Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now
50

For some reason, the 'extend it to 100 doors and eliminate 98' explanation doesn't make it any clearer for me. Rather than talk about probabilities as fractions, I explain it this way: "If you picked the car (without knowing it) on the first choice, you'll lose it by switching, whereas if you didn't pick the car, you'll gain it by switching." (stop here ...


25

Your explanation, by the way, is very elegant. As an experienced mathematician, I see immediately that it cuts right to the heart of the matter and admits no ambiguity. Unfortunately, this is precisely the quality that makes it unconvincing to others; the main confounding aspect of Monty Hall is that it ruthlessly exploits an intuitive misunderstanding of ...


18

There isn't any sure-fire method of explaining anything, and especially in math. But specifically in the case of the Monty Hall problem it has been proven by extensive experience that many individuals with otherwise above average intellectual capacities exhibit an exceptional tenacity in refusing to accept the (otherwise) widely agreed upon solution; don't ...


17

I never divide or multiply by 5. I'll double and divide by 2 or divide by 10 and double. In a similar manner, I was in China last summer and found a quicker way to calculate the temperature in Fahrenheit when given the Celsius temperature: (1) Double the temperature in Celsius, (2) subtract off 10% (something that is easy to calculate), (3) add 32.


15

Perhaps it's not the explanation that's the problem. I suggest you have them explain to you their understanding of the problem. Listening to their justification might reveal why your explanation is not gaining traction. Even in the cases where people are saying they now agree with you, you don't necessarily know that they understand the problem. It's ...


14

I've had best luck by simplifying the problem to asking whether you want to pick one door or two doors. Everyone understands they'd be better off picking two doors. At that point I tell them to just ignore the fact that Monty showed them what was behind one of the doors they didn't pick. They're still picking either one door or two doors. Edit: The more ...


11

I found the most helpful way to think about the problem is to expand it to a larger number of doors. For example, if you have them select from 100 doors, where 99 are losers and 1 is a winner. Then after the initial selection is made, eliminate 98 doors. The crux of this explanation is from one of Polya's heuristics, as explained below. The initial answer ...


11

Here are some suggestions for problem sources in English. Some of them are appropriate for very bright students studying geometry or Algebra II, but might nonetheless prove too difficult for students accelerated to this extent. -Mathematical Circles, Fomin et al. -Mathematical Problems: An Anthology, Dynkin et al. -Problems in Elementary Mathematics, ...


10

Some tricks I've seen: Tricks with notable products $(a + b)^2 = a^2 + 2ab + b^2$ This formula can be used to compute squares. Say that we want to compute $46^2$. We use $46^2 = (40+6)^2 = 40^2+2\cdot40\cdot6 +6^2 = 1600 + 480 + 36 = 2116$. You can also use this method for negative $b$: $ 197^2 = (200 - 3)^2 = 200^2 - 2\cdot200\cdot3 + 3^2 = 40000 - 1200 +...


9

(This is more an extended two-part comment than explicit answer.) Part one. Finding "puzzles" (often with a mathematical flavor) is not too difficult. Martin Gardner is probably the main go-to, but "The Art and Craft of Problem Solving" (Zeitz) or the classic "Polyominoes" (Golomb) might fit the bill, too. Some other books I like are "The Tokyo Puzzles" (...


9

I think there are a few independent questions here that can be considered teaching mathematics (I will not broach the subject of how to solve the problem). First, there is: How can you state this problem unambiguously without using artificial language? Indeed, as I see it, the reason this problem sounds paradoxical is that the phrase "one of them is a ...


8

I think $(a+b)(a-b) = a^2 - b^2$ is the most underutilized trick. A bit of practice and it's easy to see how the square is $b^2$ bigger than the rectangle, and conversely, the rectangle $b^2$ smaller than the square. So 45 squared is (40*50)+25 or 2025. And any multiplying where you can latch on to an easy head math where the true numbers are +/- the ...


8

I never understood it until my Maths lecturer explained it to me. Unfortunately I can't remember the exact way he explained it, but I'll try my best to remember. I really like your own explanation, so there is a chance that the following won't work. In that case, you may have to resort to empirical experiment. Another observation from my own experience is ...


8

The best sure-fire method for teaching this to people who don't want to learn it is to set up a Monty Hall style game for small stakes of real money. When they start losing 2/3 of the time, they will become more receptive to your explanation.


7

I devised a trick in 8th grade for converting repeating decimals to fractions. They were teaching a very long drawn-out process. My trick basically does the same thing but for some reason they wouldn't let me use it on the test! Example: $0.36298298298\overline{298} \ldots$ (0.36298 with the 298 repeating). Explanation: 1) take the complete part of ...


