Date: 01/06/2003 at 11:43:04 From: Nia Lynn Subject: Making Choices Katie, a student at the world's first orbiting school, noticed that her unit cubes floated in zero gravity. If she was careful, she could even build floating geometric shapes with them. One day Katie used the unit cubes to build a larger floating cube that was three units long on each edge. She then painted all the exposed faces of the cube red. Unfortunately, just after she finished, she sneezed, scattering the unit cubes. No cubes remained attached. When Katie gathers the cubes, how many cubes will she find that have exactly 6 faces painted red? 5? 4? 3? 2? 1? 0 faces? I know that each unit cube does not have the same number of faces painted red because some are on the corners and they should only have 3 faces painted (the top and the two sides) and there are 6 faces on the cube.
Date: 01/06/2003 at 12:35:11 From: Doctor Ian Subject: Re: Making Choices Hi Nia, To find the number of unit cubes, imagine slicing the larger cube into three layers: How many cubes are in each layer? How many layers are there? The total number of cubes has to be the product of those two numbers. Now, what about the number of painted faces on the unit cubes after the larger cube is broken up? Note that for a unit cube to have all 6 of its faces painted, it would have to have all 6 of those faces exposed while part of the larger cube. Is there _any_ position in the larger cube where that would be true? One way to approach this problem is to figure out how many different _kinds_ of locations there are. There are 8 corner locations (only 7 are visible in this view), 12 edge locations (only 9 are visible in this view), and 6 center locations (only 3 are visible in this view). We can use a table to keep track of this information: Location unit cubes -------- ---------- Corner 8 Edge 12 Center 6 The main reason for grouping locations like this is that each location is equivalent to every other location of the same kind. (If you don't see why, imagine that while you're looking away, someone picks up the larger cube and rotates it so that a different corner is facing you. Would you be able to tell?) And that means that once we figure something out regarding _any_ unit cube, we've figured it out for _every_ unit cube in the same kind of location. For example, look at one of the unit cubes located at a corner of the larger cube. As you've noticed, it will have three of its faces painted. But since any corner is just like any other corner, _every_ unit cube located in a corner will have three of its faces painted: Location unit cubes faces painted -------- ---------- ------------- Corner 8 3 Edge 12 ? Center 6 ? I'll leave the rest of the table for you to fill in. Finally, note that so far, we've accounted for 26 unit cubes. But from analyzing the layers, we know we have to have 27 unit cubes. So where is the final one? And how many of its faces would be painted? Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ For a related, interesting problem, see How Many Cube Faces Were Painted?
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