Can you solve these brain-teasers?

By now, everyone knows how difficult it is to figure out when Cheryl's birthday is.

Yes, it is that mathematics problem from a Secondary 3 Math Olympiad question.

Just to see how tough the questions could be, The New Paper on Sunday put together a set of similar questions from the Singapore Mathematical Olympiad and put three people to the test.

Q1 Four married couples, with names Albert, Bob, Charlie, Don, Elaine, Fanny, Gillian and Helen, meet for a game of chess. They form four groups of two players. The following information is given:1

(i) Bob plays against Elaine.

(ii) Albert plays against Charlie's wife.

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(iii) Fanny plays against Gillian's husband.

(iv) Don plays against Albert's wife.

(v) Gillian plays against Elaine's husband.

Who is Bob's wife?

Q2 Two pipes can be used to fill up a swimming pool. The first can fill the pool in three hours, and the second can fill the pool in four hours. There is also a drain that can empty the pool in six hours. Both pipes were being used to fill the pool. After an hour, a careless maintenance man accidentally opened the drain. How long more will it take for the pool to fill?

Q3 Jane, June, Jean and Jenny are good friends. Something fascinating can be found mathematically with their names. Each letter of their name can be associated with an integer greater than one and the sum incidentally gives their current age. Jane is 16, June is a year older while Jenny is the oldest at 19. Suppose no two different letters are associated with the same integer. Which of the following numbers is a possible value for "u"?

(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

*These questions are from the 2001, 1995 and 2003 Singapore Mathematical Olympiad (SMO) Junior Round 1 respectively.

Here's how the individuals fared:

THE ENGINEER

This 55-year-old, who wants to be known only as Mr Chin, found the questions challenging.

And that is taking into account that he has more than 30 years of experience as an engineer.

His greatest challenge was Question 1, which took him 13 minutes to complete. Even then, he did not get the correct answer.

He quickly worked through Questions 2 and 3, completing them in 3½ minutes and 8½ minutes respectively.

He had both correct.

Mr Chin says: "Doing the questions can be stressful as they are challenging.

"Even though maths feature often in my work, as an engineer, I prefer real-world problems as there is a sense of the problem being right or wrong, instead of logic questions that are more theoretical."

THE JOURNALIST

Next to take the test was Mr Ng Jun Sen, 26, TNP's Young Journalist of the Year.

The O-level math distinction student confesses that the last time he touched a math book was in university, and that was more than two years ago.

Mr Ng was determined to solve all three questions.

He says: "Even if I do it slowly, I must get it done."

He got Questions 1 and 2 right, taking 25 and five minutes to complete them respectively.

After 45 minutes, Mr Ng was defeated by the final question.

"Kids should just stick to addition and subtraction," he says in jest.

THE MATH WHIZ

Ahnt Htoo Myat was the final candidate for this test.

The 15-year-old student at NUS High School of Mathematics and Science breezed through the first two questions with logic and practice.

He took about five minutes for the Question 1 and three minutes for Question 2. But he did not get the right answer for the second one.

His challenge came at Question 3, which took him close to 40 minutes to complete.

He admits: "I was stunned initially, because I didn't know how to approach the question."

Ahnt has participated in the Math Olympiad for two years and he says that he enjoys logic-type questions, similar to Questions 1 and 3.

"I prefer questions that requires more deduction and less calculation," he adds.

THE ANSWERS
(courtesy of Singapore Mathematical Society)

Q1)

Since (i) says Bob play against Elaine, Elaine can neither be Charlie’s nor Albert’s wife. So either Bob or Don is Elaine’s husband. However, since Elaine’s husband plays against Gillian (v), her husband must be Don.
As Don plays against Albert’s wife (iv), Albert’s wife must be Gillian.
(iii) tells us Fanny plays against Gillian’s husband so Fanny plays against Albert which means (by (ii) ) that she is Charlie’s wife. That leaves Helen who must then be Bob’s wife.

Q2)

In the first hour the two pipes fill \frac{1}{3} + \frac{1}{4} = \frac{7}{12} of the pool per hour.
After the drain was opened, the pool is filled at \frac{7}{12} - \frac{1}{6} = \frac{5}{12} of the pool per hour.
So it takes one more hour to fill the pool.

Q3)

We consider each possible value of “u”. Since the names all share the same letters “J, n,e”, we can work out that the relations “u-a = 1? and “n+y = u+2?.
If u=3 then a=2 and n+y = 5. This would be impossible since (n,y) must be (2,3) or (3,2) but different letters have distinct values.
If u=4 then a=3 and n+y = 6. Again, impossible.
Similar arguments hold for u = 5 or u= 6.
If u=7, then a=6 and n+y = 9 and it is possible to have n=5, and y=4. Note that J+n+e = 10 which means (J,e)=(2,3) or (3,2) gives a possible solution.