I don't know the context around which this was presented to students, but it's much easier if you think of it as a set operations problem than a pure logic one.
Also nice to see that the age old act of providing irrelevant distracting information in word problems is still alive and well.
A neat problem, though. I'm curious whether I would have gotten it at that age.
Great question! Initially, I didn't quite understand the premise of the question which is why I couldn't get the right answer but after googling around I found the correct premise and the correct solution in this article: http://www.independent.co.uk/news/world/asia/singapore-maths...
But, I think some people still won't understand the solution without visualisation! So, here it is:
"Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively."
Great! Now we know:
* Albert knows the month
* Bernard knows the date
"Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.
The only way Albert (he knows the month remember) can say "but I know that Bernard does not know too." for definite is if the month he knows is a month that has a date which clashes with other months!
Why? Because if the month he knew had a unique date then there's a possibility that Bernard could know the month hence he couldn't know for definite that Bernard didn't know the birthday. For example if Bernard knew the date was the 19th then obviously the only month with 19th date in it is May hence we would know the date of birth is the 19th of May.
Because, of the information above we can completely disregard both the months wholly with the unique date in them which are May & June.
"Albert: Then I also know when Cheryl’s birthday is."
So, now because Albert knows the month is July is can say for definite that the full date of birth is 16 July because there is no other date in that month left.
Explanation: Because if it was August then Albert wouldn't know the date of birth as there two date for him to choose between (the 15th and 17th) but because Albert does know the date; we know the month is July and the only date left in July is the 16th!
1. Take the subset of months without a globally unique day? (Bernard doesn't know the answer)
2. Of the subset of months without a globally unique day, is there a globally unique day within that subset? (Bernard knows the answer, and therefore so does Albert)
And this is why I didn't do well in math. Whenever I got a question like this, my first thought was always along the lines of "Cheryl is fucking bitch, just tell them your goddamn birthday so I can solve a real problem".
How does your embarrassing comment add to the discussion we are having? I'm afraid the gentlemen on this forum could not care less for your childhood struggles. And just in case you're wondering, your math performance was probably hindered by your flagrant, insulting and imprecise use of language.
Also an extremely pretentious way to refer to the users of this forum. I have no reason to believe HN users are more cultured than the rest of society.
Because A knows the month and says B doesn't know the birthday, it cannot be May or June since they have unique days (Which would allow B to know the birthday). So far so good.
After this statement B know it's one of the remaining two months, July or August and that's enough for him to know the correct date hence they day he was told cannot be the 14th. leaving only July 16th, Aug 15 and Aug 17.
Since this information is enough for B to know the exact date, it clearly cannot be August, leaving only July the 16th.
In the last line, A notes that B knowing means that A now knows. This is only possible if it is July, since knowing the month isn't enough to distinguish between the two possible dates in August.
Albert: I don't know, but Bernard might.
Bernard: Then your birthday is in May!
Albert: Ha ha! I was lying about what I knew.
Bernard: So was I!
Cheryl: Then you both know my birthday now.
(May 15, 16, 19, June 17, 18, July 14, 16, August 14, 15, 17)
If Bernard knew that Albert's first comment was impossible, then by the principle of explosion his first statement is correct according to his premises. Therefore his seond statement saying that his first statement is a lie would be incorrect.
This is a recurring but stupid theme. Look people in the past/in other country/in other school know how to solve problems which are unfamiliar to us! We have no clue and they do it pretty easy. They must be very smart and learned?
Take aways:
- You never learned to solve problems, just apply familiar patterns. This is sad, you wasted youth.
- Those people probably didn't either, but they were given a different set of patterns. Doesn't make them smarter.
- Anyone with actual solving skill looks at all of you with amusement.
Agreed that the news articles tend to dramatize these differences as some daunting gap in competence, and the subjects tend to be cherry-picked to illustrate that point. Still, patterns can differ in complexity and depth too. Say it's geometry question--recognizing a 3-4-5 right triangle is one matter. Creating a general proof of perpendicularity in a 3D object that contains right triangles is objectively more advanced. (I'm referring to another hyped-up article: http://news.bbc.co.uk/2/hi/uk_news/education/6589301.stm )
I had to solve tests not unlike the chinese one from the article you refer (less complex, but closer to first one than to second) to get accepted into university - and never had any use for any of those skills.
This called "admission math" and despised by people who are into real math. It's all grinding values without understanding where they lead.
I had a whole class failing to apply basic logarithm2 in a bisection algorithm and they had no idea what's going on. They didn't see logarithm as a tool! They solved a lot of questions with half dozen of uneven logarithms, but they had no idea when they met one in a real life.
http://egeurok.ru/resh_mat/10_11kl_Mordkovich/5/43.22.jpg I'm talking about craziness like this.
What is "solving skill"? Where can I get some of that? Why do the people who have it look at the people who don't with amusement, are they jerks? Should those of us without "solving skill" rise up against them? This changes EVERYTHING people!
I don't know where. Where do you get some of foreign language knowledge? I don't have such place ready for you, yet here I am writing to you in a language that is foreign to me.
