Math... First Hand
           
 
Writing "Personal" Problems
A famous math problem holds the key to Diaphontus' age. Hints on writing math problems with similar information to engage students' interest.
The Greek mathematician Diophantus is sometimes called the father of Algebra. The following riddle is told about his age:

    "He was a boy for one-sixth of his life. After one-twelfth more, his face was covered in whiskers. He spent another seventh as a bachelor. In his fifth year of marriage, his wife bore him a son. That son lived half as many years as his father. After four years of grief at his son's death, father died as well."

Diophantus' riddle is engaging, but it is especially appealing to math teachers. This sort of problem could meet several goals.

Young students are naturally curious about their teachers. They giggle when they discover a teacher's first name, they are stunned when they spot one at the grocery store, and they frequently ask their instructor's age. As with many questions from young students, the age question usually comes at an inappropriate time - like during lecture or in the middle of an exam.

Imagine answering this question with a short puzzle that keeps the nosy student occupied and encourages him or her to practice algebra skills. Diophantus epitaph offers just that potential. This article will provide some hints for anyone interested in crafting their own riddle, and examine some alternate age puzzles.

Traditional Diophantine Riddle
Let us first review a solution to Diophantus' riddle. The symbol "a" will represent his age. The riddle can be translated algebraically thus:

1/6(a) + 1/12(a) + 1/7(a) + 5 + ½(a) + 4

The sum of these parts are equal to Diophantus whole life, so:
  • Multiply by 84, the least common denominator
  • Combine like terms
  • Subtract 75a from both sides
  • Divide both sides by 9

    To construct a similar problem, begin with the factors of the final answer. In this case, the factors of 84 are {1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84}. From this list, we know that one-half the age, one-third of the age, one-third, one-fourth, one-sixth, and so on are all integers. We also see that Diophantus was fortunate enough to pass on at an age with an abundant list of factors.

    In the first step of this solution, the least common denominator provides a strong clue to the final answer. By using both one-seventh and one-twelfth in the problem, the answer is coincidentally the same as the least common denominator of our fractions. If we limit ourselves to the fractions derived from the factor list, the answer will always be a multiple of the least common denominator. This limitation also means that for some ages it will not be possible to create elaborate problems.

    Had Diophantus expired one year earlier, his age would have been a prime number. If he was 83, his problem might have read, "He was a boy for 14 83rds of his life, a boy for 7 83rds of his life, etc." It is not very creative, but the options are limited according to the number of factors. The next section will explain some tricks around this limit, but for now, we will see how to create problems for non-prime factors.

    Identifying the building blocks
    Begin with the target age, and list all non-trivial factors (ignore 1 and the number itself in the factor list). For example, if the puzzle maker were 36 years old, he or she would identify the factors {2, 3, 4, 6, 9, 12, 18}. Use this list to create a table of the fractions that may be used and the number of years that each fraction represents (remember that we are not limited to unit fractions). Using the factor list from 36, we see this table.

    1/182 years 7/1814 years13/1826 years
    1/123 years 5/1215 years 3/4 27 years
    1/9 4 years 4/916 years 7/9 28 years
    1/6 6 years 1/218 years 5/6 30 years
    2/9 8 years 5/920 years 8/9 32 years
    1/4 9 years 7/1221 years11/1233 years
    5/1810 years11/1822 years17/1834 years
    1/3 12 years2/324 years
    Creating combinations
    Now select a combination of years that sums to 36. There are several options such as;

    "4 + 12 + 6 + 6 + 8"   "6 + 9 + 9 + 8 + 4"   "8 + 19 + 18"

    and of course, there are several other combination. Now replace several of these numbers with the fraction of the total age that they represent. For example, the first combination above becomes

    1/9(a) + 1/3(a) + 1/6(a) + 1/6(a) + 2/9(a)

    This arrangement, however, will provide an infinite number of solutions. In order to limit the problem to a unique solution, at least one of these fractions must be replaced with the number of years it represents. The combination might be rewritten as

    1/9(a) + 1/3(a) + 1/6(a) + 6 + 2/9(a)

    Of course, if you notice Diophantus's original riddle, we are not limited to integers that can be expressed as a fraction of the total age. For example, one could use the combination

    4 + 12 + 5 + 15

    4, 12, and 15 can be represented by the fractions one-ninth, one-third, and 5-twelfths respectively. Since 5 cannot be represented by one of our fractions, it can be listed in the problem. This combination could translate to

    1/9(a) + 1/3(a) + 5 + 6 + 5/12(a)

    Spinning the tale
    All that is left is to create the descriptions to match each component in the combination. For example

      Diane Diophantus spent one-ninth of her life playing at her mother's feet. The next third of her life she attended public schools. She spent five years earning her college degree, and the remaining five-twelfths of her life has been spent paying student loans.

    It may be easier to first determine the periods of time you wish to describe in your problem and then choose fractions from the table that most closely match the time periods you have chosen.

    Tricks for prime factors
    You may need to get creative to find ways to describe certain intervals. This creativity often results in a richer problem. Take, for example, a 21 year old. 10 years could be described as "Half a year less than half of his life." Seven years could be "Two years more than a third of his life."
    Looking for more classroom activities?
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