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April 24th, 2017, 6:22 pm
Forum: Brainteaser Forum
Topic: I can't solve this ...
Replies: 88
Views: 7426

### Re: I can't solve this ...

this problem is due to Vladimir I. Arnold, who claimed that he had seen it in a standard American examination and used it as an example to criticize American math education. such triangle does not exist.  one way to see it is that the vertex of the right angle lies on a circle with the hypotenuse as...
April 17th, 2017, 10:07 pm
Forum: Brainteaser Forum
Topic: wrap the cube
Replies: 42
Views: 12295

### Re: wrap the cube

here is a reference for people who are interested. paper-folding is one of  Martin Gardner's favorite topics (e.g., hexaflexagons) and he discussed a restricted version of OP's problem in his "Mathematical Games" column ( Scientific American 202, 161-170, 1960 ).  below is a screenshot from one of t...
May 15th, 2016, 3:14 am
Forum: Brainteaser Forum
Topic: Success runs with donuts
Replies: 1
Views: 1665

### Success runs with donuts

<t>henktijms,thanks for sharing this problem. i don't have an exact solution of the problem as it's stated, but had lots of fun thinking about it. below is my take:(1) your estimate of $\lambda$ looks high, e.g., a 12-day gap would be counted three times in your formula. we need to count it just...
August 15th, 2015, 5:40 pm
Forum: Brainteaser Forum
Topic: cut & assemble
Replies: 6
Views: 8044

### cut & assemble

<t>here is a cute puzzle i encountered recently (no obvious spoiler please):suppose you are given a wooden rectangular board with a point marked. your task is to cut the board into two pieces and when you put them back into a new rectangular board, the marked point is the center.follow-ups: how abou...
August 15th, 2015, 5:24 pm
Forum: Brainteaser Forum
Topic: Random walk
Replies: 1
Views: 7912

### Random walk

<t>reaverprog,i dont think you mean vertices, e.g., when n=1, there are only 2 steps, there would be no path for t=1, because the last step always ends at 0, one has P(2,2)=P(1,2)=1/2 and P(0,2)=0.the question would make sense if one considers instead the number of steps on the positive side (2t) an...
June 27th, 2015, 8:00 am
Forum: Brainteaser Forum
Topic: Stale Pi
Replies: 13
Views: 10223

### Stale Pi

<t>believe the non-integer base/radix has appeared more frequently under the name of "beta-expansion", which was introduced first in late 50's, e.g., the golden ratio base has been extensively studied and used in A/D conversion. ExSan should be interested in looking at complex bases, since they are ...
June 27th, 2015, 6:54 am
Forum: Brainteaser Forum
Topic: Secret Santa
Replies: 2
Views: 9414

### Secret Santa

<t>perico,essentially one needs to count the number of permutations without certain types of cycles. to calculate the probability (assuming all the valid draws are equally possible), denominator: the ones with no fixed point, which is known as 'derangement' (e.g., here is one hard version); numerato...
March 17th, 2015, 6:07 am
Forum: Brainteaser Forum
Topic: Prove that E0 is transcendental
Replies: 15
Views: 8106

### Prove that E0 is transcendental

<t>Grenville,i think your question is still open. here are few things that might be helpful for you to search around:(1) the base 2 case of your number E0 is known as the "Prime constant";(2) since it's apparently not normal, which is indeed a strong indication that it is transcendental, as all irra...
March 13th, 2015, 4:58 am
Forum: Brainteaser Forum
Topic: Conditional three-way coloring of a cicular disc with unit radius
Replies: 4
Views: 9693

### Conditional three-way coloring of a cicular disc with unit radius

algosh, i am recommending this book by Soifer. there are so many interesting anecdotes about the history of this so-called "chromatic number of the plane", though there are painstakingly long chapters about van der waerden that one might not find as fascinating...
March 13th, 2015, 4:37 am
Forum: Brainteaser Forum
Topic: four point problem.
Replies: 40
Views: 8317

### four point problem.

please disregard the link if you don't want spoiler.these problems and generalization have been discussed few more times here before
January 19th, 2015, 3:34 pm
Forum: Brainteaser Forum
Topic: Coin flip grouping
Replies: 3
Views: 7520

### Coin flip grouping

<t>suigeneris,this might qualify as an analytic solution but might or might not fill the 5%:the fair coin case (p=1/2) is simple, as the probability of any sequence is same: 1/2^N. to count the number of configurations given there are k groups (1<=k<=N), we count how the k-1 group-dividers (between ...
January 9th, 2015, 3:41 am
Forum: Brainteaser Forum
Topic: What's Next?
Replies: 4
Views: 6372

### What's Next?

sorry for hijacking the thread. the column sum of this sequence was recently encountered while solving this problem of project Euler. wondering if someone has a good strategy to handle the "distinctive" constraint.
November 30th, 2014, 5:02 pm
Forum: Brainteaser Forum
Topic: sin(1deg)
Replies: 43
Views: 8531

### sin(1deg)

<t>couple minor comments:if only the numerical value is needed, one could construct a polynomial such that F[sin(pi/180)]=0, like the cubic equation given by outrun above. with the additional requirement that the coefficients are rational, the minimal polynomial of sin(pi/180) with respect to Q/R ha...
November 24th, 2014, 2:48 am
Forum: Brainteaser Forum
Topic: Can someone explain?
Replies: 11
Views: 6085

### Can someone explain?

<t>there were couple of similar combinatorics problems discussed before - the common strategy is to count in two ways (see e.g., here).suppose total of N contestants, we will count the number of combination (call it S) of the pair of problems solved by some contestant:(1) every two problems were sol...
September 5th, 2014, 2:57 am
Forum: Brainteaser Forum
Topic: Coint toss game
Replies: 16
Views: 11836

### Coint toss game

<t>instead of the 2-player game, suppose there are N people participate the tossing (e.g., with three players, the turns are A,B,C,A,B,C...).same stopping criteria: whoever gets a tail after a head wins.one can prove that the 2nd player has the highest probability of winning while the 1st player has...
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