Ben

Briefly, the question can be phrased as:

what are the integer solutions to the equation

(x+1)C2 = (y+2)C3

or in other words,

x(x+1)/2 = y(y+1)(y+2)/6

I was thinking about neat ways to stack objects, and I remembered that if you have 10 objects, you can either stack them as a 4-level triangle (4 in the bottom, 3 above that, 2 above that, 1 above that) and a 3-level triangular pyramid (6 in the bottom, 3 above that, 1 on top).

Are there other numbers for which multiple such stackings are possible?

What is the probability that an n-digit number adds up to 5*n?

for n = 1, clearly this is 1/10, since 5 is the only valid number and there are 10^1 possibilities.

for n = 2, it becomes 9/100 because we have the valid numbers {19, 28, 37, ..., 82, 91} and a possible 10^2 = 100 possibilities.

What about n = 3? Is there a neat generalized solution for an arbitrary choice of n?

All contributions appreciated! Thanks for the help!

Let x and y be positive integers such that

3x + 7y

is divisible by 11. Which of the following must also be divisible by 11?

A) 4x + 6y

B) x + y + 5

C) 9x + 4y

D) 4x - 9y

E) x + y - 1

Apparently the answer is D, but it's not obvious to me how one should go about finding the answer (or, for that matter, how to do so quickly).

The simplest approach I can think of is to take an example

y = -1

x = 2

And simply plug in to eliminate. However, this is an extremely unsatisfying solution and, if possible, I would like to how to figure this out algebraically. All mode of analysis welcome; feel free to use modular arithmetic or anything else if it helps. Any input is greatly appreciated!

lim x -> 0 of (ln(1 - x) - sinx)/[1 - cos^2(x)]

I can prove this limit, messily, with some heavy-handed L'Hôpital.

My question is whether there's a quick way to solve and/or intuit the problem that I missed.

Why it came to mind (and, for that matter, the answer to the question):

http://www.youtube.com/watch?v=oDAKKQuBtDo

Prove (algebraically) that

√(1 + 2√(1 + 3√(1 + 4√(1 + ... )))) = 3

Any help is appreciated!

Is there a box (rectangular prism) which satisfies the following properties? If no such box exists, prove that this is the case. If such a box exists, what is the smallest such box (in terms of volume)?

Here are the constraints:

1. All dimensions are of (positive) integer length

that is, the sides a, b, and c are all integers

2. The diagonals of each face are of integer length

That is, a^2 + b^2, b^2 + c^2, and c^2 + a^2 are all perfect squares

3. The main diagonal of the box is of integer length

That is, a^2 + b^2 + c^2 is a perfect square

Any contributions are appreciated! You need not complete the problem in it's entirety, since I certainly don't know how off-hand. Best answer goes to the answer which makes the most headway in this problem.

Other things for consideration:

an example of a box that doesn't work: 3,4,12

Two faces certainly have integer-length diagonals since 3^2 + 4^2 = 5^2. Furthermore, the box has an integer length diagonal since 3^2 + 4^2 + 12^2 = 13^2. However, none of the remaining four faces have diagonals of integral (or, for that matter, rational) length.

It should be fairly clear that, because of the constraints, no two dimensions may be identical. I have the feeling that this is a problem that has already been considered, but I'm not sure where to look.

Well, that about sums it up. Thanks for reading!

Find 9 numbers less than or equal to 100 such that each subset has a distinct sum, or prove that it is impossible to do so.

I had thought of this when I saw the following question

http://answers.yahoo.com/question/index;_ylt=AmFzK...

It is unequivocally impossible to do so for 10 such numbers. What, then, is the largest group of numbers we can get with this property? From the above, the (exclusive) upper bound is 10. The highest (inclusive) lower bound I have found is 7, with the set {1,2,4,8,16,32,64}.

So, for those of you who made it this far, the question is whether there a set of 8 or 9 elements with the above property.

Any help or interesting contributions to this question are appreciated. Thanks!

Made some progress, don't think this brings me all the way though. Additional details upon request.

Made some progress, don't think this brings me all the way though. Additional details upon request.

What is the value of

√(1 + 2√(1 + 3√(1 + 4√(1 + ... ))))

I'm looking for an algebraic way to find this. BA to the first complete answer, or the best attempt after some time.

I found the following video:

http://www.youtube.com/watch?v=YLhgF_RJUxc&feature...

Set two mirrors together at a 60° or 36˚ angle and rotate the pair around your line of sight. Your image is preserved despite the mirrors’ rotation. This is not the case if the mirrors are set at 30˚, 45°, or 90°. Why?

I found the following equations on futilitycloset.com (a favorite blog of mine)

2^n + 7^n + 8^n + 18^n + 19^n + 24^n = 3^n + 4^n + 12^n + 14^n + 22^n + 23^n

2^n + 11^n + 6^n + 28^n + 19^n + 24^n = 3^n + 4^n + 16^n + 14^n + 27^n + 26^n

2^n + 9^n + 16^n + 30^n + 37^n + 40^n = 4^n + 5^n + 22^n + 24^n + 41^n + 42^n

1 + 5^n + 10^n + 18^n + 23^n + 27^n = 2^n + 3^n + 13^n + 15^n + 25^n + 26^n

The above equations are valid for integer values of n from 1 to 5 (and, trivially, 0). I have no idea how anyone came up with these, nor can I come up with a good explanation of why this works.

Can anyone come up with a satisfying explanation for how anyone could have gotten this or find a way to find other such equations? Thank you all in advance

If I had lived 3000 years ago and kept track of the days of the week until today, would a "Sunday" by my standards be the same as a "Sunday" by today's standards? To what point could we know that they days of the week were the same?

"A 'perfect shuffle' of a deck of 2n cards is what the name suggests: Cut the deck (perfectly) into the upper half and lower half. Then, take the top card of the upper half, followed by the top card of the lower half, followed by the second card of the upper half, followed by the second card of the upper half, and so on. Note that a perfect shuffle defines a permutation F of the cards"

Let F_n be a perfect shuffle of 2n cards. Then, the following is a formula for the new position of the kth card:

F(k) = { 2k - 1... k≤n

..........{ 2(k-n)... k>n

The goal is to find a formula for |F_n|, the magnitude of F_n for an arbitrary positive integer n.

Here is what I've found:

- A permutation F_n will keep the first and last cards in the same spot, in other words, F_n(1)=1 and F_n(2n) = 2n, always.

- The highest |F_n| can be is 2n-2 (when the permutation is broken into disjoint cycles, the length of a smaller one will always divide the length of the largest).

Any help with this would be appreciated