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What is the largest number of pieces that can cut a cake making four straight cuts with a knife?
every cut has to go through the top and bottom of the cake and I think it's more than 10
First, let's go to deceive "means that they are bad solutions: Cut the cake in half. Cut again into quarters. Line consecutive quarters and cut to the bottom (now 1/8ths) Line them again in a row and cut to the bottom (1/16ths) This gives 16 pieces, but it's a trick - you rearrange the pieces. There is another form of deception for three cuts. Here's why that way trick (trick all responses) is extra stupid. For three cuts: Cut Vertical section cut horizontally along the face of the cake (ie, cutting the upper and the lower half of the cake apart). It's stupid because if you try to do additional four sections, gives a total nonsense - it would take a fourth spatial dimension to make this trick work Goofy. --------- --------- --------- ------ --- --------- ----- ---- --------- --------- --------- If you do it the honest way, is only allowed to cut the cake in a straight line without moving any of the pieces. The maximum is 11 (not 10). Why 11? Two cuts to get the four regions. That is obvious. However, the third cut can not pass through the four regions. Can be divided only three of the four regions, thereby bringing the total to seven. The reason for this is simple - Complies with the other two cuts (lines) in one place (if they were two points in common with another cut, which would be the same line). This starts in a region, through a court to another, and then crosses again to another party. You can only cut those three regions. Similarly, the fourth court can not pass through all seven regions. The reason is again the same. You can only cut across four regions. Cut once through each other, so it starts in a region undergoing a court to another, then another, then other. This can only happen three times, since there are three existing lines can be crossed (and cross only once each). That divides the four regions, for a total 7 +4 = 11. So the answer is eleven. Proof: kellen/interspace/cakecuts4.gif http://math.colgate.edu/ ~ --------- --------- --------- --- ------ --------- --------- ----- ---- --------- --------- NOTE: I am the first to post the correct solution (11). I am a combinatorialist, and I have been a series of lectures on combinatorial geometry much more advanced. This is one of the standard problems that are interesting, but very easy to calculate if you know what you're doing (which they certainly do). --------- --------- --------- --------- --------- - -------- ---- ----- --------- --------- Note: To people rude enough to make rude comments and I get a preview down, I suggest to do the following: read the question, then try to understand the spirit of what you are ordering. Who slices of a cake than in straight cuts perpendicular to the table? The problem is almost certainly not ask for anything more. And if you want to use the three-dimensional geometry of you, the answer is still wrong - to take my argument, change lines with the planes, and obtained more than 12 ... Jeez. Who do you think you are? Divide the cake into 8 slices, and let these planes correspond to coordinates in 3D space. We want a plane through octants as many as possible. What about the plane: x - y = 2? Hit these points: (2,1,1) (2.1, -1) (-2, -3.1) (-2, -3, -1) (1 -1.1) (1, -1, -1), which means that divided at least six of the eight pieces, for a total of 14 or more. You can actually get 15 and no more: http://books.google.com/books?id=Z3TkuH5MVuAC&pg=PA261&lpg=PA261&dq=plane+cuts+in+three+space&source=web&ots=L25GlKFxrX&sig=9xKMPjP1rVXiywd_gA4eP7DmFKo Therefore, this rude person, I say: Get a clue, not being rude, and do not give thumbs down when you do not know what your talking about.
Knife Making
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Crafting a private home-made archery bow is not as difficult as one might suspect. You do not have to be a woodworker of experts to create an arc Archery functional which can be used for hunting or recreation. A variety of woods that can be used in the manufacture of an archery bow, such as yew, bamboo, wicker and lemon wood. Make sure the timber you choose is about one meter long, straight and containing absolutely no knots or rolls. It is recommended that the wood is dry but not dried to the point of crack or otherwise break when bent.
Remember: The most important tool in the process is your knife. It is imperative you use a sharp knife, so remember your sharp knife before starting your project.
To place an archery bow, the first step is to establish the natural curve of wood - each piece of wood has one, although some may be easier to distinguish from others. This is very important as the carving on the wrong side cause the arc to complement its first use. Then carve the wood into a boomerang-like shape, ie the ends of the rod should be slightly less than average. For example, if the media is one inch thick, then both ends should be about 3 / 4 inch. To ensure that your arch last long, tightly wrap wet leather strip around the center of the arch and allow to dry naturally. This gives the arc of greater durability and arrows that drives faster.
At this point, the initial shape has been carved, and the next step is to attach a chain. There are an unlimited number of types of a chain can be used in the manufacture of a bow shooting arc. This list is an exhaustive list of different types of channels can be used:
Fish-line
-Hemp
Nylon
-Clothing
-Dacron
-Kevlar
Having chosen the type of string used, you need to wax the chain to promote sustainability and protect natural attrition. Then carve a notch in each end of the arch and ring the rope around each end. Make sure the rope tied tight now, as a string that is flexible and loose is almost useless in archery.
An archery bow would be nothing without the arrows. It is important for the construction of the arrows to fit inside the frame, but not too long, otherwise you will break the contact. The arrows can be basically long sticks, however, each end is what really makes them fly. The final out should be fitted with a steel arrowhead. For optimal flight, three feathers attached to the end closest to you.
If you follow these simple steps, you're well on your way to becoming an expert manufacturer of archery bow.
Ian Pennington is an accomplished niche website developer and author. To learn more about making an archery bow [http://choosingabow.info/how-to-make-an-archery-bow], please visit Choosing a Bow [http://choosingabow.info] for current articles and discussions.
