Search Loci: Convergence:
In H. Eves, Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Using Historical Problems in the Middle School
A Problem from Alcuin of York
A final problem comes from medieval Europe. Alcuin of York (c. 735-804), the educational advisor to Charlemagne, left a collection of mathematics problems in a work entitled Propositions for Sharpening Youthful Minds. Among the problems was the following:
This problem turns out to be a particularly “rich” one. I gave this problem to groups of my 8th grade students as part of a problem solving project using historical problems. The solutions given are those that the students found.
One group came up with the following solution.
A second group arrived at a different solution.
This is essentially the solution given by Alcuin of York. It is interesting to note the appeal of this logic. I attended a workshop at a N.C.T.M. Regional Meeting in which about 25 participants were divided into cooperative learning groups. All groups came up with the above solution, with no dissenters.
One student arrived at the following solution.
S + D + M = 960.
S = 3M D= 7/5M.
3M + 7/5 M + M = 27/5M = 960. Therefore M=177 7/9.
Son gets 533 3/9. Daughter gets 248 8/9. Mother gets 177 7/9.
This solution uses the same logic as Nicolas Chuquet, who presented a similar problem in his Triparty of 1484:
Chuquet's solution is that the inheritance should be divided in the ratio of 4:2:1.
As a postscript, when a related problem appeared in the column, Ask Marilyn, in Parade Magazine, 10/05/03, the response given was “The estate should be divided according to the law. If the conditions of the will cannot be met, it is invalid. Without a will , the widow would receive half or a third of the man’s estate, and the children would share the remainder.” Interestingly enough, Robert Recorde, in his Grounde of Artes (c. 1542), wrote, "If some cunning lawyers had this matter in scanning, theywould determine this testament to be quite voyde, and so the man to die untestate, because the testament was made unsufficient." Is this thirteen hundred years of progress in solving this problem?
More on this problem may be found in D. E. Smith, History of Mathematics, vol. II, pp. 544-546 (New York: Dover, 1958) and in Singmaster and Hadley, "Problems to sharpen the young", Mathematical Gazette, No. 475 (March, 1992), 102-126. The nineteenth century arithmetic text, Edward Brooks, The Normal Written Arithmetic (Sower, Barnes and Potts, 1863) contains similar problems.