Wikipedia:Reference desk/Archives/Mathematics/2013 November 13

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November 13

Hilbert's third problem

What is known about the problem if it is generalized to dimensions higher than 3? Is it always false? 68.0.144.214 (talk) 02:39, 13 November 2013 (UTC)

A three dimensionsl polyhedron can be embedded in any space with more than 3 dimensions, and adding more dimensions does not increase the ways in which that polyhedron can be dissected by planes or hyperplanes. So I think if you take two polyhedra which are not scissors-congruent in three dimensions, and then embed then in a higher dimensional space, they will not be scissors-congruent in that higher dimensional space either. Which then gives you a counter-example to the conjecture that all polyhedra in the higher-dimensional space are scissors-congruent. Gandalf61 (talk) 08:55, 13 November 2013 (UTC)

I don't know anything about this subject beyond the linked article (not my thing), but here are links that discuss higher dimensional objects: [1], [2], [3], [4], [5], [6], [7] (see around pg.139, and [8].Phoenixia1177 (talk) 09:57, 13 November 2013 (UTC)

Interesting number sequences

Howdy. I'm working on a project to assemble a database of "interesting" numbers - or at least, properties of numbers that the average person could understand. My main problem is that I'm low on ideas of properties that most people would recognise by name. I'm hoping you guys and gals can give me some suggestions?

So far I've got:

• Prime numbers
• Fibonacci numbers
• Square numbers (n^2)
• Cubic numbers (n^3)
• Powers of 2 (2^n)
• Palindromic numbers
• "Binary" numbers
• Numbers consisting of one digit repeated
• Numbers containing digits in ascending/descending order (eg 12345, or 654)

I'm aware some of these aren't strictly mathematical, but it's more about numbers that are either part of a "famous" sequence (household-name level of fame), or are visually interesting. If anyone has any suggestions I'd love to hear them. Organics^LRO 11:10, 13 November 2013 (UTC)

I don't have any specific suggestions, but you may want to have a look at [9] (online ency. of int. seqs.) if you're not already familiar with it- you might be able to get some ideas there.Phoenixia1177 (talk) 11:14, 13 November 2013 (UTC)

Thanks for the suggestion, I'm familiar with it and have had a flick through, but a lot of it is beyond me, and without some kind of "most visited" list it's hard to know what to go for. Organics^LRO 12:54, 13 November 2013 (UTC)

You may find "The Penguin Dictionary of Curious and Interesting Numbers" worth a read, since it deals with a similar topic (though focussing on individual numbers). MChesterMC (talk) 14:02, 13 November 2013 (UTC)

I had a flick through the preview on Google Books and it does look like an interesting read. Many thanks! Organics^LRO 09:59, 14 November 2013 (UTC)

Maybe triangular numbers, perfect numbers, and Pythagorean triples. Duoduoduo (talk) 15:28, 13 November 2013 (UTC)

And Lucas numbers. Duoduoduo (talk) 15:34, 13 November 2013 (UTC)

I don't know how I forgot triangular numbers! All very good choices, many thanks. Organics^LRO 09:59, 14 November 2013 (UTC)

• You should certainly look at our article on the interesting number paradox. (For those who don't know, the paradox is that it is possible to prove by induction that all natural numbers are interesting. Because, if there are any uninteresting numbers, the principle of induction implies that there must be a smallest such number. But being the smallest uninteresting number is quite an interesting property.) The article discusses some ramifications of the concept of "interestingness", and gives a couple of useful references. Looie496 (talk) 16:01, 13 November 2013 (UTC)

I've heard of that paradox, and it's one of the first things that crossed my mind when I finished assembling the sequences I listed above. My database has a minimum value of 100 (because of reasons), and the smallest "uninteresting" number so far is... 102. =/ Organics^LRO 09:59, 14 November 2013 (UTC)

Does anything listed at 102 (number) make it interesting to you? :-) Katie R (talk) 19:49, 15 November 2013 (UTC)

• Integers that are not definable in fewer than twelve words. Count Iblis (talk) 19:19, 13 November 2013 (UTC)

Hah, I think I remember that from an episode of QI? Because once you know the smallest integer which uses more than twelve words, you can define it as "the smallest integer not definable in fewer than twelve words" which is only 10 words. I like it, but I think one paradox is enough for me ;) Organics^LRO 09:59, 14 November 2013 (UTC)

How are you structuring this? Would you just look up primes, then be presented with prime number seq and some values? Or are you "clustering" related types together? For example, there are Triangle numbers, Square numbers, etc., but they're all Figurate numbers. By the same token, there's lots of numbers that come from summing divisors (a lot, actually), the same goes for all the various families of primes, and many other cases. What I'm getting at, would it be useful to suggest interesting families of sequences, or do you just want sequences?Phoenixia1177 (talk) 05:50, 14 November 2013 (UTC)

Well, without getting into too much detail - I'm putting it all in a database, so you'd search for a number and it will report back its' properties. I haven't been clustering properties so far, but by all means suggest some families and I'll gladly take a look. I'm trying to avoid anything reliant on pairs or groups of numbers, e.g. Home primes, although facinating, are defined as "the home prime of x is y", which isn't really a property of x, and although possibly a property of y it isn't really what I'm going for. Organics^LRO 09:59, 14 November 2013 (UTC)

I created a template a short while ago which groups some types positive integers: {{Classes of natural numbers}}. This excludes numbers where all members in the sequence are prime, those types are listed in {{Prime number classes}}. There is some degree of overlap between the types in the two templates. -- Toshio Yamaguchi 11:39, 15 November 2013 (UTC)

Ah, most excellent! Of course the first thing I realise when looking at the first template is that I forgot powers of 10... Facepalm. Organics^LRO 12:58, 15 November 2013 (UTC)

calculating probability

Not homework, I'm just too stupid to figure it out myself. There are a total of X original given numbers. A bettor selects Y numbers out of X. Then, Z numbers are selected out of X on a random basis (Z > Y). By what formula can one calculate the probability that the bettor will have T guesses, that is, out of the Y numbers, T will match Z numbers? Qoin (talk) 16:02, 13 November 2013 (UTC)

I believe this is the Hypergeometric distribution. When calculating the probability, it makes no difference whether the Z "winning" numbers are chosen before or after the Y selected numbers. So you have a population of Z winners and X-Z losers from which you're choosing Y samples without replacement, then the distribution of the number of winners is the definition of the hypergeometric distribution. --RDBury (talk) 17:20, 13 November 2013 (UTC)

Thank you. Qoin (talk) 06:31, 14 November 2013 (UTC)