1. **Problem statement:** We analyze the pattern of blocks and balls in figures 1, 2, 3, and find formulas for the number of balls, blocks, and total in figure $n$. 2. **Observing the number of balls:** From the table, balls in figures 1, 2, 3 are 1, 3, 5 respectively. 3. **Pattern type:** The number of balls forms an arithmetic sequence increasing by 2 each time: 1, 3, 5, ... This is an odd number sequence. 4. **Formula for balls:** The $n$th odd number is given by $$T_n = 2n - 1$$ 5. **Observing the number of blocks:** Blocks in figures 1, 2, 3 are 4, 9, 16 respectively. 6. **Pattern type for blocks:** These are perfect squares: $2^2=4$, $3^2=9$, $4^2=16$. 7. **Formula for blocks:** The number of blocks in figure $n$ is $$T_n = (n+1)^2$$ 8. **Completing the table:** - Number of balls for figures 4, 5, 10, 21, 100: - Figure 4: $2(4)-1=7$ - Figure 5: $2(5)-1=9$ - Figure 10: $2(10)-1=19$ - Figure 21: $2(21)-1=41$ - Figure 100: $2(100)-1=199$ - Number of blocks for figures 4, 5, 10, 21, 100: - Figure 4: $(4+1)^2=5^2=25$ - Figure 5: $(5+1)^2=6^2=36$ - Figure 10: $(10+1)^2=11^2=121$ - Figure 21: $(21+1)^2=22^2=484$ - Figure 100: $(100+1)^2=101^2=10201$ - Total = balls + blocks: - Figure 4: $7 + 25 = 32$ - Figure 5: $9 + 36 = 45$ - Figure 10: $19 + 121 = 140$ - Figure 21: $41 + 484 = 525$ - Figure 100: $199 + 10201 = 10400$ 9. **Finding $n$ for 819 balls:** Solve $2n - 1 = 819$ $$2n - 1 = 819$$ $$2n = 819 + 1 = 820$$ $$n = \frac{820}{2} = 410$$ So, figure number $n=410$. 10. **Simplified formula for total:** Total $T_n = $ blocks $+$ balls $$T_n = (n+1)^2 + (2n - 1)$$ Expand and simplify: $$T_n = n^2 + 2n + 1 + 2n - 1 = n^2 + 4n$$ So, $$\boxed{T_n = n^2 + 4n}$$ **Final answers:** - Balls: $T_n = 2n - 1$ - Blocks: $T_n = (n+1)^2$ - Total: $T_n = n^2 + 4n$ - For 819 balls, $n=410$