Quadrillerator

Suppose an equilateral triangle inscribed in a square has the altitude from the 'middle' vertex extend to intersect the fourth side at S. Like what Dragan has at answers.yahoo.com/question/?qid=20121102102617AAZOUjB only without any areas specified.

Now make a quadrilateral with the equilateral triangle and S. This leaves four triangles at the square's corners. If one of these four triangles is similar to the triangle at the square's opposite corner, what is the smallest angle among all four corner triangles?

Please, no derivatives and no series expansions. Just basic trig and algebra such as

Law of sine/cosines, sin/cos/tan of sum/difference formulas, etc.

I can show that cos(x) > 1 - x²/2 + x⁴/4, but this

is not sufficient for the above relationship.

Let x be the angle in radians that a radius of the unit circle with center at O = (0,0) makes with the x axis. Let P be the endpoint of this radius, let B=(1,0), C=(0,1), and let A be the projection of P onto the x-axis, and let D be the intersection of the extension of OP and the perpendicular to OB through B. Thus, PA=sin(x)=s, OA=cos(x)=c, BD=tan(x)=t, and x=∠POB=arc(PB).

Which of the following trigonometric inequality chain is always true for 0 < x < 1/2:

t > x > s > (1-c)/s > (x-s)/(1-c) > x²/2 > 1-c > xs/2 > x-s

Find & prove, 1st quadrant integer (x,y) solutions to cos²(x) - cos²(60) = cos(x)*cos(y)

I have a student (female, 12th grade) in the (math / computer) class I am teaching, and she has the interesting property that she is quite bright but thinks in very concrete terms. I've encountered this type of person before (but normally on a social basis) and I'm always shocked at the contrast (smart vs. non conceptual thinking).

For example, in addressing an aspect of the Tower of Hanoi problem, she resorted to cutting up a sheet of paper to make squares of various sizes to model the problem (and as a result did better than most of the people in the class - very innovative). In a problem involving e, she used the value 2.718 in lieu of the symbol that had already been given for it and later made the identification of another quantity to it by virtue of the numeric amounts matching rather than seeing that they were both the same expression.

My question is: Does anyone have experience with nurturing conceptual thinking into a person like this? In particular, are there problems or exercises I might give to this end? They need not necessarily be homework - they could also be in class.

Find an n (n>1) digit square integer where each digit is the same (base 10), or show that no such integer exists.

Given a triangle with sides a, b, and c, what kind of bounds (inequalities) can be placed above and/or below a² + b² + c², in terms of R, the radius of the circle that circumscribes the triangle?

Is there a monomial f(x) with all integer coefficients, the greatest common divisor of which is 1, such that the monomial cannot be factored (over the integers) but nevertheless produces a composite number for each integer x?

What is the first year after 2000, whose last digit is 7, and which may be expressed as a sum of squares of consecutive integers?

Call a sine monomial any monomial satisfying the following conditions: (A) one of the solutions is sin(πr) for some rational r, (B) the high order coefficient is positive, (C) there is no common divisor of all the coefficients except ±1, and (D) no root of the monomial is the root of a smaller order sine monomial.

What are the sine monomials of degree 3 or less?

I'd like to find all examples of A*cos(a) + B*cos(b) + C*cos(c) = K

where A, B, C, K are integers whose greatest common divisor is 1,

and where a, b, and c are closed form expressions without inverse trigonometric functions.

Example: 2cos(60) = 2cos(π/3) = 1

But not: 15cos (cos⁻¹(3/5)) + 20cos(cos⁻¹(4/5)) = 25

Define T(n) = (1+1/n)^n

Earlier we saw that T(n) is monotonic increasing and has a limit in http://answers.yahoo.com/question?qid=200905161645...

Show, without integrating/differentiating, that this limit is the indicated sum. In other words, show that T(n) gets arbitrarily close to ∑[k=0 to n] 1/k!

Prove (1+1/n)^(n+1/2)>e for all positive integers n.

Preference given to answers that have the least

amount (preferably none) of integration/differentiation.

In http://answers.yahoo.com/question?qid=200905060233... we saw that it is possible to arrange 7 planar points so that whenever any three are chosen, at least two of them will be exactly one unit distance apart:

Take two identical rhombuses with sides of unit length, and have a 60⁰ vertex of each be a common point, O. Now rotate one of the rhombuses about O till its opposite vertex is one unit from the opposite vertex of the other rhombus.

