The Dogs-and-Ducks Problem

Recently, while reviewing some grade 5 olympiad questions for a local company, which specializes in conducting olympiad math programs in elementary schools, I came across the following nonroutine dogs-and-ducks question.



The number of ducks is 10 fewer than the number of dogs. 
The dogs have three times as many legs as the ducks. 
How many ducks and how many dogs are there?



When I realized that my mental calculated answer differs from the one given by the problem poser, I thought maybe my incorrect solution might serve as a "logic exercise" to tickle others' mathematical bones to point out the flawed reasoning. My faulty argument followed something along these lines:

Since there are 10 fewer ducks, and each duck has 4 legs, there are 10 × 4 = 40 fewer duck legs.



From the model drawing,



2 units = 40


1 unit = 20



Number of duck legs = 1 unit = 20


Number of ducks = 20 ÷ 2 = 10



Number of dog legs = 3 units = 3 × 20 = 60


Number of dogs = 60 ÷ 4 = 15



A quick check shows that the answers don't satisfy the given conditions in the question. Although the dogs have three times as many legs as the ducks, however, the number of ducks is only 5 fewer than the number of dogs, which is expected to be 10.




When the why is harder than the how



The lesson here is to identify the illogical step, although seasoned problem solvers would probably not commit this kind of error or blunder. 

In general, it's one thing not to make an error, and get the correct answer; it's another thing to be able to explain someone else's error.

At a deeper level, it's sometimes conceptually harder to explain someone's flawed reasoning to an incorrect answer than just know how to solve a problem. Think of mathematical fallacies and paradoxes, where in a number of cases the illegal step is anything but obvious.




A fabricated faux solution

Alternatively, if we argued that a dog has 2 more legs than a duck, we'd then have a model that looks like this:

In this case, students would easily realize that the answers are also incorrect, when they find out that the number of dogs isn't an integer.


Two quick-and-dirty solutions

Why don't you try solving the dogs-and-ducks question on your own first, before comparing your solution with the ones I've worked out? If yours is an alternative (or elegant) solution, the mathematical brethren couldn't wait to reading it!

Method 1

Since a dog has twice as many legs as a duck, and there are 3 times as many dog legs as duck legs, there are 3/2 times as many dogs as ducks.

A model depicting the above relationship may be drawn as follows:

From the model drawing,

1 unit = 10

Number of ducks = 2 units = 2 × 10 = 20

Number of dogs = 3 units = 3 × 10 = 30

Check: Ducks: 20 × 2 = 40 legs

Dogs: 30 × 4 = 120 legs = 3 × 40 legs

Method 2

Given: Number of dogs = Number of ducks + 10

Number of duck legs = 2 × Number of ducks

Number of dog legs = 4 × Number of dogs

Given: Number of dog legs = 3 × Number of duck legs

A quick-and-dirty model representing the above information may be drawn as follows:

From the model drawing,

3 × 2 units = 6 units = 4 units + 10 + 10 + 10 + 10 

2 units = 10 + 10 + 10 + 10

1 unit = 10 + 10 = 20

1 unit + 10 = 30

Therefore, there are 20 ducks and 30 dogs.

Method 3 (Sakamoto method)

Those of you who are versed with the three-step Sakamoto method in solving word problems, may proceed as follows:

1. Grasp the relation

Let ④ represent the number of duck legs.

Duck legs           Dog legs

④                     ⑫

__________________________

Number of ducks     Number of dogs

④ ÷ 2 = ②                 ⑫ ÷ 4 = ③

                                             – 10

__________________________

1              :              1

                             + 10

2. Diagram

3. Number sentences

③ – ② = ① = 10

② = 2 × 10 = 20 (ducks)

③ = 3 × 10 = 30 (dogs)

The number of ducks is 20, and the number of dogs is 30.

Two bonus problems

If you like being challenged by similar dogs-and-ducks or chickens-and-rabbits problems, may I direct you to solving two more questions at Yan's One Minute Math Blog?

Happy Mathematical Problem Solving!


© Yan Kow Cheong, March 3, 2013

20 Things You Probably Didn't Know about Singapore Math

The media love to paint a positive or negative picture of life's successes and failures, and Singapore's success in mathematics education is no different. Here are some unwritten, often undesirable, factors contributing to Singapore's mathematical success.