"Why do the people who have it look at the people who don't with amusement?"
Because people who don't have it consistently praise the wrong thing.
You get foreign language learning by looking at examples of the patterns used in a language and learning how to apply them. By your logic there should be a general "language skill" that anyone could use to immediately apprehend any language. No such thing exists.
When I started to learn English any non-normal form like 'ye olde' or 'lotsa' threw me into stupor. Now I can derive meaning from context while gaining better understanding.
What are the "actual solving skills" that allow you to generally solve problems that you have never seen a prototype of before? Pretty much all learning is recognizing what types of patterns there are and how to apply them. People who are good at solving logic puzzles like this tend to have seen a lot of them before. In the real world "applying the method used to solve a similar problem" is a solving skill used to solve actual problems. If there are generalizable skills, why aren't we teaching them? And how would we teach them if not by showing examples of problems where they are applied?
We are not teaching actual solving skills because it's hard to check for them.
You can't just give a new problem to a bunch of kids and expect them to yield a measurable result. Some may be already familiar with this kind of problems, other may be only able to get ideas when bouncing them against other kids. It's a mess.
Solving a hundred of similar questions? That gives you real performance score - how children concentrate under stress while doing boring repetitive tasks.
While I share an awful lot of your antipathy to grading there's an entire discipline devoted to how to measure capabilities and knowledge, psychometrics.
In mathematics you don't understand things, you just get used to them -- John von Neumann
The "actual solving skills" you speak of are a product of talent and application, in various proportions. What I know of the psychology of expertise speaks strongly against your thesis. Experts see things effortlessly that others have to really work at, and things others can't see without massive handholding because the basic chunks of understanding they work with are so much larger than those of novices or the merely competent.
Problem solving skills are developed. You are not born with them. Some people have talent, they learn faster and most of them hit the wall where they have to grind later and some people have sufficiently little talent that trying to learn a topic or skill is a waste of their time but everybody has to work at it.
I would be interested to see this written formally (at least the premise, if not also the steps) -- is there a Boolean notation for this sort of thing?
Perhaps a combination of boolean logic and set theory? For instance let X be the set of possible dates and let m, d, be functions which, respectively, return the month and the day of any date in X.
Now the three statements can be translated to the following logical statements (this isn't necessarily a good thing to do, but you can):
I was great at this and would have solved it before that age (not actually a brag). I would never be in that olympiad. I was not good at maths when it turned from numbers to letters.
When solving this I didn't apply any form of maths, but non-articulated abstract reasoning I would have no idea how I'd put on paper.
What I'm saying is: Is this really math? I'm great with puzzles, not math. They are not translatable for me.
Fair enough then. I guess I got hung up on not being able to model it on paper, or relate it to any maths I've studied, but I can't deny your definition.
This is most definitely math, it is more related to logic than geometry / calculus though so the mathematics behind it might be somewhat unfamiliar to you if you haven't followed any logic oriented courses.
Imagine you're Bernard and you've just been handed the day of the month. If you got 14, 15, 16, or 17 then you don't know the month, but if you got 18 or 19 then you know that the month is May/June.
I shouldn't have tried to fit that on one slide :)
Consider the case if the birthday is May 19th:
- Bernard knows the date, and he knows that May 19th is the only possible "19th", so he knows the answer immediately.
- Albert would know it's in May, and that May 19th is possible. So, he knows there's a chance that Bernard knows the birthday immediately.
Since in the question, Albert is confident that Bernard doesn't know the birthday immediately, we can conclude that the month cannot be May. The exact same logic applies for June and the 18th.
The only possible months are July and August, because every possible date in July and August also appears in another month.
58 comments
[ 2.8 ms ] story [ 124 ms ] threadI'm certainly going to need a lot more practice if I want to place in Singapore's top 40% of 14/15 year olds
Also nice to see that the age old act of providing irrelevant distracting information in word problems is still alive and well.
A neat problem, though. I'm curious whether I would have gotten it at that age.
But, I think some people still won't understand the solution without visualisation! So, here it is:
Well, I started of by setting a table like this: https://i.imgur.com/VhJ1gSZ.png
Next we are told:
"Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively."
Great! Now we know:
"Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.The only way Albert (he knows the month remember) can say "but I know that Bernard does not know too." for definite is if the month he knows is a month that has a date which clashes with other months!
Why? Because if the month he knew had a unique date then there's a possibility that Bernard could know the month hence he couldn't know for definite that Bernard didn't know the birthday. For example if Bernard knew the date was the 19th then obviously the only month with 19th date in it is May hence we would know the date of birth is the 19th of May.
Because, of the information above we can completely disregard both the months wholly with the unique date in them which are May & June.
So, we now have a new table which we can update the hit, misses and clashes on: https://i.imgur.com/LoBJWa1.png
"Bernard: At first I don’t know when Cheryl’s birthday is, but I know now."
At this point Bernard knows that the choice is either between July or Aug and he goes ahead and says he now knows what the full date of birth is!