6

Because of an exercise we ran every year, I accidentally memorised $\log_{10}2=0.301$, and I could find many logarithms quickly in my head using the log laws. One holiday I decided to take it a little further: If you memorise the base 10 logarithms of 2, 3 and 7, you can quickly deduce the logs of all the other digits in your head and amaze people with ...


6

To tell if a number is divisible by 2, ask whether the last digit even. 3, if there is one digit, ask whether it is 0, 3, 6, or 9, otherwise add the digits and ask whether the sum is divisible by 3. 4, if the second last digit is even, ask whether the last digit is 0, 4, or 8, otherwise ask whether it the last digit is 2, 6. 5, ask whether the last digit is ...


6

Speaking from my own experience, a stage I had to go through before understanding any explanations of the logic and math involved was indignation at being baffled by a cheap fairground trick that probably goes back centuries. Also the eventual recognition that what put me, the punter, at a disadvantage to the operator was the fact that s/he is working to a ...


6

While your explanation is correct, and easily understood by people like us, it's a bit too terse (a quality we like) for a lay person to understand. You can simply expand on it. And most importantly ask the audience questions along the way. Every explanatory sentence should have a question that goes a long with it. The key is for you to identify where the ...


6

I find that most people who THINK they understand the Monty Hall Problem, actually don't. For about 5 years I was one of them, until a further insight made me understand it better. More of that at the end. However, I came up with a super-exaggerated version that seems to give people pause for thought at least. In the UK, our lottery has a probability of 14,...


6

Here's my take on explaining it. (And it usually involves a drawing for me :-) ) Assuming the host opens a door the odds of making the right choice without switching the door is 1/3 because it was made before having the knowledge the host provided. Once you decide to make the switch here is a breakdown of the probability of losing VS winning:


6

I work with gifted elementary school students, but one of my favorite sites, nrich has challenging problems that you could use for older gifted students. Try looking at secondary problems for stages 4 and 5. Here are some suggested problems to see if you'll like the site: You can also look at the Post-16 Curriculum on nrich here. See description below. ...


6

I had a teacher friend who learned in college that word searches were worthless in most respects, but crosswords should have some intrinsic value because it requires you to read the definition and connect it with the proper vocabulary word. So, I did some research and found two things worth sharing: This study reports 90% of students said crossword puzzles ...


5

The squares of the first couple of numbers using only the digit one show a nice pattern and are thus easy to remember. They are collected in the list below. $$\begin{align} 11\cdot11&=121 \\ 111\cdot111&=12321 \\ 1111\cdot1111&=1234321 \\ 11111\cdot11111&=123454321 \\ 111111\cdot111111&=12345654321 \\ 1111111\cdot1111111&=...


5

There are many examples of games that children play (in the same way that kids might enjoy mazes) which have deep mathematics underlying them. An oft-cited example of this is the game of Hex: In the board above, the red and blue players take turns putting down hexagons of their own color. Whoever first traces a connected path between his or her two sides (...


5

This book of Art Benjamin is fantastic. There is a new version of it, but I cannot remember the title. I should mention that although the book contains many specific "tricks for calculation" as requested by the OP, I think the value of the book is to give a taste for mental constructions for calculation that feel an awful lot like what mathematicians do ...


5

Some possibilities coming from algebra are https://mathoverflow.net/questions/93276/a-game-on-noetherian-rings and http://arxiv.org/abs/1205.2884 though it depends on how much algebra the students will have been introduced to at that point. Another possibility is the game Hex (http://en.wikipedia.org/wiki/Hex_(board_game)) since here the impossibility of a ...


5

The big problem with the "Monty Hall" problem is that there are many problems that sound superficially the same, but have different solutions. The terms of the game have to be stated very precisely. As an example, Marily vos Savant's statement of the problem as it is quoted in the Wikipedia article is imprecise. It doesn't state whether the game host must ...


5

I see three issues: the limitations of the English language, information leakage, sampling. Issue 1 was addressed by Ryan Reich. In brief, there is no natural English phrase corresponding to the mathematician's "I have two children, at least one of whom is female." Real people just don't talk that way. Issue 2: in real life, when people talk about ...


4

For propositional logic (especially truth tables and implication) I suggest Raymond's Smullyan's wonderful book, What is the name of this book?. From memory: one example has a prosecutor in a court say to the defendant: 'If you committed the crime, then you must have had an accomplice'. The defendant hotly denies this. But $A \implies B$ is false only if $A$...


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