Question

predict the number of squares in the next figure.

Explanation:

Answer:

To determine the number of squares in the next figure, we analyze the pattern:

• First figure: \( 1 \) square (a single square).
• Second figure: \( 1 + 4 = 5 \) squares (1 center + 4 arms, each with 1 square).
• Third figure: \( 5 + 4 = 9 \) squares? Wait, no—wait, looking again: Wait, the first figure is 1, second is \( 1 + 4 = 5 \)? Wait, no, the second figure (middle) has 5? Wait, no, let's count:

Wait, the first figure: 1 square.

Second figure: center square, plus 1 above, 1 below, 1 left, 1 right: total \( 1 + 4 = 5 \)? Wait, no, the middle figure (second) has 5? Wait, no, the third figure (right) has: center row (horizontal) 5 squares, vertical column: top 2, bottom 2, plus center? Wait, no, let's count the third figure: horizontal row: 5 squares. Vertical column: top 2, bottom 2, and the center square is shared. So total: \( 5 + 4 = 9 \)? Wait, no, first figure: 1, second: 5, third: 9? Wait, no, wait the third figure: horizontal: 5, vertical: 4 (top 2, bottom 2, but center is already in horizontal). Wait, no, let's count the third figure's squares:

Top: 2, middle horizontal: 5, bottom: 2. Wait, 2 + 5 + 2 = 9? Wait, first figure: 1 (1), second: 1 + 4 = 5 (1 center, 1 up, 1 down, 1 left, 1 right? No, the second figure (middle) has: center, up, down, left, right: 5. Third figure: center, up (2), down (2), left, right (wait, horizontal is 5: left, center, right? No, horizontal is 5 squares: left, left-middle, center, right-middle, right. Vertical: top (2: top, top-middle), center, bottom (2: bottom-middle, bottom). Wait, no, the third figure has: horizontal row (5), vertical column (5) but overlapping at center. So total: \( 5 + 5 - 1 = 9 \)? Wait, first figure: 1 (1), second: 5 (1 + 4), third: 9 (5 + 4)? Wait, no, the difference between first and second is 4, second and third is 4? Wait, first: 1, second: 5 (1 + 4), third: 9 (5 + 4), so the pattern is \( 1 + 4(n-1) \), where \( n \) is the figure number. For \( n=1 \): 1, \( n=2 \): 5, \( n=3 \): 9, so \( n=4 \) would be \( 9 + 4 = 13 \)? Wait, no, wait the third figure: let's count again. The third figure (right) has: top 2, middle horizontal 5, bottom 2. Wait, 2 + 5 + 2 = 9. The second figure (middle) has: top 1, middle horizontal 3, bottom 1. 1 + 3 + 1 = 5. The first figure: 1. So the horizontal row: 1, 3, 5 (odd numbers: \( 2n - 1 \) for \( n=1,2,3 \)). The vertical column: 1, 3, 5? Wait, no, first figure: vertical column is 1 (only center). Second figure: vertical column (top, center, bottom) 3. Third figure: vertical column (top 2? No, wait the third figure's vertical: top 2, center, bottom 2? No, the third figure's vertical has 5 squares? Wait, no, the third figure's vertical: top square, then another square below it, then center, then another square below center, then another square below that. Wait, that's 5? Wait, no, the third figure's vertical: 5 squares (top, top-middle, center, bottom-middle, bottom). And horizontal: 5 squares (left, left-middle, center, right-middle, right). So total: 5 (horizontal) + 5 (vertical) - 1 (center overlap) = 9. Second figure: horizontal 3, vertical 3, overlap 1: 3 + 3 - 1 = 5. First figure: horizontal 1, vertical 1, overlap 1: 1 + 1 - 1 = 1. So the pattern for horizontal/vertical length: 1, 3, 5 (odd numbers, \( 2n - 1 \) for figure \( n \)). So for figure 4, horizontal and vertical length would be \( 2(4) - 1 = 7 \). Then total squares: \( 7 + 7 - 1 = 13 \). Wait, but let's check the counts:

Figure 1: \( n=1 \), length=1: \( 1 + 1 - 1 = 1 \).

Figure 2: \( n=2 \), length=3: \( 3 + 3 - 1 = 5 \).

Figure 3: \( n=3 \), length=5: \( 5 + 5 - 1 = 9 \).

Figure 4: \( n=4 \), length=7: \( 7 + 7 - 1 = 13 \).

Yes, that makes sense. The number of squares increases by 4 each time (from 1 to 5: +4, 5 to 9: +4, 9 to 13: +4). So the next figure (fourth) has 13 squares.

Wait, but let's confirm the third figure: the right figure has 9 squares? Let's count: top two, middle horizontal five, bottom two. Wait, 2 + 5 + 2 = 9. Yes. Second figure: top one, middle horizontal three, bottom one: 1 + 3 + 1 = 5. First figure: 1. So the vertical arms: first figure, vertical arm length 1 (only center). Second figure, vertical arm length 2 (top 1, bottom 1, plus center: total 3? Wait, no, the vertical arm in second figure: top 1, center, bottom 1: 3 squares. Horizontal arm: left 1, center, right 1: 3 squares. Total: 3 + 3 - 1 = 5. Third figure: vertical arm: top 2, center, bottom 2: 5 squares. Horizontal arm: left 2, center, right 2: 5 squares. Total: 5 + 5 - 1 = 9. So vertical and horizontal arms each have length \( n \) (number of squares from end to center), so total length \( 2n - 1 \). For figure 1: \( n=1 \), length \( 2(1)-1=1 \). Figure 2: \( n=2 \), length \( 3 \). Figure 3: \( n=3 \), length \( 5 \). Figure 4: \( n=4 \), length \( 7 \). So vertical and horizontal each have 7 squares, overlapping at center, so total \( 7 + 7 - 1 = 13 \).

Thus, the number of squares in the next figure is \(\boldsymbol{13}\).