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A.K.A. catch 22.

Circles aren't bigger or smaller than squares until constraints are added. If the constraint is a given perimeter, circles are bigger (area wise) If the constraint is maximum and minimum values of x and y, then squares are bigger. The present constraints are of the second type.

The four squares connect at corners defining an arbitrary quadrilateral. Connecting the centers of opposite squares, show that the segments are perpendicular.

How much of the area of a 3 4 5 right triangle can be covered by three non overlapping circles completely contained by the triangle?

A man is the owner of a winery who recently passed away. In his will, he left 21 barrels (seven of which are filled with wine, seven of which are half full, and seven of which are empty) to his three sons. However, the wine and barrels must be split so that each son has the same number of full barrels, the same number of half-full barrels, and the same number of empty barrels. Note that there are no measuring devices handy. How can the barrels and wine be evenly divided?

That rings the bell. Well earned. Another form is nC4 + n-1C2 + n

You were having fun playing with your red and blue balls each identical in weight and type with diameters of 4 inches. Some other person snatched your five balls and through them into the top of a large bin. The bin was 10 feet tall and full of 1000 large balls each with diameter of 2 feet. The bin has a unique function in that every time it receives five balls (regardless of size and weight) the bin closes and shakes violently for five minutes. 1. Assuming the bin was full of these large balls where the tops of the balls matched the top of the container and each ball touches the other balls without bending, how wide is the container? (A range of possible answers will suffice, if necessary) 2. To get your balls back you must ask the technician where they are located, these technicians are easilly bored and will not look far from precisely where you tell them. Where is the best place to have them open the bin to find your balls? (Assume that no balls fall out when the container is open)

How can you build pig pens so you can put nine pigs in four pens such that each pen has an odd number of pigs?

You've been sentenced to death in an obscure foreign country which has a strange law. Before the sentence is carried out, two papers -- one with "LIFE" written on it and one with "DEATH" written on it -- are folded up and placed in a hat. You are permitted to pick out one of the papers (without looking), and if you choose the one with "LIFE" written on it, you are set free. Otherwise, the death sentence is carried out. On this occasion, a mean-spirited acquaintance of yours, bent on your demise, has substituted the paper with "LIFE" written on it with another one with "DEATH" written on it. This person gleefully informs you of what he has done and that you are doomed to die. You are not permitted to speak to anyone about this misdeed, nor will you have a chance to switch the papers or the hat yourself in time. How will you avoid certain death?

TPVM YXKD WL VJB BXTJ MPK DMVEID TEMPXWM NXUEJY? I don't care if you can translate the cryptogram. I want to know if you can solve the question.

Are you saying what is the greatest path?

The school children were returning to their homes when they met the mathematical milkman, who propounds the following problem: In one of the two cans there is milk which is so rich with cream that it becomes absolutely necessary to dilute it with a little water to make it wholesome. Therefore, in the other can there is some pure spring water, now I proceed to pour spring water from can No. 1 into can No. 2 sufficient to double its contents, and then repour from No. 2 into No.1 enough of the mixture to double the contents. Then to equalize matters, I again pour from No. 1 into No. 2 to double the contents of No. 2 and find the same number of gallons of milk in each can, although there is one more gallon of water in can No. 2 than there is milk, so I want you to tell me how much more water than milk is there in can No. 1?

Verdict Is Guilty Verdict is not Guilty Committed Crime 80% 20% False Negative Innocent of Crime 10% False Positive 90% The above chart shows the rate that individuals are correctly found guilty or innocent of crimes in a particular jurisdiction. If 1% of the population of residents in this county were tried in court, what are the chances that someone was correctly charged with crime?

You have 8 coins that all look exactly the same, two of them are fake,one of these two fake coins is 0.1gm more than a normal coin, whereas the second coin is 0.1 gm less than a normal coin. With howmany weighings can you find them out using a balance scale?

I owe it to the Den to post at least one of these without error. Possibly this one does that. Give the longest route (sequence of city numbers) that visits all the cities (a) not returning to starting city (sum of 7 distances - starting point matters) (b) returning to starting city (sum of 8 distances - starting point does not matter) This puzzle has more choices than the first one. Cities lie clockwise on the perimeter of a 6x6 square: 6--O------O-------+--------O |3 4 5| | | | | 4--+ + | | | | | | 2--O2 6O | | | | |1 8 7| 0--O-------+-------O-------O | | | | 0 2 4 6 +----+---+---+ |City| x | y | Distances: +----+---+---+ 8.485 1-5 3-7 | 1 | 0 | 0 | 7.211 2-5 3-6 3-8 4-7 | 2 | 0 | 2 | 6.325 1-4 1-6 2-7 4-8 5-8 | 3 | 0 | 6 | 6.000 1-3 1-7 2-6 3-5 5-7 | 4 | 2 | 6 | 5.656 4-6 | 5 | 6 | 6 | 4.472 2-4 2-8 | 6 | 6 | 2 | 4.000 1-8 2-3 4-5 5-6 | 7 | 6 | 0 | 2.828 6-8 | 8 | 4 | 0 | 2.000 1-2 3-4 6-7 7-8 +----+---+---+

is this right?

