18.2.2 A 10x10x10 cube is dipped in paint. How many small cubes have paint? (3+ Solutions)
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- Опубліковано 18 жов 2024
- For maximum benefit, I highly recommend following this series, The Art of Math: A No-Nonsense Guide, in the order I designed it using this playlist: • The Art of Math: A No-...
(0:45) What to FOCUS ON in this video
(1:38) Criteria for a "GOOD APPROACH"
(2:07) Solution 1b.2 (FORWARDS, VERSION 3)
(2:35) Terminology: FACES, EDGES, VERTICES
(4:38) Solution 1a (FORWARDS, VERSION 1)
(7:41) Solution 1b.1 (FORWARDS, VERSION 2)
(9:45) Solution 2 (BACKWARDS)
(11:49) FURTHER PRACTICE: HAVE FUN WITH A TIMER and compare the different solutions across different problem variations (9x9x9, 11x11x11, 4x5x6)
Enjoy!
You are great at explaining! Thank you so much :)
Thanks! My pleasure!
Nice Explanation. Thanks! :)
My pleasure!
That's a really good explanation
Thank you!
Hey, this is a fairly late question, but aren't there 8*8 small cubes for one face and then that's multiplied by 6?
Good question. If I understand your question correctly, I understand the calculation you're referring to, but there are actually 10x10 cubes on each side. You're probably doing 8x8 cubes per face so that you don't include the outer edge of cubes that might be shared between faces. That's fine. You can totally do 8x8x6 faces. However, in this case you still haven't counted the cubes along each of the edges. One good way to visualize this is to add another 8 cubes per edge x 12 edges + 8 corner cubes. Then, you'd be correct. So the entire calculation would be 8x8x6+8(edge cubes for each edge)*12 (edges, 4 on the top, 4 on the sides, 4 on the bottom)+8 (corner cubes)=384+96+8=488.
Mentioned in the video, but just want to repeat that a faster way at arriving at 488 (although I think there is value in practicing various approaches and perspectives of visualizing the problem) is to do 10*10*10 (all the cubes)-8*8*8 (the inner cubes that have no paint (i.e. working backwards)=1000-512=488.
@@itsinis How are there a thousand cubes in total...a cube has six faces...one face has a hundred cubes so six hundred in total, right? I have a feeling I'm wrong but not sure why.. Thank you for the fast reply.
@@Yes-ls3wk There are 2 issues with your thinking:
1. The cube is SOLID, not hollow so that's why it's 10x10x10. There are small cubes *inside the painted surface (even though they will not get painted).
2. Even if we thought it were a hollow cube, it still would not be 600 cubes on the surface. The cubes along the edges are being overcounted. You'd need to do 600 minus the overcounted cubes.
There are 8 cubes per edge that are not corner cubes and 8 corner cubes, so this gives 600-8*12 (edge but not corner cubes)-2*8 (corner)=488.
The "2" int he 2*8 is tricky because you can't just subtract out the 8 corners once. Since they are part of 3 faces they have been TRIPLE counted. Since we want to count them ONCE, we need to subtract them twice.
Hope that helps. Please feel free to continue asking questions if you're confused.
ilya