Brain Teasers
Cubes, Cubes, Everywhere!
In the far future, you find yourself a cube-maker. To make cubes, you would take a 12 by 12 by 12 Rubik's cube-like object. Normally, you would touch each tiny one by one by one cube and they would fall off. However, with new quantum technology, it has become more interesting. You may say the total number of possible cubes (not necessarily with side length one) in the twelve by twelve by twelve Rubik's cube, and hocus-pocus, cubes of all sizes would appear. Assuming that each side in the tiny cubes has length one and that you may only make cubes of integer side length, what number should you say for the new technology to work?
Hint
For 1^3+2^3+... n^3, use (1+2+...n)^2.Answer
6084.First, try a easier problem. Try a two by two by two Rubik's cube. There are 8 tiny cubes and one 2 by 2 by 2 cube. It should come to your notice that you take 2^3+1^3. But just to check, you look at a standard Rubik's cube with side length 3. There are, predictably, 27 one by one cubes, 8 two by two cubes, and one 3 by 3 cube. That was 3^3 +2^3 + 1^3. Now you can come to the conclusion that for any Rubik's cube with side length n, there are n^3+(n-1)^3 + ... 2^3+1^3 total cubes with integer side length. Therefore, there are 1^3+2^3+...12^3 cubes in the Rubik's cube with side length 12. The hint gives the formula for adding a sequence of cubes. Thus, you would get (1+2+3+4+...+11+12)^2 = 6084 cubes.
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Sorry 14mt this is not my forte. Am sure it is a good teaser for someone who has this type of knowledge - it's just not in my data base. Thanks for submitting.
dude ppl are too lazy to figure this one out or are they? hmmmmm......find out next time on math problems! Here is next weks preview:
Math: hey science wait up
Science: what now math?
Math: I have a big problem i need your help
Science: sorry man you know i cant do that stuff last time i tried to solve your problem i had to design a calcualtor for you and i got betrayed when you stole that technology!
Math: o sorry man so can i borrow 10 dollars i need to buy a new geometry set
Science; go to the math answer sheet!
Math: how dare you im going to ask history to help!
Sciencek fine!!
Math: hey history!!!
History: Hey math i had alot of fun last night
Math: yes i love the end of the film when george washington figure out the math equation
History: yes that was a beutiful moment
Math: so.......can yo help me i have a problem
History: sorry i need to go study the World war!
Math: i can help you add that! worldwar 1 + world war 2 = world war 3!
History:What! The future how dare you!!
Will math and historyever resolve their problem find out next time on Math problems!!
Math: hey science wait up
Science: what now math?
Math: I have a big problem i need your help
Science: sorry man you know i cant do that stuff last time i tried to solve your problem i had to design a calcualtor for you and i got betrayed when you stole that technology!
Math: o sorry man so can i borrow 10 dollars i need to buy a new geometry set
Science; go to the math answer sheet!
Math: how dare you im going to ask history to help!
Sciencek fine!!
Math: hey history!!!
History: Hey math i had alot of fun last night
Math: yes i love the end of the film when george washington figure out the math equation
History: yes that was a beutiful moment
Math: so.......can yo help me i have a problem
History: sorry i need to go study the World war!
Math: i can help you add that! worldwar 1 + world war 2 = world war 3!
History:What! The future how dare you!!
Will math and historyever resolve their problem find out next time on Math problems!!
WOW! that was difficult but i finaly got it! nice job.
i didn't understand that at all
For this answer, you have to assume that individual cubes can be used many times over - for more than 1 unique cube, be it a 1x1x1, 2x2x2x, 3x3x3, etc. If each cube can only be used for 1 unique cube - and not by many different cube sizes simultaneously, then the answer would be much different!
Instructions were not clear as to whether you had to extract a whole pile of cubes at one go or just extract one at a time an count the ways. Nice use of patterns though. I liked it!
and, while ive never taken apart a rubiks cube, im almost positive it doesnt have a cube in the center of the 3by3, but instead something that all the cubes are conencted to. better, more detailed explanation necessary, but good teaser, i like the formula, i didnt know that.
nice formula.
but the problem wasn't clear. i only understand what i need to do after i saw your answer. and after that, its easy.
but the problem wasn't clear. i only understand what i need to do after i saw your answer. and after that, its easy.
This was fun!!
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