Brain Teasers and Maths Puzzles

[Tormuse]

I'm not familiar with Bayes' rule, so that all went over my head. You got the right answer, though. (What do those vertical lines in your calculations mean?)

The way I had the answer explained to me was that for each coin, with three coin tosses, there are 2 X 2 X 2 = 8 possibilities. (For the two headed coin, you can't tell the sides apart, but they are still two separate outcomes) That makes 16 total possibilities when you consider both coins. Out of all those possibilities, 9 of them have three heads, and 8 of those 9 are the two-headed coin, making the answer 8/9. (Rounded to two decimal places, that becomes 0.89)

Back when I was in high school, that answer filled me with rage and I argued with my math teacher that the answer should be 1/2, because the act of flipping the coin is independent of the act of choosing the coin, so the results of the coin flip shouldn't affect the probability of which coin you chose. His response was to say, "well, what if the coin comes up tails? Then the chance that you picked the two-headed coin becomes zero, which means the result of the coin flip *does* affect the probability of the coin you chose!"

I have to admit that he has a point, but that answer has still never really sat well with me and I have similar feelings about the Sunday girl puzzle. I guess I just don't really like probability problems; they mess with my feelings of common sense.

[Zyx]

Many people have similar feeling about Bayesian statistics, it goes against intuition. It is quite important thing in cases where you have a very rare thing and a not very perfect way to detect it. The common example is a rare disease and a screening that is 99% accurate. You'd think the test is very accurate, but in reality most of the positive results are false positives (ie. the patient doesn't have the disease even though the test says so). In research done doctors, very few grasp the probabilities even though they should understand them when using the disease screening test.

In probability, P(A) means the probability of event A. P(A|B) means the probability of event A given B, in this case probability that the coin has two heads given that you got three heads in a row. Similarly P(B|A) means the probability that you get three heads given that you have a coin with two heads, naturally this is 100%.

One way to test out most probability things is of course mapping out all the alternatives in a decision tree.

Someone else can come up with the next puzzle.

[Tormuse]

Well, that drug test example helps put things in perspective. Now, all I can think about is this XKCD cartoon.

I can't think of any puzzles right now, but if no one else can think of any, then maybe I'll Google one or something.

[eMTe]

I have one, rather light and relaxing, puzzle.

How to cut the cake into eight pieces with three straight cuts? (You're not allowed to move the cake or otherwise manipulate the pieces)

[Tormuse]

My first thought was to wonder what the shape of the cake is, (Round? Square?) but then it occurred to me that it doesn't matter, because you didn't say the pieces have to be equal.

My second thought was that you need to make the three cuts so that you maximize the number of points of intersection between them. If all three cuts are done vertically, then you can only get 7 pieces, so you need to take advantage of the third dimension and make a horizontal cut. So, my solution is to make two vertical cuts that cross in the middle and the third cut goes horizontally through the middle of the cake.

Ooh! Bonus! If the cake is symmetrical, all 8 pieces will be equal!

[eMTe]

Yup!

One funny thing about this puzzle is that you start to think about it as an abstract exercise - so you draw the flat wheel in your mind or on the paper and begin to draw lines. Only a moment later you realize that cake is three dimensional and the puzzle is in fact a simple action, common in real life.

[Tormuse]

Yeah, the only problem is that using this method, only half the pieces have icing on them.

[eMTe]

What's the problem if you have the whole cake for yourself?

[eMTe]

This one is rather light and not very serious, however it's a good exercise in logic, so if you don't have anything better to do you can elaborate on the subject.

Who is right? The one who stole the right or the one who was stolen the right?

[eMTe]

Time to revive the topic.

You are a landscape specialist, and have been asked to design a garden for a math professor. He wants four trees that are all equidistant from each other. How do you place the trees?

[Zyx]

He can want whatever he wants but I'm going to put them where-ever I want.

In a pyramid formation.

[Pater Alf]

How about planting them in the middle of the garden, so that they all touch each other?

[eMTe]

In a pyramid formation.

Dat ist korekt.

However I must add that you forgot to say that to arrange trees in a pyramid formation one must first create an artificial hill in the middle of the garden.

As if a math teacher was rich enough to build a personal hill.

[Zyx]

How about planting them in the middle of the garden, so that they all touch each other?

This is a math teacher, so we assume the distance is calculated from the center of these equally thick, perfectly round trees.

[eMTe]

My guess the creator of teh puzzle was math teacher playing Minecraft.

[Tormuse]

(Alternative solution to the above puzzle: Dig a hole and plant one of the trees at the bottom to form an inverse pyramid!)

I want to revive this topic because I saw a really good logic problem earlier today which I'll transcribe here. (I'm gonna edit the wording a bit so the meaning is clearer; It took me a while to figure out, but it's definitely solvable)



Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.

May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17

Cheryl then talks to them separately and tells Albert the month and she tells Bernard the day number.

Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know too.
Bernard: At first I didn't know when Cheryl's birthday is, but now I know.
Albert: Then I also know when Cheryl's birthday is.

When is Cheryl's birthday?

[eMTe]

Somebody here follows IFLS, I see.

[Tormuse]

(I don't know what that is; my brother sent me this puzzle)

Does anyone want to take a crack at this puzzle? I just realized that it's been here a month! I can give a hint if you like?

[eMTe]

I don't engage my brain during sunny weather, as a rule.

Then your brother follows IFLS. The puzzle or its version or version of the version was very popular recently.

[Tormuse]

Now that there's some activity on these forums, maybe it's time to bring back this topic! It looks like I was the last one to post a puzzle (two years ago!) and nobody solved it! The funny part is that it's been so long, that I don't remember the solution, so if no one else wants to try, then maybe I can try to solve it all over again.

EDIT: Okay, I couldn't resist figuring it out for myself. I'll spoilerify my answer in case anyone else wants the fun of figuring it out too.

So Albert says he doesn't know the date, which makes sense because each month has multiple days, so he can't know which one it is, but the fact that he knows that Bernard also doesn't know is useful information. This means that the month can't be May or June, because each of those months has a day number that is unique to it. (19 for May and 18 for June)

Bernard, hearing Albert say this, now knows that the month must be July or August. Since he says he now knows, we know it can't be the 14th, because that number is shared between July and August. This narrows down the options (as far as we know) to July 16, August 15, or August 17.

And now that Albert has heard that Bernard knows, he says he now knows. This means that it can't be August, because if the month had been August, there would be no way for Albert to know whether it's the 15th or the 17th, therefore, the answer must be July 16.

tldr; Cheryl's birthday is July 16.