Let's discuss Mathematics

[Samson]

Is communication allowed? I guess not or it should be trivial.

 

[Harv]

Is communication allowed? I guess not or it should be trivial.

I grabbed a wording of the problem from the Widipedia article.

The director of a prison offers 100 death row prisoners, who are numbered from 1 to 100, a last chance. A room contains a cupboard with 100 drawers. The director randomly puts one prisoner's number in each closed drawer. The prisoners enter the room, one after another. Each prisoner may open and look into 50 drawers in any order. The drawers are closed again afterwards. If, during this search, every prisoner finds their number in one of the drawers, all prisoners are pardoned. If even one prisoner does not find their number, all prisoners die. Before the first prisoner enters the room, the prisoners may discuss strategy — but may not communicate once the first prisoner enters to look in the drawers. What is the prisoners' best strategy?

100 prisoners problem - Wikipedia

en.wikipedia.org

So far, I reduced N=4 and only got a 1/6 probability of success - so I am still missing something.

 

[Kyriakos]

Should be directly related to the difference between random (=non tied) chances of x participants, and tied ones. For the simplest example, while if you have only two boxes, two players, and one attempt to get the correct box, in a random/untied attempt you have 1/2 for each player and consequently 1/4 for both winning (or both losing), if you tie the strategy (say they agreed to choose the box that corresponds to who goes first, who goes second) you now have a 1/2 chance of both winning or both losing, since effectively only one attempt is made (no room for one winning and one losing).

I think the result for (untied/no strategy) attempts of number x, in a set of options of number n, should be (n+n-1+...+n-x)/[(n)(n-1)...(n-x)], which gets simplified to (2n-x)(1/2)/(n!/x!). Of course this is when the position of "winning number" is random itself. So in the problem you posted you can incorporate that it isn't random at all, but specific for each player, leading to cumulative states and thus strategies to exploit them.

 

[Kyriakos]

Scaling triangles is a very popular part of new proofs (of established geometrical theorems, such as Ptolemy's), because it is both fun and easy to do.
I applied it to an (otherwise trivial) issue inspired by a recent basic video.

The point in the video (which is way too forced, since you can immediately calculate the needed area by simpler methods; eg since in similar triangles the ratio of corresponding sides is stable, height to smaller side is still 9/6=3/2=>(3/2x)^2+x^2=36 etc ), is to use the square of the ratio of corresponding sides of non-linked similar triangles as the ratio of their area. But this can be proven more elegantly (imo), by scaling:
a) you scale one (not already linked) similar triangle so as to have a common side with the other similar triangle. This means their common side is scaled by sideA/sideB (A is of the other triangle, B of the one scaled).
b) since they are similar triangles, when they link up (as in the image) they will inevitably form a right angle (this allows for circular calculations too, since it is known -Thales theorem etc- that if two linked up triangles have a right angle formed, the over-triangle has its base be the diameter of a circle, and all its vertices be points on the periphery).
c)In such a linked similar triangles formation, obviously the ratio of their respective areas is proportional to just the ratio of their bases. But due to (a) this gets multiplied by the same ratio, so ends up being the square of the ratio.

 

[Samson]

I am doing some tutoring of my niece, and I have a couple of questions, perhaps just the right words to google would help. I can find lots of maths stuff online, but none that describe these concepts.

The first is what is the maths words for the difference in the meaning of the minus sign between these two expressions: y = -1 and y = x-1. If I was talking computer programming I would say the first is a feature of the variable and the second is an operator. What is the maths words for that difference?

The second is a bit harder to explain. The first thing they teach them at secondary school is sequences, such as "Find the formula for 2, 5, 10, 17". A method is to find the difference between subsequent terms, and then find the difference between them The rules for how this work are similar to the rules of differentiation. Is there anything that explains this link? What are the words to search for for more reading about this?

