According to the order as shown in a figure, a figure is made by adding a square whose one side is 1cm one by one.
For example, the figure which added the 23rd square becomes as follows.
(1) Find the number of all the squares in the figure which added the 9th square.
For example, in the case of the figure which added the 6th square, a square will be eight pieces in all.
(2) Find the number of all the squares in the figure which added the 75th square.
(3) What number of square is added when there are 76 pieces of square in all ?
For example, the figure which added the 23rd square becomes as follows.
Answer the following questions.
For example, in the case of the figure which added the 6th square, a square will be eight pieces in all.
(2) Find the number of all the squares in the figure which added the 75th square.
(3) What number of square is added when there are 76 pieces of square in all ?
Answer
The square whose one side is 1 cm in this figure is nine pieces.
The square whose one side is 2 cm is 2 × 2 = 4 pieces.
The square whose one side is 3 cm is one piece. In total there are 9 + 4 + 1 = 14 pieces.
There are eight squares 1cm one side in length and nine in width and are three in a line upper right.
The square whose one side is 1 cm is 75 pieces.
The square whose one side is 2 cm is 7 × 8 = 56 pieces in an 8 cm long and 9 cm wide rectangle.
Furthermore, as shown in a figure, two pieces (red and blue) are made using the grid of 73 ~ 75.
Sum total 56 + 2 = 58 pieces.
The square whose one side is 3 cm is 6 × 7 = 42 pieces in an 8 cm long and 9 cm wide rectangle.
Furthermore, since one piece of square is made using the grid of 73 ~ 75 as shown in a figure, it is sum total 42 + 1 = 43 pieces.
The square whose one side is 4 cm is 5 × 6 = 30 pieces.
The square whose one side is 5 cm is 4 × 5 = 20 pieces.
The square whose one side is 6 cm is 3 × 4 = 12 pieces.
The square whose one side is 7 cm is 2 × 3 = 6 pieces.
The square whose one side is 8 cm is 1 × 2 = 2 pieces.
The above is totaled, it is 75 + 58 + 43 + 30 + 20 + 12 + 6 + 2 = 246 pieces.
The number of the squares whose one side is 1 cm, 2 cm, 3 cm, 4 cm, and 5 cm will be sum total 25 + 16 + 9 + 4 + 1 = 55 pieces calculated by 5 × 5 = 25 pieces, 4 × 4 = 16 pieces, 3 × 3 = 9 pieces, 2 × 2 = 4 pieces, and 1 × 1 = 1 piece, respectively.
When the 26th piece is added, one 1 cm square will increase.
When the 27th piece is added, two 1cm square and one 2 cm square will increase.
When the 28th piece is added, three 1 cm, two 2cm and one 3cm square will increase.
When the 29th piece is added, four 1 cm, three 2cm, two 3cm and one 4cm square will increase.
When the 30th piece is added, five 1cm, four 2cm, three 3cm, two 4cm and one 5cm square will increase.
When the 30th piece is added, it will increase by 5 + 4 + 3 + 2 + 1 = 15 pieces, and will become sum total 55 + 15 = 70 pieces.
(1) 14 pieces
(2) 246 pieces
(3) the 33rd
(2) 246 pieces
(3) the 33rd
Solution
(1) The figure which added the 9th square becomes as it is shown in a figure.
The square whose one side is 1 cm in this figure is nine pieces.
The square whose one side is 2 cm is 2 × 2 = 4 pieces.
The square whose one side is 3 cm is one piece. In total there are 9 + 4 + 1 = 14 pieces.
(2) Looking at the leftmost column, it turns out that 1, 4, 9, 16, 25, 36, 49, 64, 81 -----.
The figure with 75th square is what the 65th ~ the 75th square was added to the big square made with the 8 × 8 = 64 pieces square as shown in a figure.
The figure with 75th square is what the 65th ~ the 75th square was added to the big square made with the 8 × 8 = 64 pieces square as shown in a figure.
There are eight squares 1cm one side in length and nine in width and are three in a line upper right.
The square whose one side is 1 cm is 75 pieces.
The square whose one side is 2 cm is 7 × 8 = 56 pieces in an 8 cm long and 9 cm wide rectangle.
Furthermore, as shown in a figure, two pieces (red and blue) are made using the grid of 73 ~ 75.
Sum total 56 + 2 = 58 pieces.
The square whose one side is 3 cm is 6 × 7 = 42 pieces in an 8 cm long and 9 cm wide rectangle.
Furthermore, since one piece of square is made using the grid of 73 ~ 75 as shown in a figure, it is sum total 42 + 1 = 43 pieces.
The square whose one side is 4 cm is 5 × 6 = 30 pieces.
The square whose one side is 5 cm is 4 × 5 = 20 pieces.
The square whose one side is 6 cm is 3 × 4 = 12 pieces.
The square whose one side is 7 cm is 2 × 3 = 6 pieces.
The square whose one side is 8 cm is 1 × 2 = 2 pieces.
The above is totaled, it is 75 + 58 + 43 + 30 + 20 + 12 + 6 + 2 = 246 pieces.
(3) When the square 1cm one side is put as the square 5cm one side, the number of all squares is counted.
The number of the squares whose one side is 1 cm, 2 cm, 3 cm, 4 cm, and 5 cm will be sum total 25 + 16 + 9 + 4 + 1 = 55 pieces calculated by 5 × 5 = 25 pieces, 4 × 4 = 16 pieces, 3 × 3 = 9 pieces, 2 × 2 = 4 pieces, and 1 × 1 = 1 piece, respectively.
Next, while one piece is added from the 26th piece, it would be counted the sum total will be 76 pieces by how many 1cm squares are added.
When the 26th piece is added, one 1 cm square will increase.
When the 27th piece is added, two 1cm square and one 2 cm square will increase.
When the 28th piece is added, three 1 cm, two 2cm and one 3cm square will increase.
When the 29th piece is added, four 1 cm, three 2cm, two 3cm and one 4cm square will increase.
When the 30th piece is added, five 1cm, four 2cm, three 3cm, two 4cm and one 5cm square will increase.
When the 30th piece is added, it will increase by 5 + 4 + 3 + 2 + 1 = 15 pieces, and will become sum total 55 + 15 = 70 pieces.
Furthermore, when the 31st piece is added, the square of 1 cm increases by one piece, it is the same as above, and it turns out by the 32nd piece 1 + 2 = 3 pieces increase and by the 33rd 1 + 2 + 3 = 6 pieces increase.
Therefore, the number of square when it becomes 76 pieces in all is the 33rd.
Therefore, the number of square when it becomes 76 pieces in all is the 33rd.