20. About sixty percent of students know how to differentiate and integrate un-pathological functions by grades 9 and 10—they read two-year "Additional Mathematics" plus four-year "Elementary Mathematics."

19. About ninety percent of students would have had a math tutor by the time they reach grade 6—private tuition is a multi-million-dollar business in Singapore because school teachers know tutors and anxious parents would eventually fill in the gap.

18. Some sixty percent of students complete their secondary education in four years, which includes reading calculus, trigonometry, and proofs in plane geometry.

A title that promotes
the model method
SingaporeMath.com

A Singapore wallet-friendly
 grade 4 title of 
a six-book series 

17. On average, most students would practice three to four assessment [supplementary] math titles every year, up to grade six, mostly purchased by parents and recommended by tutors, because local textbooks ill-prepare them for school tests and exams.

16. An estimated 60% of students in every cohort dislike math, because it’s taught in a boringly sterile manner in schools, and often by boring math teachers who are simply teaching math to the test.

15. Statistical anecdotal evidence suggests that as high as 80% grades 1-6 teachers prefer to teach other subjects to math—the painful truth is that grades 5-6 math (with their share of challenging word problems) are harder to teach than grades 7-8 math.

14. Most K-6 math teachers are non-college graduates; interestingly, they’re also known to be better math teachers than their peers armed with a university degree.

A Formula for Singapore's Math Success = 20% Textbook + 30% Teacher + 30% Tuition + 20% Parental Involvement

13. The better math teachers and tutors aren’t teaching in the top schools, but rather in neighborhood ones, with far less-ideal facilities and resources.

A dear pseudo-monograph
about the
Singapore model method 

12. Since the 2000s, the standard of math education in Singapore has dropped significantly, due to the recruitment of non-math majors—many have a degree in Accountancy, Engineering, or Computer Science. This means that many wouldn’t have been exposed to a rigorous treatment of college math (abstract algebra, topology, or complex analysis).

11. The majority of math teachers moonlight, often compromising their day-time jobs—the better ones teach in tuition or enrichment centers, or give private tuition, often charging obscenely.


10. Up to 70% of Singapore students are probably one grade higher than their peers in the United States—for instance, a primary 2 student in a good neighborhood school in Singapore would have covered at least 60% of what a grade 3 student in the US had read, based on the textbooks’ contents from both countries.

9. Other than those few expensive-cheat math titles written by some lecturers or tuition centers' owners, most Singapore-published math textbooks and assessments are value-for-money titles vis-à-vis the expensive, thick, colorful—inch-deep, mile-wide—textbooks published in the US.

8. The power and beauty of the Singapore model (or bar) method is mostly appreciated by those outside Singapore, as compared to an unappreciative lot of local math teachers. Since the late eighties, they've been inundated with an unhealthy number of assessment (supplementary) books, aimed at promoting the visual heuristic—today, most local math teachers treat the model method as a hype or a bore.

7. Most elementary math teachers and graduates-parents have difficulty drawing a model when faced with a grade 5 0r 6 challenging word problem, preferring to use algebra instead to solve them—without peeking at the model-or bar-method solutions, most parents are unable to help their grades 5-6 children with their school homework.

6. Most math teachers feel uncomfortable or ill-prepared to coach their own students for math contests and competitions, leaving the task to trainers from private companies, or to coaches from mainland China.

A decade-old
Singapore bestseller
http://tinyurl.com/3cctqcg
http://tinyurl.com/3uy5hou

5. Local mathletes are trained to answer questions that would defeat most secondary teachers, who are primarily drill-and-kill specialists employed to produce exam-smart students to outperform their peers from other Commonwealth countries.


4. Singapore-published textbooks and assessments are mostly written or ghostwritten by foreign-born authors, most of whom have never taught in primary or secondary schools, or by lecturers supervising trainee-teachers.

3. Most Singapore-published math titles are "edited" by non-Singapore citizens, who only have a smattering understanding of the local educational system.

2. An unhealthy number of school textbooks are rewritten or ghostwritten by editors for their PhD authors—many titles-conscious general editors or consultants are notoriously known to contribute quasi-zero input and to collect an undeserved royalty or lump sum payment.

1. Singapore is a haven for assessment math titles, but a hell-on-earth for mathophobics who are forced and terrorized by parents and tutors to go through hundreds of non-routine or challenging word problems, so that they'd remain ahead of the competition. The only consolation is Singapore's top ranking in TIMSS; more medals at contests and competitions, and more university places at top universities.

