Hello everyone. I have one hard math problem and I do not know how to solve it. would you help me to solve this problem? Thank you
An 11[times]11[times]11 wooden cube is formed by gluing together 11[cubed] unit cubes. what is the gratest number of unit cubes that cam be seen from a single point?
Love this question!
I will take a viewpoint where I can see 3 of the faces, (can't do better than that)
I then set up a pattern
2by2by2
I see 2^2 cubes from the front row + 1 block along the edge between the other two faces + 2(1)^2 from the remaining top and side = 7
3by3by3
I see 3^2 cubes from the front row + 2 blocks along the edge between the other two faces + 2(2)^2 from the remaining top and side = 19
4by4by4
I see 4^2 cubes from the front row + 3 blocks along the edge between the other two faces + 2(3)^2 from the remaining top and side = 37
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11by11by11
I see 11^2 cubes from the front row + 10 blocks along the edge between the other two faces + 2(10)^2 from the remaining top and side = 331
Hold a cube so that your line of sight is parallel to the diagonal from one vertex of the cube to the opposite xertex. This allows you to see three faces of the cube, and no more. Starting with a 1x1 cube, work your way up to 2x2, 3x3, 4x4, etc., and determine the number of individual cubes you see. Remember that the cube at the vertex nearest to your eye counts as one cube, not three cubes, since it is on each face. This results in 1 cube for a 1x1x1 cube, 3+3+3+1=10 cubes for a 2x2x2 cube, 8+8+8+1=25 for a 3x3x3 cube, 15+15+15+1= 46 for 1 4x4x4 cube, and so on. This results in the following data with n = the number of cubes to the side and N = the total number of cubes visible.
n...1...2...3...4...5...
N...1..10..25..46..73...
Diff.9...15..21..27
Diff...6...6...6
With the second differences being constant, the expression defining the total number of cubes visible for any "n" sided cube is of the form an^2 + bn + c.
Using the data derived so far, we can write
a(n^2) + b(n) +c =N
a(1)^2 + b(1) +c =1 or a + b + c = 1
a(2)^2 + b(2) +c =10 or 4a + 2b + c= 10
a(3)^2 + b(3) +c =25 or 9a + 3b + c= 25
Solving, a = 3, b = 0 and c = -2 yielding the general expression for the numbder of cubes visible in a cube of nxnxn sides
N = 3n^2 - 2
Therefore, an 11x11x11 cube has 361 visible cubes.
Clearly, this could have been solved quicker by recognizing that each visible surface shows 3n^2 cubes. since the apex cube counts as only one cube, the total seen becomes N = 3n^2 - 2.
Correction
My previous statement
"Clearly, this could have been solved quicker by recognizing that each visible surface shows 3n^2 cubes. since the apex cube counts as only one cube, the total seen becomes N = 3n^2 - 2."
should have read
Clearly, this could have been solved quicker by recognizing that each visible surface shows n^2 cubes. Since the apex cube counts as only one cube, the total seen becomes N = 3n^2 - 2.
lim x infnitive (4x-1/2x+9)
Of course, I'll be happy to help you solve this math problem!
To determine the greatest number of unit cubes that can be seen from a single point in an 11x11x11 wooden cube, we need to consider the different layers of cubes within the larger cube.
First, visualize the cube as being made up of layers. The outermost layer forms a frame consisting of 11x11x11 - 9x9x9 = 1331 - 729 = 602 unit cubes. These cubes are visible from any point within the cube as they are on the surface.
Next, consider the next layer inside the outer frame. This would leave us with a new cube shape, 9x9x9, made up of 729 unit cubes. From a single point within this new cube, 8x8x8 = 512 unit cubes can be seen. However, keep in mind that not all of the unit cubes in the first layer of the larger cube are visible from this point since some are blocked by the second layer.
To count the total number of cubes visible, we sum the cubes from the outer frame (602) and the second layer (512), giving us a total of 602 + 512 = 1114 unit cubes visible from a single point within the 11x11x11 wooden cube.
Therefore, the greatest number of unit cubes that can be seen from a single point within the cube is 1114.