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The Two Cubes (Posted on 2013-08-28) Difficulty: 3 of 5
+-------------+                               +-------------+
|\            |\                             /|            /|
| \     ○     | \                           / |          / |
|  \          |  \                         /  |          /  |
|   \         |   \                       /   |         /   |
|    +-------------+                     +-------------+    |
|   |        |    |                     |    |        |  □ |
+----|--------+    |                     |    +--------|----+
 \   |         \   |                  |   /         |   /
  \  |    ■     \  |                     |  /     ■    |  /
   \ |           \ |                     | /           | /
    \|            \|                     |/            |/
     +-------------+                     +-------------+
(i) Two cubes are shown above which are exactly alike, that is, they have the same number of a given figure.
(ii) The hidden faces indicated by the arrows have the same figure on them.
(iii) Only the four figures ○, ●, ■ and □ (white circle, black circle, black square, white square) appear on the six faces of each of the two cubes.

Which figure - white circle, black circle, white square, or black square - appears on the faces indicated by the arrows?

No Solution Yet Submitted by K Sengupta    
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Solution solution Comment 1 of 1

With four symbols, either one must be present in triplicate or two in duplicate. Either way, two or three must be present only once. By examining three of them we're assured that one of them will be present only once.

If the black circle is present only once, then the hidden vertical faces in the right diagram are white circle and black square, with the white circle being opposite the white square and the black square being opposite the black square that already appears in that diagram. The arrow on the left points to the white circle, and the unseen bottom, being the face pointed at by the arrow in the left hand diagram must also then be a white circle as we are told both arrows point to the same shape.  This certainly counts as one possible solution.

Suppose the black square appears only once. If it were placed on top, the sequence clockwise on the sides would be black circle, white circle, black circle, white square. The arrow on the left of the right hand diagram would point to a white circle, but the arrow on the right side of the left hand diagram would point to the second black circle.  This can't be the case.

To get a third possibility for a symbol appearing only once, lets take the white circle seen at the top of the left hand diagram. Looking down from the top, one would see clockwise, black square, black circle, but they're certainly not those seen on the right diagram, as that would be the view of the white square. The white circle couldn't be the left hidden side pointed at by the arrow, as that needs to be part of a matching pair. If the white circle were on the back square in the right diagram, the left side would be the black square and the bottom would be the black circle--the left side (arrow) would be the black square. But the two black circles would be opposite one another, so the arrow in the left diagram would be a black dot--not a black square.  That leaves the possibility of the lone white circle being on the bottom of the right hand diagram. To get the clockwise black square, black circle seen from the bottom, the black square would need to be on the back (opposite the visible black square) and the black circle on the left (arrow pointed) side. That makes the full ring around the sides, seen clockwise from the bottom, black square, black circle, black square, white square, thus having only one black circle in the sequence, but the same sequence seen clockwise from the top in the left diagram (as the white circle is at the top there), has its black circle visible and can't be the arrow-pointed invisible right side.

The first scenario must be the correct one, and the arrow points to a white circle. There are two white circles and two black squares. The black squares are on opposite faces of the cube and the white circles are on adjacent faces of the cube. There is only one black circle and only one white square.

  Posted by Charlie on 2013-08-28 13:30:29
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