Two weeks ago, a Singapore TV presenter shared a Mathematics question on his facebook feed. It involved a character called Cheryl and her friends Albert and Bernard.
As good friends go, Albert and Bernard wanted to find out about Cheryl’s birthday and perhaps send over some greetings via facebook or such. Cheryl was coy though, and only told each of them separate pieces of information about her birthday. Albert and Bernard had to work together to figure out her birthday before they could even get close.
Love triangles are fascinating. This one is no different and it captured the imagination of the population. It went viral in a flash. For a few days, no conversations I had with my friends went by without any mention of Cheryl and her birthday. The international news networks picked it up and CNN, BBC and the New York Times all ran articles on it. It even has its own Wiki page.
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The question was tough. So tough that it prompted outrage initially when the question was thought to be from the Primary 5 examination. It was later clarified that the question originated from the Singapore and Asian Schools Maths Olympiad (SASMO), a competition styled set up meant for older kids.
Here is the question reproduced
Albert and Bernard just become friends with Cheryl and they want to know when her birthday is. Cheryl gives them a list of ten possible dates.
12th May, 16th May, 19th May, 17th Jun, 18th Jun, 14th Jul, 16th Jul, 14th Aug, 15th Aug and 17th Aug
Cheryl then tells Albert and Bernard separately the month and day of her birthday respectively.
Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.
Bernard: At first I do not know when Cheryl’s birthday is, but now I know.
Albert: Then I also know when Cheryl’s birthday is.
Discovering Cheryl’s birthday.
Here is how the logic goes.
Statement 1. Albert does not know when Cheryl’s birthday is. He knows that Bernard does not know too. If Cheryl had told Bernard that her birthday is either on the 18th or the 19th, Bernard would have known, because these are unique dates. Hence, 18th June and 19th May are out.
In addition, because Albert knows that Bernard does not know, the month May and June are also out.
This is the crucial part of the process that tripped myself and a lot of people up. Albert must think one step ahead. Or rather, in order to solve the puzzle, we must think one step ahead for Albert. It is not good enough to know what Albert or Bernard knows, we also need to understand the implications of what the individuals does NOT know.
Statement 2. Bernard now knows that it can only be July or August. Since the 14th involves both month and is not a unique date, it is to be eliminated. (Remember, Bernard now knows the birthday. If it is the 14th, there is no way he can tell).
Statement 3. There are 3 dates remaining – 16th July, 15th Aug and 17th Aug, Since Albert has been told the month, the only way for him to know for sure would be that there is only one day in the month. Hence the answer is 16th July.
The Most Important Thing
Howard Marks is an American billionaire investor and the founder of Oaktree Capital. He writes a memo to his shareholders. All the memos are available here. In terms of wisdom per square foot, the memos are definitely in the same league as Warren Buffett’s Berkshire shareholder letters.
In 2011, he published a book called The Most Important Thing – Uncommon Sense for the Thoughtful Investor. The very first chapter is called Second Level Thinking. In this chapter, Marks differentiates between First and Second Level Thinking.
First Level Thinking is simplistic and superficial. Just about anyone can do it. Second level thinking deep, is deep, complex and convoluted. It takes many more factors into account.
First Level Thinking vs Second Level Thinking
First level thinking says – It is a good company; let’s buy the stock. Second level thinking says – It is a good company, but everyone thinks it is a good company. The stock is overrated and overpriced; let’s sell the stock.
First level thinking says – The outlook calls for low growth and rising inflation. Let’s dump our stocks. Second level thinking says – The outlook stinks, but everyone else is selling in panic. Buy!
First level thinking says – I think the company’s earnings will fail; sell. Second level thinking says – The company’s earnings will fail, but it is already priced in. If it falls less than what people expect, the pleasant surprise will lift the stock. Buy.
From an investing standpoint, Marks shares some thought on the second level process. They include thinking about not just one, but an entire range of future possible outcomes. Instead of shutting down other outcomes, he weigh the likelihood of each materialising. Second level thinking also involve being aware of the general consensus, how much the predicted outcome differs from the masses, and how would prices react in each different scenario.
Because it is so prevalent, First Level Thinking is insufficient at producing any result over and above the masses. Yet because it is so easy, just about everyone engages in first level thinking. A vicious cycle is perpetuated.
Thinking in levels.
As an investor, first level thinking is all about what you know. Second level thinking is all about knowing what others know (or do not know) and drawing your conclusions about the market by taking that into consideration.
It is exactly like how Albert knows what Bernard does and does not know. And how Bernard knows the same about Albert. Only when they put all the information together can they both come to derive Cheryl’s birthday. They will never be able to derive the answer if they remain in Level 1 thinking.
Maths Olympiads are not meant to be easy. Neither is investing. Do not be deceived.
As Howard Marks quotes Charlie Munger
It is not supposed to be easy. Anyone who finds it easy is stupid.
PS. Speaking of birthdays, over at Cheerfulegg, Lionel just turned the big three o. Check out his (epic) post about life lessons here.