SOLUTION: A rectangular cube with sides measuring 4cm, is made with 1cm³ blocks. Two straight tunnels of 4 cubes each are taken out, as illustrated. All exposed surfaces are painted includi
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Question 1197011: A rectangular cube with sides measuring 4cm, is made with 1cm³ blocks. Two straight tunnels of 4 cubes each are taken out, as illustrated. All exposed surfaces are painted including inside the tunnels. How many small cubes have been painted on exactly 3 faces?
Please view the diagram/illustration as here on my google drive link:
https://drive.google.com/file/d/1YkGkoCOHEX0ZksF9r8UWdviXre3j3RDd/view?usp=sharing
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!
Answer: 18
========================================================
Reason:
For now let's ignore the tunnels and consider a fully intact 4x4x4 Rubik's Cube.
The cube has four levels which I'll call A,B,C,D.
Think of these as floors of a four story building.
I'll have level A at the top floor and D at the bottom.
The order could easily be reversed without affecting the answer. This is because of symmetry.
For each level, we have 4*4 = 16 cubes.
On level A...
the 1st row goes from block A1 to A4
the 2nd row goes from block A5 to A8
the 3rd row goes from block A9 to A12
the 4th row goes from block A13 to A16
Similar labeling applies for levels B,C and D.
It might help to make a table like thisthen stick letters A,B,C, or D before the numbers (either on paper or just mentally) so you can remember where all the blocks are positioned.
In this configuration where we haven't formed any tunnels, I hope you agree that the four corners on the top floor (A1,A4,A13,A16) and the four corners on the bottom floor (D1,D4,D13,D16) are the only blocks that have 3 faces painted.
That means we have 4+4 = 8 blocks with 3 faces painted so far.
Check out this page
http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/SeitzBrian/EMT669/painted.cube/painted.html
to see a diagram where the professor has painted 3 faces at each corner. The diagram is around the middle of the page.
After we dig out the tunnels, we will add to this count.
This is where things get a bit tricky (in my opinion).
The 3D nature of the problem is a big reason why.
It might help to draw out the four levels of blocks as shown in the diagram below.
Levels A,B,C, and D are shown side by side.
I've unstacked the blocks to "flatten" out the drawing a bit.
Imagine this is a sort of birds-eye-view of the blocks.
The empty spaces represent where the tunnels are carved out.
The blue faces are the original exterior faces. These are the exterior painted faces before the tunnels were carved out. The pink faces are the result of forming the tunnels. These are the tunnel floor, ceiling and wall faces.
The white faces are not painted as they are glued to another white face, and it's hidden from exterior view (no matter how much you rotate the 3D figure).
Unfortunately I can't think of a way to have this shown in 3D, to rotate the figure around and such. So you'll have to use your imagination.
The blocks marked with an X represent blocks that have exactly 3 faces painted. Note the four corner blocks mentioned on level A, and the four on level D.
The other 5 X's on level A are due to the tunnels.
Block A2 for instance has a hidden pink face in the back because of the tunnel along blocks B2,B6,B10,B14
Counting the X's in the diagram we have:
9 X's on level A
3 X's on level B
0 X's on level C
6 X's on level D
In total there are 9+3+0+6 = 18 blocks with exactly 3 painted faces after the two tunnels form.
Let me know if I missed any blocks that have 3 painted faces. Also, feel free to ask further questions.
There might be some clever formula for this type of problem, but I'm blanking on it. Another tutor may know.
Answer by ikleyn(52803) (Show Source): You can put this solution on YOUR website!
.
A rectangular cube with sides measuring 4cm, is made with 1cm³ blocks.
Two straight tunnels of 4 cubes each are taken out, as illustrated.
All exposed surfaces are painted including inside the tunnels.
How many small cubes have been painted on exactly 3 faces?
~~~~~~~~~~~~~~~~~
My count is 16 blocks, having three faces painted.
My reasoning is as follows.
First, lets consider the 4x4x4 cube with no tunnels.
I want to explain you ---how to count, analyze and classify the small cubics of this whole cube---.
There are 8 corner blocks of the 4x4x4-cube. They all have 3 faces painted.
Next, there are 12 edges of the cube, and along each of these edges,
we have 2 = 4-2 small cubics, that have 2 (two) painted faces each.
The rest of small cubics that ARE at the surface of the large cube, have 1 (one)
painted face.
The rest of the small cubics, that are INSIDE the large cube and are NOT
at the surface, do not have painted faces, at all.
--------------------
- - - - Now let's introduce our tunnels.
- - - - From now I consider large cube with the tunnels.
8 small corner cubics still have 3 painted faces each.
It gives the number 8 to my counter.
Which additional cubics have 3 faces painted ?
Look at your picture, to which you refer to.
You see large 4x4-face of the cube.
Focus on the small face in the upper row, 3rd from the left.
It is easy to understand that this unique cubic, to which this face does belong, has 3 painted faces:
one in the tunnel and 2 on the surface. Thus we identified one such cubic.
Next, focus on the small face in the 2nd row from the top, which (face) is last in this row, counting from the left.
It is also easy to understand that this unique cubic, to which this face does belong, has 3 painted faces:
one in the tunnel and 2 on the surface. Thus we identified second such cubic.
Thus, we identified 2 small cubics at the ENTRANCE of the tunnel.
It is easy to understand that TWO OTHER similar cubics are at the EXIT of the tunnel.
So, with this tunnel, we identified 4 (four) small cubics having 3 faces painted.
Absolutely in the same way and the same logic tells us, that there are other 4 small cubics
associated with the entrance and the exit of the other tunnel.
Thus, we have 8 + 4 + 4 = 16 3-face-painted small cubics.
Doing this way, we completed analyzing 3-face-painted cubics.
There is no other 3-face-painted cubes.
ANSWER. There are 16 3-face-painted cubics in this problem.
--------------
Solved.
It is how my logic works.
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