Evelyn has formed a solid by sticking identical cubes: she places a cube at the top, 3 cubes in the second row, 6 cubes in the third row ... etc first,what is the number of cubes in the forth row? second thing, Evelyn has soaked all solid in a pot of painting.How many Cubes will have exactly one single color face?
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First, the number of cubes in the fourth row is 10.
Second, all of the cubes will have exactly one single color face.
Réponse :
The number of cubes in each row follows a pattern where the number of cubes in the nth row is given by n^2. So the number of cubes in the fourth row will be 4^2 = 16 cubes.
To find the number of cubes with exactly one single color face after painting, we can first observe that each cube has six faces. Since we do not know the orientation of the cubes, we need to consider the possible number of faces that can be painted.
For the cubes in the top row, all six faces are exposed and can be painted. For the cubes in the second row, only five faces can be painted, since one face is covered by the cube above it. Similarly, for the cubes in the third row, only four faces can be painted.
Continuing in this pattern, we can see that for the cubes in the nth row, only (6 - n) faces can be painted. Therefore, the total number of cubes with exactly one single color face after painting is:
6 (number of cubes in the top row) + 5 (number of cubes in the second row) + 4 (number of cubes in the third row) + 3 (number of cubes in the fourth row) + ... + 1 (number of cubes in the sixth row)
= 6 + 5 + 4 + 3 + 2 + 1
= 21 cubes
Explications étape par étape :