Indeed, Singapore firsts in math education comes with a high price and with much pain and suffering for students, teachers, tutors, parents, writers, editors, and publishers. Not to say, tens of thousands of students who feel shortchanged and alienated by the culture of mathematics challenge, resulting in poor self-esteem and a dislike for the subject!

© Yan Kow Cheong, November 2, 2011.

Math for Girls ONLY

Since it started participating in the International Mathematical Olympiad (IMO) in 1985, when it informally sent only two contestants, the China team of six contestants has been Number One fourteen times. An outstanding achievement, considering that the most populated nation is relatively new to this high-level mathematical competition among high-school students, as compared to other countries like Russia and other ex-communist countries, with their rich culture of decades-old Olympiad and competitive mathematics.

Note that China, which hosted the 31st IMO in 1990, didn’t send a team to the IMO when the event was held in Taiwan in 1998. The first IMO, which was held in Romania in 1959, has since been held annually, except in 1980.

Since 1986, the China team has never had a female student. To encourage more female mathletes, the China Mathematical Olympiad Committee launched the China Girls’ Mathematical Olympiad in 2002. The top two winners will be admitted directly into the national training team of about 20 to 30 students, from which six students will be finally selected to form the China IMO team.

In 2007, the first girl who was winner of China Girls’ Mathematical Olympiad was selected to enter the 2008 China national team and won a gold medal at the 49th IMO, in Hanoi, Vietnam.

A Sample of Questions

Let’s look at a sample of these high-school contests problems, which were posed in recent China Girls’ Mathematical Olympiads.

Eight persons join a party.

(1) If there exist three persons who know each other in any group of five, prove that we can find that four persons know each other.

(2) If there exist three persons in a group of six who know each other in a cyclical manner, can we find four persons who know each other in a cyclical manner? [2006, #4]

A positive integer m is called good, if there is a positive integer n such that m is the quotient of n over the number of positive integer divisors of n (including 1 and n itself). Prove that 1, 2,… ⋯ , 17 are good numbers and that 18 is not a good number. [2007, #1]

Find all positive integers n such that 20n + 2 can divide 2003n + 2002. [2002, #1]

Let ABC be an obtuse triangle inscribed in a circle of radius 1. Prove that triangle ABC can be covered by an isosceles right-angled triangle with hypotenuse √2 + 1. [2004, #3]

Find all pairs of positive integers (x, y) satisfying x^y = y^x–y. [2002, #6]

Given that a × b rectangle with a > b > 0, determine the minimum length of a square that covers the rectangle. (A square covers the rectangle if each point in the rectangle lies inside the square.)

Let x and y be positive real numbers with x³ + y³ = x – y. Prove that x² + 4y² < 1. [2005, #5]

Let a, b, c be integers each with absolute value less than or equal to 10.

The cubic polynomial

f(x) = x³ + ax² + bx + c

satisfies the property

｜f(2 + √3)｜< 0.0001.

Determine if 2 + √3 is a root of f. [2007, #7]

If China had taken the lead to host a Girls’ Mathematical Olympiad to support its female mathletes, it wouldn’t too late for other countries to follow suit to encourage more girls to take part in math contests and competitions. This could only have a positive effect on the enrollment of both female undergrads and postgraduate students, who would be motivated to take up more advanced math courses in university.

A Girls’ IMO

Due to some countries’ high pools of talented and gifted mathletes, perhaps the IMO committee would re-look at its invitation guidelines, by inviting more than one team from countries like China, Russia and the United States, to give more opportunities for high-school students to be a medalist at the IMO.

An Individual Category

There could be an open individual category so that potential medalists-mathletes who don’t make it to their national team (due to the limit number of six contestants per country) could still compete on their own. And, to encourage more girls to compete, there could also be an IMO for female mathletes, in the spirit of the China Girls’ Mathematical Olympiad.

References

Xiong, B. & Lee, P. Y. (eds.) (2009). Mathematical olympiad in China (2007-2009): Problems and solutions. East China Normal University Press & World Scientific.

Xiong, B. & Lee, P. Y.  (eds.) (2007). Mathematical olympiad in China: Problems and solutions. East China Normal University Press & World Scientific.

© Yan Kow Cheong, April 18 2010.