A traveler just came back from where they observed the most 'miraculous' thing. He visited four small towns in four days. Each town was exactly five miles from each other. Never in his life could he have expected to see four towns that were all the same distance from each other. How was this feat achieved?

Resource Desk Table Chair Availability Lumber 8 board ft 6 board ft 1 board ft 48 board ft Finishing 4 hours 2 hours 1.5 hours 20 hours Carpentry 2 hours 1.5 hours 0.5 hours 8 hours Selling Price $60 $30 $20 Given the constraints listed in terms of time and wood available, how many of each object should be produced to maximize revenue?

The product, the quotient, and the difference of two real numbers are all the same. Find the sum of the two numbers.

"What time is it, Rory?" asked Cory one lazy day. "When I checked my watch this morning, the hour hand was where the minute hand is now, and the minute hand was one minute before where the hour hand now sits. I notice both hands are now at exact minute divisions." What is the time now? When did Rory check this morning?

King Chester awarded a triangular piece of land to his favorite court jester. The three sides measured 150 2/3 yards, 195 3/4 yards, and 45 yards 3 inches. His wife had long been asking him to have a piece of land where she could build a home, a garden, and a temple in each corner. The jester told his wife the good news but was surprised that shew as unhappy. Can you tell why she was unhappy?

When Jack went back for a late-night snack, he bought three items off the rack. Zack rang up the snacks and said "5.70, Jack." "Wait, Zack, you multiplied the prices instead of adding!" "Multiply, add; it still comes out the same. Pay up." What were the prices of Jack's three items?

I think one of the problems with this problem is that we are missing the assumption that standard ice trays hold 12 ice cubes

That's not what's missing--you said in the OP that each tray holds a dozen.

As a swimmer jumps off a small bridge and begins to swim upstream, her swim cap comes off and floats downstream. Ten minutes later she turns around, swimming downstream with the same effort, past her original bridge. At the next bridge, 1000 meters away from the first, she catches the cap. What was the speed of the current? Of the swimmer?

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How will you make a slice on a cube ice so that it will melt quickest?

A detector with largest flat surface is to be installed inside a lab cubicle that has 1 sq.m opening and 1m deep underground to maximize capture of particles. Is the shape of detector a square or a circle?

I have lots of photo prints, in two sizes: (a) 4 x 6 inches, and (b) 5 x 7 inches. I put photos up on the wall, each one can be vertical or horizontal, so that they tile into a big rectangle (with no overlapping or cutting, of course). i) Prove I can't make a 19 x 19-inch "photo-square." II) Show me how to tile the 29 x 29 photo-square.

You have an old-fashioned refrigerator with a small freezer compartment capable of holding seven ice cube trays stacked vertically. But there are no shelves to separate the trays, and if you stack one tray on top of another before the ice cubes in the bottom tray are fully frozen, the top tray will nestle into it, and you won't get full cubes in the bottom tray. You have an unlimited supply of trays, each of which can make a dozen cubes. What's the fastest way to make full-sized ice cubes?

Many of my algebra and precalculus students think the 'inverse function' of f(x), often written f^(-1)(x), is the same as the reciprocal 1/f(x) (mistaking the -1 for an exponent). This (as I am obliged to remind them) is almost always false. But can you find at least one function whose inverse is also its reciprocal? Tiebreaker: Find as many as you can!

suppose there is a three digit number M where 100*a + 10*b +1*c = M where a, b, and c are digits What is the minimum value that can be found from M/(a+b+c)?

You just held up a bank on the corner of 6th & 7th avenue and walked to your get away car but upon entering the car, you've been spotted on the CCTV that alarmed the patrol cars on the 3 corners of the area that is bounded by high walls along the perimeter . If all the vehicles are as fast as each other and all streets and avenues are of same span of one-way roads, where is the best intersection for the get away car to be waiting for you in order to evade the police cars in hot pursuit?

Winoc sells four types of products. The resources needed to produce one unit of each and the sales prices is listed below: Resource Product 1 Product 2 Product 3 Product 4 Raw Material 2 3 4 7 Hours of labor 3 4 5 6 Sales Price ($) 4 6 7 8 Currently, 4,600 units of raw material and 5,000 labor hours are available.To meet customer demands, exactly 950 total units must be produced. Customers also demand that at least 400 units of product 4 be produced. What is the solution that maximizes Winoc's sales revenue?

Find pairs of positive integers where 2a = 3b + 5. Prove that you have found them all.

A regulation golf ball is spherical and has 384 dimples, arranged in a triangular pattern. Most of the dimples are surrounded by six other dimples, but some are surrounded by only five. How many dimples have only five neighbors?

How to fill it?