Simple one!

Probability Level 3

A 3 3 by 3 3 cube is formed by gluing together 27 27 standard cubical dice.What is the minimum sum of all the numbers shown on the surfaces of the cube?

Details and Assumptions \text{Details and Assumptions}

A standard dice has the sum of the opposite faces equal to 7 7 .


The answer is 90.

Anik Mandal by Anik Mandal , 21, India

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1 solution

Curtis Clement
Feb 24, 2015

Each face is split up into 9 pieces (3 by 3) and the middle piece must always have a 1 on it. Now there are 12 cubes with 2 faces exposed each. These faces can total to a minimum of 3 each. Similarly, there are 8 corner cubes that have exposed faces that sum to 6 (1+2+3). This gives a sum of ( 1 × 6 ) + ( 12 × 3 ) + ( 8 × 6 ) = 90 (1\times \ 6) + (12\times \ 3) + (8\times \ 6) = \boxed{90}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

It helps to have a rubik's cube on your desk. Then this boils down to a counting problem.

6-center cubes (1 face exposed), 1 dot 12-edge cubes (2 faces exposed), 1 and 2 dots 8-corner cubes (3 faces exposed), 1, 2, and 3 dots 1-middle cube (no faces exposed), no dots =27

6(1)+12(1+2)+8(1+2+3)=90

Scott Ripperda - 6 years, 2 months ago

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