This tell's us that the date is a unique from the updated table because if it wasn't a unique date then he couldn't have possibly known the month!
So we can update our table again: https://i.imgur.com/UUyNQQC.png
"Albert: Then I also know when Cheryl’s birthday is."
So, now because Albert knows the month is July is can say for definite that the full date of birth is 16 July because there is no other date in that month left.
Explanation: Because if it was August then Albert wouldn't know the date of birth as there two date for him to choose between (the 15th and 17th) but because Albert does know the date; we know the month is July and the only date left in July is the 16th!
Hence the correct answer is 16th July!
Hope that helped!
1. Take the subset of months without a globally unique day? (Bernard doesn't know the answer)
2. Of the subset of months without a globally unique day, is there a globally unique day within that subset? (Bernard knows the answer, and therefore so does Albert)
After this statement B know it's one of the remaining two months, July or August and that's enough for him to know the correct date hence they day he was told cannot be the 14th. leaving only July 16th, Aug 15 and Aug 17.
Since this information is enough for B to know the exact date, it clearly cannot be August, leaving only July the 16th.
Take aways:
- You never learned to solve problems, just apply familiar patterns. This is sad, you wasted youth.
- Those people probably didn't either, but they were given a different set of patterns. Doesn't make them smarter.
- Anyone with actual solving skill looks at all of you with amusement.
This called "admission math" and despised by people who are into real math. It's all grinding values without understanding where they lead.
I had a whole class failing to apply basic logarithm2 in a bisection algorithm and they had no idea what's going on. They didn't see logarithm as a tool! They solved a lot of questions with half dozen of uneven logarithms, but they had no idea when they met one in a real life. http://egeurok.ru/resh_mat/10_11kl_Mordkovich/5/43.22.jpg I'm talking about craziness like this.
"Why do the people who have it look at the people who don't with amusement?"
Because people who don't have it consistently praise the wrong thing.
When I started to learn English any non-normal form like 'ye olde' or 'lotsa' threw me into stupor. Now I can derive meaning from context while gaining better understanding.
You can't just give a new problem to a bunch of kids and expect them to yield a measurable result. Some may be already familiar with this kind of problems, other may be only able to get ideas when bouncing them against other kids. It's a mess.
Solving a hundred of similar questions? That gives you real performance score - how children concentrate under stress while doing boring repetitive tasks.
Grading is evil.
http://en.Wikipedia.org/wiki/Psychometrics
While devising excellent tests is difficult, okay ones aren't that hard either.
The "actual solving skills" you speak of are a product of talent and application, in various proportions. What I know of the psychology of expertise speaks strongly against your thesis. Experts see things effortlessly that others have to really work at, and things others can't see without massive handholding because the basic chunks of understanding they work with are so much larger than those of novices or the merely competent.
Problem solving skills are developed. You are not born with them. Some people have talent, they learn faster and most of them hit the wall where they have to grind later and some people have sufficiently little talent that trying to learn a topic or skill is a waste of their time but everybody has to work at it.
Now the three statements can be translated to the following logical statements (this isn't necessarily a good thing to do, but you can):
P(B) = #m^-1(m(B)) > 1 ∧ ∀x∈m^-1(m(B)) : #d^-1(d(x)) > 1
Q(B) = ∃!x∈X: d(x)=d(B) ∧ P(x)
R(B) = ∃!x∈X: m(x)=m(B) ∧ Q(x)
your task is then to find a birthday B in X such that P(B), Q(B), and R(B) hold. Or equivalently find the unique element of {x∈X : P(x)∧Q(x)∧R(x)}.
Edit: Still not 100% sure if P(B) is correct.
When solving this I didn't apply any form of maths, but non-articulated abstract reasoning I would have no idea how I'd put on paper.
What I'm saying is: Is this really math? I'm great with puzzles, not math. They are not translatable for me.
Is it abstract (can be retold about other unrelated objects without changing solution)? Yes.
Does it have a definite answer? Yes.
Do you use reason to get from A to B in simple concrete steps? Yes.
It's math then.
For examples, look at page 16 of the 2013 exam: https://www.maths.ox.ac.uk/system/files/attachments/test13.p...
and page 16 of the 2014 exam: https://www.maths.ox.ac.uk/system/files/attachments/test14.p...
https://docs.google.com/presentation/d/1YVe2WymBiu_zBwZ9Dnqu...
Consider the case if the birthday is May 19th:
- Bernard knows the date, and he knows that May 19th is the only possible "19th", so he knows the answer immediately.
- Albert would know it's in May, and that May 19th is possible. So, he knows there's a chance that Bernard knows the birthday immediately.
Since in the question, Albert is confident that Bernard doesn't know the birthday immediately, we can conclude that the month cannot be May. The exact same logic applies for June and the 18th.
The only possible months are July and August, because every possible date in July and August also appears in another month.
Hope that makes more sense!
One example is "Sherlock" by Everett Kaser.
http://www.kaser.com/
It's like giving someone a Sudoku problem on a Math test. If you'd never seen it before, you'd probably fail, but if you'd seen it before, it's easy.