 

[AdrienIer]

For the first one, unfortunately I haven't really done maths (or more importantly taught maths) in english so I can't help.
For the second one I assume it's linked to the study of sequences (arithmetic, geometric, arithmetico-geometric etc), where in your example you can see that U(0) = 2, U(n+1) = U(n) + 2n + 1. So if you're looking for the general term (the forumla) you tend to study that recurrence relation, and U(n+1) - U(n) (because U(n) is the sum of U(k+1) - U(k) for k = 0 to n-1).

 

[Kyriakos]

For the first one, wouldn't you anyway (for school stuff) be expressing those properties on the real line? In which case, afaik in secondary school (particularly if they are not taught vectors yet) "-a" is defined as the symmetrical point of a with the axis of symmetry being y (and as mentioned, a being on the x axis). x-a, accordingly, is the symmetrical point of a-x, ie afaik in secondary school "-" is only defined in the context of symmetry. Functions will be presented to some extent in the second year (the obvious ones, for proportional, inversely proportional relations, maybe the hyperbola and f(x)=ax+b) but there "-" again would be taught by symmetry, now as a line or linear segments instead of points.
For the second one, are you asking about a way to link those formulas to calculus? It is a Δx, which afaik they will use (typically without insight, however) to solve some basic questions in secondary school physics.

 

[AdrienIer]

For the first one, wouldn't you anyway (for school stuff) be expressing those properties on the real line? In which case, afaik in secondary school (particularly if they are not taught vectors yet) "-a" is defined as the symmetrical point of a with the axis of symmetry being y (and as mentioned, a being on the x axis). x-a, accordingly, is the symmetrical point of a-x, ie afaik in secondary school "-" is only defined in the context of symmetry.
For the second one, are you asking about a way to link those formulas to calculus? It is a Δx, which afaik they will use (typically without insight, however) to solve some basic questions in secondary school physics.

You'd be surprised to see the different ways maths are taught in different countries. In France you'd probably not see numbers in term of symmetry on the real axis as a basic skill for 10 to 15 year olds.

 

[Kyriakos]

You'd be surprised to see the different ways maths are taught in different countries. In France you'd probably not see numbers in term of symmetry on the real axis as a basic skill for 10 to 15 year olds.

In Greece that is how the books present them - at least the current/almost current books Starting with when the kids are 10 (although what I wrote above is about 11-14).
I recently read all of the local first half of secondary school math books. The topics include basic functions (the end of secondary school features the emblematic general solution to the single-variable polynomial of second degree), basic stats, trigonometry, vectors and general polynomial properties (eg division of polynomials), square roots, factorization of select phrases with exponent of 2 and 3, and some secondary stuff like a couple of irrational numbers, some stereometry and homothety and turning numbers with repeated decimal part into fractions. Basically at the end of middle school you'd be able to use an astrolabe (and very likely not being aware that you can, due to most kids just memorizing what they read for the next test) ^^

 

[Samson]

For the first one, unfortunately I haven't really done maths (or more importantly taught maths) in english so I can't help.
For the second one I assume it's linked to the study of sequences (arithmetic, geometric, arithmetico-geometric etc), where in your example you can see that U(0) = 2, U(n+1) = U(n) + 2n + 1. So if you're looking for the general term (the forumla) you tend to study that recurrence relation, and U(n+1) - U(n) (because U(n) is the sum of U(k+1) - U(k) for k = 0 to n-1).

This is the BBC bit about it, which is pretty much their silabus. The sort of way they solve it is illustrated in this picture:



They do not explicitly say it, but this is something like differentiation, and going the other way is something like integration, but the rules are not quite the same.

You can explain why the rules are the way they are quite visually, with wooden cubes making the shapes. It seems a small step to go from that to calculus, but I cannot find a description in those terms.

For the first one, wouldn't you anyway (for school stuff) be expressing those properties on the real line? In which case, afaik in secondary school (particularly if they are not taught vectors yet) "-a" is defined as the symmetrical point of a with the axis of symmetry being y (and as mentioned, a being on the x axis). x-a, accordingly, is the symmetrical point of a-x, ie afaik in secondary school "-" is only defined in the context of symmetry. Functions will be presented to some extent in the second year (the obvious ones, for proportional, inversely proportional relations, maybe the hyperbola and f(x)=x+b) but there "-" again would be taught by symmetry, now as a line or linear segments instead of points.

