A headache-inducing logic problem from Singapore's Math Olympiad went viral earlier this year, sparking online debates, a Twitter hashtag, and even a song that mimics the process of elimination that leads to the correct answer.
The task — figuring out a girl named Cheryl's birthday with seemingly little information — was maddening at first glance.
But the problem wasn't nonsense: it's actually a test of logical reasoning skills. And questions like these help explain how Singapore's students have come to rank as some of the best problem-solvers in the world — by being taught math differently, and well.
A 2005 study from the American Institutes for Research praised Singapore's method of teaching math, saying it was much better than the American method. On reason was that word problems and real-world examples were used not just to show students that math is important outside the classroom, but to illustrate how math works.
Here's the problem Singaporean high school students were asked to solve, reworded slightly for clarity:
Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl marks 10 possible dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, or August 17.
Then Cheryl tells Albert the month of her birthday, but not the day. She tells Bernard the day of her birthday, but not the month. Then she asked if they can figure it out.
Albert: I don't know when Cheryl's birthday is, but I know Bernard doesn't know either.
Bernard: At first I didn't know when Cheryl's birthday is, but now I know.
Albert: If you know, then I know too!
When is Cheryl's birthday?
This problem is meant to test logical and analytical reasoning skills. Students have to work backward from the information they're given to the solution. It's not something that can be solved by a formula, but requires rigorous logical and technical skills.
So when is Cheryl's birthday?
Good news if you're struggling: this is not from a general textbook. It was designed for the top 40 percent of high school math and science students in the country for use in a math competition, and was a relatively challenging question for that competition, according to the Singapore and Asian School Math Olympiad, the group that wrote the question.
Bernard, who knows the date, has more information in this situation than Albert, who knows the month. No matter what Cheryl told Albert, there's no way Albert could figure out her birthday.
Bernard, though, might have been able to figure out Cheryl's birthday on his own, but only on two out of Cheryl's 10 possible dates.
If Cheryl told Bernard she was born on the 18th or 19th, he would know her birthday right away. That's because the 18th and 19th only show up once in Cheryl's list of possibilities. The rest of the dates are duplicates: the 14th could be in July or August, the 15th could be in May or August, the 16th in May or July, and the 17th in June or August. But the 18th could only be in June, and the 19th only in May.
But Bernard didn't know when Cheryl's birthday was at first — so she wasn't born on the 18th or 19th.
How did Albert know that Bernard didn't know? When Cheryl told him the month, she must have said July or August, because every possible date in July and August is also in another month. If she had told Albert she was born in May or June, she might have been born on May 19 or June 18, and Albert wouldn't be certain that Bernard was in the dark.
As soon as Albert says that, though, Bernard figures it out. He knows Cheryl must have told Albert her birthday is in July or August, because that's the only way Albert can be certain that Bernard doesn't know her birthday. Narrowing the months down to two possibilities is all it takes for him to find the answer. That means the date Cheryl gave Bernard must not be a possibility in both July and August. It's only a possibility in one of the two months.
We already knew Cheryl's birthday isn't the 18th or 19th. Eliminating the dates that are in both July and August knocks out the 14th. That leaves three possible birthdays out of the original 10: July 16, August 15, or August 17.
But now Albert knows, too. That means Cheryl must not have told him she was born in August, because he'd still be confused; two of the possible dates are in August. The only remaining possibility is July 16.
The best math and problem-solving students in Singapore are really good
These problems show up in Singapore for a reason: they're meant to strengthen and test students' problem solving skills — and they seem to work.
"This kind of problems trains a person to analyze a problem in order to come to a logical solution," Henry Ong, the director of Singapore and Asian Schools Math Olympiad, wrote in a statement. "We are not saying this problem is for every student (since it involves rather sophisticated reasoning)."
Singapore has a disproportionate share of top problem-solvers. Its 15-year-olds had the highest scores, tied with Korea, on the problem-solving portion of the Programme for International Student Assessment, a standardized test administered to students in developed countries in 2012. Almost 10 percent of students performed well enough to be considered "highly skilled problem-solvers" — the highest share in the world. In the US, just 2.7 percent of students tested that well.
Students might be so good at problem-solving because they're very good at math. Students in Singapore scored second in the world, behind students in Shanghai, on the math portion of the PISA. Forty percent of Singapore's students are considered top performers in math, compared with just 8 percent in the US.
In another international test of math abilities, the Trends in International Mathematics and Science Study, Singapore has been ranked at or near the top every time it's been given since 1995.
Why Singapore's students are so good at math
In the US, teachers teach a concept, move on to the next one, then return to the earlier concept and teach it again, later that school year or the next year. In Singapore, students are expected to fully understand a concept before moving on: they start learning fractions two years before they learn decimals, for example. Teachers still review, but in much less depth.
Students in Singapore also start out solving problems with concrete objects. Then they move on to drawing pictures to represent the problem. Then they solve the problems in the abstract, using only numbers.
Here's an example of how drawing out a word problem works:
These methods are supposed to help students develop number sense — an understanding of how numbers are composed of other numbers, and a deeper understanding of why math works the way it does. In the US, the American Institutes for Research report found, students too often learned math as a rote series of steps without understanding what they were doing or why.
Can Singapore's methods work in the US?
Because Singapore has a national curriculum, it might seem easy to export the secrets of its math success: just give Singapore's textbooks and lesson plans to American students.
Doing just that has in fact become increasingly popular. Two parents who moved to the US from Singapore in the late 1990s began importing and distributing Singapore textbooks, at first for homeschoolers, and later for schools. The textbook publishing giant Houghton Mifflin Harcourt later adapted a series of Singapore textbooks for use in the US. Singapore math textbooks have been used in Los Angeles; Montgomery County, Maryland; Seattle; and dozens of smaller districts.
The central concepts of Singapore's approach — teaching fewer concepts in more depth and developing number sense — are front and center in the Common Core standards.
Studies of schools that have used Singapore's textbooks show mixed results: the people who love it really love it, and test scores go up. But it's challenging for teachers, and many schools who try the textbooks eventually give up. In the early 2000s, four schools in Montgomery County, Maryland, tried the Singapore approach, but quickly abandoned it due to poor planning and preparation and concerns about how much the switch would cost.
Some Los Angeles schools use, and love, the Singapore textbooks, crediting them for an increase in standardized math test scores. But they haven't been adopted districtwide.
Just importing textbooks isn't enough, several reports have found. The 2005 report from the American Institutes for Research noted there are other possible explanations for Singapore's success.
Singapore is a tiny country — 5.4 million people, about the same size as Finland — with a highly centralized education bureaucracy. Its standardized tests are more challenging, and schools are judged not only for how many students are proficient but for how much progress they made. (The US only judges proficiency.)
And standards for teachers are in Singapore are higher. The 2005 report noted that the Praxis test, the standard certification exam for American teachers, had multiple-choice math questions. In Singapore, those questions would be at best at a sixth-grade level, it noted.