wellkow coffee-milk miture problem. Takig Póla s cousel to heart, we first solved a slight etesio of that problem ad the posed a more ambitious geeralizatio. A log joure through a algebraic thicket ucovered a formula that gave us, i tur, the abilit to aswer a multitude of other questios. The mathematics eeded to aswer those questios raged from the evaluatio of algebraic epressios to the use of logarithms to solve equatios. Our aalsis of the coffee-milk miture riddle also illustrates the timeless advice of Póla s masterpiece How to Solve It, a work that has eriched the lives of geeratios of problem solvers. Cotiuousl i prit sice 45, it will likel serve future geeratios of mathematics teachers ad their studets Vol. 0, No. September 05 MATHEMATICS TEACHER 57