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• $450 Tuition Credit & Official CAT Packs FREE November 15, 2018 November 15, 2018 10:00 PM MST 11:00 PM MST EMPOWERgmat is giving away the complete Official GMAT Exam Pack collection worth$100 with the 3 Month Pack ($299) The entire exterior of a large wooden cube is painted red  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: Hide Tags Manager Joined: 06 Jun 2013 Posts: 54 Concentration: Entrepreneurship, General Management The entire exterior of a large wooden cube is painted red [#permalink] Show Tags Updated on: 14 Jul 2013, 09:38 11 51 00:00 Difficulty: 55% (hard) Question Stats: 69% (02:39) correct 31% (02:26) wrong based on 691 sessions HideShow timer Statistics The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces? A. 6n^2 B. 6n^2 – 12n + 8 C. 6n^2 – 16n + 24 D. 4n^2 E. 24n – 24 _________________ With all the best! Originally posted by TheGerman on 14 Jul 2013, 09:36. Last edited by Bunuel on 14 Jul 2013, 09:38, edited 1 time in total. Edited the question. Most Helpful Expert Reply Math Expert Joined: 02 Sep 2009 Posts: 50570 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 14 Jul 2013, 09:48 12 16 TheGerman wrote: The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces? A. 6n^2 B. 6n^2 – 12n + 8 C. 6n^2 – 16n + 24 D. 4n^2 E. 24n – 24 Say n=3. So, we would have that the large cube is cut into 3^3=27 smaller cubes: Attachment: Red Cube.png [ 79.39 KiB | Viewed 21524 times ] Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26. Answer: B. _________________ Most Helpful Community Reply Intern Joined: 14 May 2014 Posts: 41 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 21 May 2014, 10:14 3 7 Number of cube inside = (n-2)^3 (n-2)^3 cubes have no colored faces. Remaining cubes will have at least one face colored red. Remaining cubes = n^3 -(n-2)^3 = n^3 -( n^3 - 8 - 3*n*2(n-2)) =n^3 - (n^3 - 8 - 6n^2 +12n) =6n^2-12 n +8 Answer is B _________________ Help me with Kudos if it helped you " Mathematics is a thought process. General Discussion Math Expert Joined: 02 Sep 2009 Posts: 50570 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 14 Jul 2013, 09:52 6 7 TheGerman wrote: The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces? A. 6n^2 B. 6n^2 – 12n + 8 C. 6n^2 – 16n + 24 D. 4n^2 E. 24n – 24 Similar questions to practice: if-a-4-cm-cube-is-cut-into-1-cm-cubes-then-what-is-the-107843.html a-big-cube-is-formed-by-rearranging-the-160-coloured-and-99424.html a-large-cube-consists-of-125-identical-small-cubes-how-110256.html 64-small-identical-cubes-are-used-to-form-a-large-cube-151009.html Hope it helps. _________________ Intern Joined: 02 May 2012 Posts: 20 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 14 Aug 2013, 03:41 3 1 Obviously bunuel's solution is mindblowing and the best approach we need in such PS questions ...But just while brainstorming trying to find the solution to this problem i reached here ...Try to visualise that all the smaller cubes which lie on the exterior face of the larger wooden cube have one or more faces painted red...rest all other cubes which lie beneath the first layer of cube wont have any faces painted red... Now if we can visualise the situation .....we can see that if we remove the external layers of cube ..we will be left with cubes having none of their faces red coloured...and if we remove this external layers of cube we are basically removing one cube from each side symmetrically ...so we will be left with a cube having dimensions of (n-2).... so basically we will be left with (n-2)^3 cubes ... now if we want to find the number of cubes in the larger cube having one or more faces as red we can deduct (n-2)^3 from n^3... so number of cubes painted red =n^3 - (n-2)^3 =n^3-(n^3-8-6n^2+12n)= 6n^2-12n+8.......B this solution is fairly easy too just need little bit of visualization........ Bunuel please correct me if i am wrong somewhere ... Manager Joined: 21 Jun 2011 Posts: 66 Location: United States Concentration: Accounting, Finance WE: Accounting (Accounting) Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 11 Dec 2013, 05:29 Bunuel wrote: TheGerman wrote: The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces? A. 6n^2 B. 6n^2 – 12n + 8 C. 6n^2 – 16n + 24 D. 4n^2 E. 24n – 24 Say n=3. So, we would have that the large cube is cut into 3^3=27 smaller cubes: Attachment: Red Cube.png Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26. Answer: B. Hi Bunuel, As usual great explanation. I just have one question , what if we choose n to be 4 or 5. The visualization then becomes little difficult. What would then a better approach solve such question. Math Expert Joined: 02 Sep 2009 Posts: 50570 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 12 Dec 2013, 03:04 davidfrank wrote: Bunuel wrote: TheGerman wrote: The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces? A. 6n^2 B. 6n^2 – 12n + 8 C. 6n^2 – 16n + 24 D. 4n^2 E. 24n – 24 Say n=3. So, we would have that the large cube is cut into 3^3=27 smaller cubes: Attachment: Red Cube.png Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26. Answer: B. Hi Bunuel, As usual great explanation. I just have one question , what if we choose n to be 4 or 5. The visualization then becomes little difficult. What would then a better approach solve such question. You could apply the same logic: Say n=5. So, we would have that the large cube is cut into 5^3=125 smaller cubes. Out of them (5-2)^3=27 little cubes won't be painted red at all and the remaining 125-27=98 will have at least one red face. Now, plug n=5 and see which one of the options will yield 98. But you can use n directly: Total = n^3 Not painted: (n-2)^3 Difference = n^3 - (n-2)^3 = 6n^2 – 12n + 8. Hope it helps. _________________ SVP Status: The Best Or Nothing Joined: 27 Dec 2012 Posts: 1827 Location: India Concentration: General Management, Technology WE: Information Technology (Computer Software) Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 22 May 2014, 22:41 2 Just for purpose of simplification, this formula can be used $$a^3 - b^3 = (a-b) (a^2 + ab + b^2)$$ $$= (n - n + 2) (n^2 + n(n-2) + (n-2)^2)$$ $$= 2 (n^2 + n^2 - 2n + n^2 - 4n + 4)$$ $$= 2 (3n^2 - 6n + 4)$$ $$= 6n^2 - 12n + 8$$ _________________ Kindly press "+1 Kudos" to appreciate Manager Joined: 25 Apr 2014 Posts: 114 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 14 Sep 2014, 07:57 Hi Guys, When I try to substitute n = 4, it seems that both B and C works. Please help! Math Expert Joined: 02 Sep 2009 Posts: 50570 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 14 Sep 2014, 16:00 maggie27 wrote: Hi Guys, When I try to substitute n = 4, it seems that both B and C works. Please help! For plug-in method it might happen that for some particular number(s) more than one option may give "correct" answer. In this case just pick some other numbers and check again these "correct" options only. _________________ Retired Moderator Joined: 19 Apr 2013 Posts: 596 Concentration: Strategy, Healthcare Schools: Sloan '18 (A) GMAT 1: 730 Q48 V41 GPA: 4 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 04 Mar 2015, 08:51 A nice example of plug in method usage. _________________ If my post was helpful, press Kudos. If not, then just press Kudos !!! Intern Joined: 03 Oct 2014 Posts: 1 GMAT 1: 650 Q44 V36 GPA: 3.39 WE: Research (Health Care) Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 05 Mar 2015, 08:54 Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26. Am I missing something here? 54 - 36 + 8 = 10 (not 26) Math Expert Joined: 02 Sep 2009 Posts: 50570 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 05 Mar 2015, 09:05 sushi574 wrote: Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26. Am I missing something here? 54 - 36 + 8 = 10 (not 26) 54 - 36 + 8 = 26, not 10. _________________ Intern Joined: 25 Mar 2015 Posts: 7 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 09 Aug 2015, 02:48 kundankshrivastava wrote: Number of cube inside = (n-2)^3 (n-2)^3 cubes have no colored faces. Remaining cubes will have at least one face colored red. Remaining cubes = n^3 -(n-2)^3 = n^3 -( n^3 - 8 - 3*n*2(n-2)) =n^3 - (n^3 - 8 - 6n^2 +12n) =6n^2-12 n +8 Answer is B Is "Number of cube inside = (n-2)^3 (n-2)^3 cubes have no colored faces." a formula? What's the logic/reasoning behind it? How does one derive this during the exam! Intern Joined: 13 Sep 2015 Posts: 8 