Prove that the solution given is unique or find all solutions (with a proof that those are all the solutions).

Arrange 7 points in a plane so that if any three are chosen, at least two of them will be a unit distance apart (or prove that it can't be done).

This was a bonus question on a friend's daughter's elementary school math competition.

Best response will favor an answer delineating the deductive process.

Suppose I have a COM object such as

Set ws = CreateObject("WScript.Shell")

In some contexts (for example, when a scheduled script runs under SYSEM rather than the current user), I get an error when a try to access properties of such COM. I presume that it arises from a permissions issue and I'd like to get further information.

My question:

Is there a way to determine the permissions associated with a COM object so that I can conclusively track down what is happening?

In http://answers.yahoo.com/question/?qid=20090314172... we saw that we could characterize the square root of the 2 x 2 identity matrix I₂ as either ±I₂ or

[ ±√(1 - bc), b ]

[ c, ∓√(1 - bc) ]

which latter could also be written as 1/sin(x) *

[ sin(y), -(cos(x) + cos(y)) ]

[ (cos(x) - cos(y)), -sin(y) ]

I'd like to characterize all the square roots of the 3 x 3 identity matrix (I₃) in an analagous fashion.

If we let c = cos(w), s = sin(w) then

cos(3w) = c(4c² - 3)

sin(3w) = s(4c² - 1) and

sin(6w) = 2sc(4c² - 1)(4c² - 3)

If w=π/9 so that sin(3w) = sin(6w) then

s(4c² - 1) = 2sc(4c² - 1)(4c² - 3) so 1 = 2c(4c² - 3) or

8c³ - 6c - 1 = 0

By construction we know c = cos(20) = cos(π/9) is a solution

Find (with proof) the other two roots.

No negative square roots, please.

I is the (2 x 2) identity matrix, of course.

This is motivated by http://answers.yahoo.com/question/?qid=20090312193...

If there are more than just a few such matrices, you should be able to express them in some reasonable (parameterized, for example) way.

This question is about what stategy I can employ for hard sudoku puzzles.

I solve the vast majority of Sudokus that I see (published in various newspapers) in my head by using one of three principles.

The one that is far and away most common is what I think of as striking out (within one of the 3x3 subsquares). I consider a subsquare and pick a number that does not appear in that subsquare. From each instance of that number either to the side or directly above/below my selected box, I draw an imaginary line from that number through my box "striking out" unfilled squares. If only one square is left, then I can fill it in with the selected number. I don't do this mechanically. Rather, I look for common numbers that impinge upon a subsquare.

This strategy can be extended. As a simple for example, suppose that the right column of a subsquare is blocked (has been filled in) with, say, non 3s. If you have the 3 impinging on the left column from outside the subsquare, that forces the 3 of the subsquare to be in the middle column, which in turn forces the remaining subsquare's 3 to be in the right column.

There are two other situations that are far less common, and I look for these when I am otherwise stuck with my main strategy above. 2) Pick an arbitrary square. If the numbers impinging on that square from both the column, row, and subsquare that square is in amount to all but one of 1-9, then the square may be filled in with the missing number.

3) Consider a row or column. If a number not in that row or column impinges on all but one of the non filled in squares of the row/column, then that last square will have the given number. This is a variation on the first strategy applied to rows/columns instead of subsquares and is relatively rare.

With these 3 strategies I write nothing down, nor do I have to remember the boatloads of special, fancy named strategies that they talk about on sudoku websites.

However, there is a harder level of sudoku. The only place I've regularly encountered these harder sudokus is at http://apps.facebook.com/challengesudoku

on the 'Harder' level (levels are Easy, Medium, Hard, Harder). You can start one of these by Creating a game, and someone will usually join within one minute. Some of them (over 50%) cannot be solved by the above 3 principles and require a more involved logic. Specifically, http://www.sudokusolver.co.uk/

cannot show a next step because a more advanced strategy is needed.

Therefore (finally) my question is what is the next level of strategy to follow? In other words, how do you expand on the strategy that I've delineated above?

To be clear, I'm after a way of looking at the harder puzzles: what should I be scanning for?

There are some advanced strategies such as at http://www.scanraid.com/Death_Blossom (though most of this site appears to be undergoing revision), but that doesn't answer my question of what I should be looking for because it doesn't tell me what the next most common type of scenario is.