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705-805 (Hard)|   Algebra|   Geometry|      
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Bunuel
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All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cu 09 May 2019, 21:35
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75% (hard)

Question Stats:

62% (02:50) correct 38% (03:04) wrong based on 144 sessions

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All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

(A) 312

(B) 343

(C) 625

(D) 729

(E) 1680
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D
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All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cu 22 Jul 2019, 20:06
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Its easy with Equations !! :angel:



Let us call the six sides of our cube a,b,c,d,e and f (where a is opposite d, c is opposite e, and b is opposite f).
Thus, for the eight vertices, we have the following products:
abc , abe , bcd , bde , acf , cdf , cef and def.
Let us find the sum of these products:
abc + abe + bcd + bde + acf + cdf + aef + def
Factorizing >
b (ac + ae + cd + de) + f (ac + ae + cd + de)
=(b+f) (ac+ae+cd+de)
=(b+f) (a (c+e) + d (c+e))
=(b+f) (a+d) (c+e) ---- Eqn 1

We have the product. Notice how the factors are sums of opposite faces.

Here is one property -- Product is maximum when numbers are equal.
Ex: consider number 6.
I can write 6 = 5+1 or 4+2 or 3+3
But product of these numbers 5*1,4*2 or 3*3 is maximum only for the case when numbers are equal i.e. 3*3

Similarly,
For 2,3,4,5,6 & 7
We need to make 3 groups (as in equation 1) so that groups are equal (as we learned in above property)
In this case it is (7+2) * (6+3) * (5+4)
=9*9*9
=729 ---

Ans D



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All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cu 21 Jul 2019, 10:45
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File comment: PLEASE FIND THE ATTACHED FILE.

I think, logic is that we need to put big numbers next to each other.

(P.S: I got the wrong solution first)
06064B46-6DFD-41AF-9651-B88FD93FD716.jpeg
06064B46-6DFD-41AF-9651-B88FD93FD716.jpeg [ 761.3 KiB | Viewed 6780 times ]

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All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cu 09 May 2019, 23:53
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Bunuel
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

(A) 312

(B) 343

(C) 625

(D) 729

(E) 1680
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to maximize value of product ;
(7+2)*(6+3)*(5+4) ; 9*9*9 ; 729
IMO D
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All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cu 21 Jul 2019, 09:23
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Archit3110
Bunuel
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

(A) 312

(B) 343

(C) 625

(D) 729

(E) 1680

to maximize value of product ;
(7+2)*(6+3)*(5+4) ; 9*9*9 ; 729
IMO D
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Hi Archit3110,

Can you elaborate your solution.
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All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cu 21 Jul 2019, 09:43
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stne ; i did not solve the question considering the product of three nos and then adding the sum of each vertice as then arranging the no on each face of cube would have been a task
the question has asked us to find greatest possible value so to find the largest value need to multiply the greatest no with smallest value and so on ..
total 3 such pairs are there (7+2)*(6+3)*(5+4) ; 729


stne
Archit3110
Bunuel
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

(A) 312

(B) 343

(C) 625

(D) 729

(E) 1680

to maximize value of product ;
(7+2)*(6+3)*(5+4) ; 9*9*9 ; 729
IMO D
Hi Archit3110,

Can you elaborate your solution.
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All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cu 17 Aug 2025, 12:18
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