There is a square mat board one side 5cm as shown in the figure and it is partitioned into grid square 25.
I pile up cubes of one side 1cm on the square by the number written in the each square.
In faces of the cube, attach a red square seal of one side 1cm on each surface visible from the outside.
I do not attach the seal on other surfaces (the surface in contact with the mat board or cubes , etc.).
(1) Find the number of seal attached.
(2) Find the number of cubes just three seals attached .
I pile up cubes of one side 1cm on the square by the number written in the each square.
In faces of the cube, attach a red square seal of one side 1cm on each surface visible from the outside.
I do not attach the seal on other surfaces (the surface in contact with the mat board or cubes , etc.).
(1) Find the number of seal attached.
(2) Find the number of cubes just three seals attached .
Answer
Solution
Since there is no seal when it is seen from down side, it is 0.
Since the number of the faces of the cube which is visible from front and rear side in the case of this problem is the same, only one side is counted.
Moreover, since the number of the faces of the cube which is visible from right and left side is the same, only one side is counted.
The red number in Fig. 1 is the number of the faces of the cubes which is visible from front and right sides, the sum total is (5 + 5 + 4 + 5 + 5) × 2 + (5 + 4 + 5 + 5 + 5) × 2 = 96.
Adding the number of the faces which are in sight from up side is added, it is 96 + 25 = 121.
Therefore, the grand total of the number of seals is 121 + 26 = 147 seals.
(2) In order to count the number of the cube in which three seals are attached exactly, divides this solid into five layers and count the number from the top layer or the 5th layer in order.
There is no cube which three seals attached in the 5th layer. As shown in Fig. 2, there are three pieces in the 4th layer.
As shown in Fig. 3, there are five pieces in the 3rd layer.
As shown in Fig. 4, there are five pieces in the 2nd layer.
As shown in Fig. 5, there are two pieces in the 1st layer.
There are in total 3 + 5 + 5 + 2 = 15 pieces.
(1) 147 seals
(2) 15 pieces
(2) 15 pieces
Solution
(1) In order to enumerate the number of seals attached, the following procedure is taken.
First, the number of the cube faces of this solid made by piling up cubes is seen from up-down, front-rear and right-left sides and counted.
Next, the number of the faces which are not in sight even if it is seen from up-down, front-rear and right-left sides is counted.
Since a seal is attached on the upper face of every cube which is on the top of cubes piled up in each grid, the number of seal is 5 × 5 = 25 seals.
First, the number of the cube faces of this solid made by piling up cubes is seen from up-down, front-rear and right-left sides and counted.
Next, the number of the faces which are not in sight even if it is seen from up-down, front-rear and right-left sides is counted.
Since a seal is attached on the upper face of every cube which is on the top of cubes piled up in each grid, the number of seal is 5 × 5 = 25 seals.
Since there is no seal when it is seen from down side, it is 0.
Since the number of the faces of the cube which is visible from front and rear side in the case of this problem is the same, only one side is counted.
Moreover, since the number of the faces of the cube which is visible from right and left side is the same, only one side is counted.
The red number in Fig. 1 is the number of the faces of the cubes which is visible from front and right sides, the sum total is (5 + 5 + 4 + 5 + 5) × 2 + (5 + 4 + 5 + 5 + 5) × 2 = 96.
Adding the number of the faces which are in sight from up side is added, it is 96 + 25 = 121.
Next, the blue number in Fig. 1 shows the number of the faces which are not in sight from up-down, front-rear and right-left sides and the sum is 26.
Therefore, the grand total of the number of seals is 121 + 26 = 147 seals.
(2) In order to count the number of the cube in which three seals are attached exactly, divides this solid into five layers and count the number from the top layer or the 5th layer in order.
There is no cube which three seals attached in the 5th layer. As shown in Fig. 2, there are three pieces in the 4th layer.
As shown in Fig. 3, there are five pieces in the 3rd layer.
As shown in Fig. 4, there are five pieces in the 2nd layer.
As shown in Fig. 5, there are two pieces in the 1st layer.
There are in total 3 + 5 + 5 + 2 = 15 pieces.