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 26 Nov 2015, 14:25 1 The (N-2) approach is right. However 2 is again, an assumption. If I take a side x, the the volume of inner cube becomes $$(N-2x)^3$$ . Here's how: Assuming volume of larger cube =$$N^3$$ Volume of smaller cubes formed = $$N^3/n^3$$ or$$(N/n)^3$$ Thus side of smaller cubes =$$N/n$$= $$x$$(say) Therefore, volume of smaller cube =$$(N-2x)^3$$ Notice that all the squares formed in this cube won't be touched by the paint. Therefore, total volume of cubes that will have a face painted = $$N^3-(N-2x)^3$$ And no. of cubes with this volume can be found by dividing the above equation by $$x^3$$. If we start- $$(N^3-(N-2x)^3)/x^3$$ = $$(8x^3 + blah blah..)/x^3$$ = 8 + something At this point, we can notice that only one option consists 8 as a constant. Otherwise, if you choose solve it, you get the same result. Intern Joined: 13 Jun 2016 Posts: 19 The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 01 Oct 2016, 23:16 Im confused about this topic, could someone please help me explain the logic behind, why there's only 1 cube in the middle that's unpainted? If we look at a rubik's cube ... theres 6 faces .... 9 "smaller cubes" on the face of the cube = 54 smaller cubes Or on another logic; 3^3 = 27 smaller cubes ----> Rubik's cube has 6 faces , therefore 27/6 = 24 (24 "smaller cubes" on the face of the rubik's cube) and then 3 remaining on the "inside". Please help me get this logic right in simple language as possible! Bunuel or anybody else. Thank you! Senior Manager Joined: 26 Dec 2015 Posts: 258 Location: United States (CA) Concentration: Finance, Strategy WE: Investment Banking (Venture Capital) Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 15 May 2017, 07:20 kundankshrivastava wrote: Number of cube inside = (n-2)^3 (n-2)^3 cubes have no colored faces. Remaining cubes will have at least one face colored red. Remaining cubes = n^3 -(n-2)^3 = n^3 -( n^3 - 8 - 3*n*2(n-2)) =n^3 - (n^3 - 8 - 6n^2 +12n) =6n^2-12 n +8 Answer is B how did you get the formula for: Number of cube inside = $$(n-2)^{3}$$? - something you need to memorize? - i can't even visualize it...a cube has 6 faces (top/bottom, +4 around). if we assume a particular cube has 3 tiny cubes per row & column, we can conclude there will be an "innermost" tiny cube that need not be painted red (it would be buried inside the cube). how do we know that in order to find this innermost cube, the formula is: $$(n-2)^{3}$$ more in-depth analysis would be much appreciated Manager Joined: 23 Dec 2013 Posts: 159 Location: United States (CA) GMAT 1: 710 Q45 V41 GMAT 2: 760 Q49 V44 GPA: 3.76 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 22 May 2017, 19:52 TheGerman wrote: The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces? A. 6n^2 B. 6n^2 – 12n + 8 C. 6n^2 – 16n + 24 D. 4n^2 E. 24n – 24 For this problem, pick the smallest n that will satisfy the problem. In this case, n =3 satisfies the criterion. You can then draw a cube with 27 individual pieces. Only the middle cube will be unpainted. So you want to then backsolve from the answer choices until you find 26. EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 12844 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: The entire exterior of a large wooden cube is painted red [#permalink] Show Tags 07 Dec 2017, 14:53 Hi All, This question can be solved by TESTing VALUES. Let's TEST N = 3 (if you think about a standard Rubik's cube, then that might help you to visualize what the cube would look like). So now every "outside" face of the Rubik's cube has been painted. There's only 1 smaller cube of the 27 smaller cubes that does not have paint on it (the one that's in the exact middle). Thus, 26 is the answer to the question when we TEST N=3. There's only one answer that matches.... Final Answer: GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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21 May 2018, 01:48
I did as Bunuel, but it took 8 minutes to come to a correct answer, hope more practice will help me reducing time
Re: The entire exterior of a large wooden cube is painted red &nbs [#permalink] 21 May 2018, 01:48
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