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1. ## Squares on a plywood

Imagine a piece of plywood with an array of evenly spaced nails forming small squares and consider that each square has side lengths of 1 unit. A simple closed shape is formed with a rubber band. If you knew the number of nails used in the perimeter and the perimeter itself, how could you predict how many squares can be counted inside this rubber band shape? for example say the rubber band shape is outlining these nails: * - * - * - * - * | / * * * * | / * - * - * Perimeter = 8 + 2*sqrt(2) Nails = 12 Squares: 6 squares (5 - 1x1 and 1 - 2x2) --- the result of 6, at a minimum, is what we are trying to predict.
2. ## Squares on a plywood

All you showed is the bound is wider than one thought.
3. ## Fill the vacancy

The premature retirement of the beloved leader Aldan has left a vacancy on the ruling triumvirate of the Malthusian city of Zocor. There are two political parties, The Meat Lovers Party (MLP)and the Vegetarian Party of the People (VPP). At present each party has one representative in the triumvirate, with the MLP representative Arctan being the most senior.The other ruling member being Arcsin. The arcane voting rules for replacing Aldan are as follows. There is a large Praesidium with n people in it. At the moment all the members are in the VPP. Each member has an integer valued priority. The selection process will now go in rounds: At the beginning of a round Arcsin will propose some subset S of the remaining candidates C of the Praesidium. (Initially C will be the whole Praesidium). Arctan then has two choices. He can devour S, but then he is forced to decrease the priority value of each member of C \ S by one. Alternatively, he can devour C \ S and decrease the priority values of the members of S by one. The process continues until one of two things happen: The whole Praesidium has been eaten and then Arctan can choose any member of his party to fill the vacancy. The other possibility is that someone will reach priority zero and then Arcsin can choose any zero priority person to fill the vacancy. Given that the initial priorities are a1, a2, .... an, which party will get to fill the vacancy?
4. ## Squares on a plywood

So then if we know N and P we should be able to bound squares x by a two values. Are those values always consecutive?
5. ## the distinguished matrix

Imagine you have several distinguishable rows composed of several distinguishable columns The intersection of the rows and columns either have a 1 or a 0. Each row sums to the same value and the question is how many of the columns can you eliminate assuming the the 1's in each row are randomly distributed across the columns Example, there are 30 rows and 20 columns with each row containing 7 randomly dispersed 1's. How many columns can be eliminated reducing the total in each row by no more than 2.

I agree
7. ## Arc length = area

Find a function where the arc lenth and area between any two randomly defined points is the same. There are two.
8. ## An interesting limit

Find the Limit as n goes to infinity for: (1^n + 2^n + 3^n + 4^n....+ n^n) ---------------------over--------------------- (n^1+ n^2 + n^3 + n^4 ... + n^n)
9. ## infinite powers

Say we have the function: y=x^x^x^x^x..... Find an x value for which the derivative of this function converges. If you are really clever you'll find the interval that converges.
10. ## An interesting limit

Hmmmm, my answer was the reciprocal of yours. Maybe I am wrong. Can you support your answer?
11. ## A special Factorial relationship

What is (-1/2)! x (-1/2)!
12. ## Complex form

Write the complex form (a + bi) for: Sqrt ( i )
13. ## Area of an infinitely shaded region

Suppose you have a triangle that has 2-1 inch lengths. Divide this triangle into half by drawing a line from vertex between the two identical sides, choose one of the sides randomly and shade it. The non-shaded side is cut in half again. Choose one of these sides randomly and cut it in half again shading one random piece. If this pattern of cut, shade, cut, cut, shade, cut, cut, shade cut, cut,.... was to be continued forever, what would be the area of the shaded region?
14. ## Area of an infinitely shaded region

I agree that it will vary but we can bound the area.
15. ## Dice Game

Two students play a game based on the total roll of two standard dice. Student A says that a 12 will be rolled first. Student B says that two consecutive 7s will be rolled first. The students keep rolling until one of them wins. What is the probability that A will win?
16. ## HTML help

I am attempting to develop a website for a university project where I am supposed to create a quiz that stores the users name and answers then decides if the answers are sufficient for mastery. If the student does well enough then a 'certificate' is printed to the screen for the user to print out their recognition. If they do not then it reports back their answers so that the student knows to go back and study. The problem is, is that we are supposed to build this website in google sites. i have little qualms with google sites for the project is almost complete except for the fact that google sites and javascript do not seem to get a long to well; which i believe is the key to solving the quiz problem i am having. Every time i attempt to enter javascript into the html editor on the site i get an error and find that the google site edited my code pretty incredibly. However, HTML seems to have little issue with editing the site. Is there a way to build my quiz in html to meet my above requirements? if not or if this isn't efficient does any one have any ideas as to how i could make my quiz according to the specs? The quiz needs to be 20 questions. So i am currently, and painstakingly so, making a giant tree like web to cover all the combination outcomes but clearly this is quite extensive and exhaustive. So any help would be appreciated.
17. ## Poisonous apples

There are two bowls that you and a challenger must eat from. After flipping a coin you were selected to pick the bowl that each would eat from. In the first bowl there are three out of five poisonous apples. In the second bowl, there are two out of five poisonous apples. Whoever eats from the first bowl must eat two apples at random from the bowl. Whoever eats from the second bowl must eat three random apples from the second bowl. Which bowl should you pick to eat?
18. ## Operating on three people with two surgical gloves.

Dr.Will wants to operate for three different persons who were wounded. But he had only two surgical gloves. There is no allowed blood contact between the three persons. How can Dr.Will operate for the three people with two pair of surgical gloves?
19. ## Square in a circle

Suppose i have a circle. I cut off its arcs such that it became the biggest possible square i could make from that circle. What's the ratio of the edge of the circle to the middle of the edge of the square (assume minimum length) to the radius of the circle.
20. ## Square in a circle

Maybe one of you answered this question: :) I should have included this pic though...
21. ## Help Bob find Alice's P(x)\$%^

Alice and Bob are playing the following game: Alice has a secret polynomial P(x) = a_0 + a_1 x + a_2 x^2 + … + a_n x^n, with non-negative integer coefficients a_0, a_1, …, a_n. At each turn, Bob picks an integer k and Alice tells Bob the value of P(k). Find, as a function of the degree n, the minimum number of turns Bob needs to completely determine Alice’s polynomial P(x).

yes.
23. ## the distinguished matrix

No they just need to be distinguishable.
24. ## Poisonous apples

I meant them to be fractions so five in each
25. ## Mining Gold

Mining gold in a particular region is hard work. The metal only appears in 1% of rocks in the mine. But your friend Old Joe created a detector he’s been perfecting for months and it is finally ready. To your astonishment it always detect gold if gold is present. Otherwise it will have a 90% accuracy rate in detecting that a particular rock does not have gold. Working with Old Joe, You guys scan a large rock and determine that it gives a positive result. In loading it up, Old Joe realizes that both of you can't fit into the vehicle. He offers to sell his share to you for \$200. You know that a rock of gold that size is worth easily \$1000. Is that a fair price? Assume the vehicle remains with the proper owner.
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