I do not think I ever really learned this in terms of symmetry. Talking about the difference in terms of the real line makes sense, a region of the line compared to a motion along it.

 

[Kyriakos]

In the local books, there actually is some calculus presented in the second year of secondary education... Although it's only in the case of the circle. Still, rather impressive (but also explicitly tied to Archimedes, who is mentioned for it).
They use this (a bit erratically) for parts of the stereometry chapter.

As for "-" being taught in the context of symmetry, one bonus there is that it can allow the student to get some early insight on min/max formulas. Those can (also) be used to actually construct other formulas which typically are not proven in the school book (a case I have in mind are those for value of x,y for the vertex of a parabola).

 

[Kyriakos]

Is there actually any proof of the law of cosines that doesn't use the Pythagorean theorem?
Moreover, would the law of cosines ever have meaning in a system which does not explicitly have the Pythagorean theorem as an earlier theorem?

 

[a pen-dragon]

Wikipedia does list some proofs that do not use the Pythagorean theorem explicitly.

OTOH, the geometrically most useful definition of the cosine I know, using the unit circle, directly implies this theorem.

As for your second question, I think that basically it is equivalent to asking if a more general theorem needs a more specialized one to make sense. IMHO, to this question the answer is no, meaning yes to your question.

 

[Kyriakos]

Wikipedia does list some proofs that do not use the Pythagorean theorem explicitly.

OTOH, the geometrically most useful definition of the cosine I know, using the unit circle, directly implies this theorem.

As for your second question, I think that basically it is equivalent to asking if a more general theorem needs a more specialized one to make sense. IMHO, to this question the answer is no, meaning yes to your question.

I did have a look at wiki, but the only proof there that (supposedly) doesn't use the pythagorean theorem is on 3d forms... Can you provide a proof of this where it is obvious the PT is never used?
Besides, the law of cosines historically arose from establishing relations between the sides in non-orthogonal triangles, to orthogonal triangles tied to those - the early form of the law of cosines, after all, is there in Euclid's second book, proposition 12. It seems that law of cosines is only nominally a generalization of PT (the formal generalization of PT is Ptolemy's theorem), and in reality it's an equivalent statement.

 

[a pen-dragon]

Can you provide a proof of this where it is obvious the PT is never used?

The best I can offer is the one via three altitudes: https://en.wikipedia.org/wiki/Law_of_cosines#From_three_altitudes. It uses the definition of a cosine on the unit circle.

Since, IMHO, the definition of the cosine is so close to the Pythagorean theorem, arguably no proof that fulfills your criterion exists.

 

[Kyriakos]

^Thank you, I will have a look later today and post back if needed

 

[Kyriakos]

How would you define, generally, the type of problem that implies a parabolic function? Eg inverse proportionality is tied to the hyperbola, proportionality to a linear function that goes through where the two axis meet, etc.

 

[Samson]

How would you define, generally, the type of problem that implies a parabolic function? Eg inverse proportionality is tied to the hyperbola, proportionality to a linear function that goes through where the two axis meet, etc.

The obvious answer is quadratic relationships.

 

[Kyriakos]

I was looking for a description, though
As in "the variable is restricted by x things in y ways" etc.
Basically a description which would allow one to instantly tell when a problem leads to a parabola, by looking at the relations described in natural language.

 

[Kyriakos]

Can you think of a simpler (by this I just mean having fewer lines, and employ a different approach; eg it could be through establishing the point where two functions connect - the critical step would be to express those two- or by using calculus) way to calculate what the radius of the smaller circle will be? In this case, the square is of side 4.

The two (distinct) equations, that allow to solve the system of two unknowns, are on the one hand the circle-to-circle analogy (homothety), on the other the boundary of the circumscribed to the larger circle square.