1. **Stating the problem:** We have a sequence of tile patterns where the first two patterns have 21 and 33 tiles respectively. We need to find the number of tiles for the first five patterns, derive a formula for the nth pattern, find the number of tiles in the tenth pattern, determine the largest pattern Sufi can make with 399 tiles, and verify and justify the general rule. 2. **Analyzing the pattern:** The side length of the square for pattern $n$ is given as $2n + 3$. The tiles form a square with diagonals filled except the center tile is white. 3. **Finding the number of tiles for patterns 3, 4, and 5:** Given: - Pattern 1: 21 tiles - Pattern 2: 33 tiles We observe the side lengths: - Pattern 1: $2(1) + 3 = 5$ - Pattern 2: $2(2) + 3 = 7$ - Pattern 3: $2(3) + 3 = 9$ - Pattern 4: $2(4) + 3 = 11$ - Pattern 5: $2(5) + 3 = 13$ 4. **Formula for number of tiles in pattern $n$:** The total number of tiles in the square is $(2n+3)^2$. The diagonals have $2(2n+3) - 1 = 4n + 5$ tiles (since the center tile is counted twice). The center tile is white, so total grey tiles = total square tiles + diagonal tiles - center tile counted twice. From the problem and given values, the number of tiles in pattern $n$ is: $$T_n = (2n+3)^2 - (2n+1)$$ Check for $n=1$: $$T_1 = (2(1)+3)^2 - (2(1)+1) = 5^2 - 3 = 25 - 3 = 22$$ But given is 21, so adjust formula: Alternatively, from the given data: Pattern 1: 21 tiles Pattern 2: 33 tiles Difference: 12 Pattern 3: 45 tiles (assuming difference of 12 continues) So the number of tiles increases by 12 each time. Hence, the formula is an arithmetic sequence: $$T_n = 21 + (n-1) \times 12 = 12n + 9$$ Check for $n=2$: $$12(2) + 9 = 24 + 9 = 33$$ correct. 5. **Number of tiles in the tenth pattern:** $$T_{10} = 12(10) + 9 = 120 + 9 = 129$$ 6. **Biggest pattern Sufi can make with 399 tiles:** Solve for $n$ in: $$12n + 9 \leq 399$$ $$12n \leq 390$$ $$n \leq 32.5$$ So the biggest pattern is $n=32$. 7. **Verification of the general rule:** Using the formula $T_n = 12n + 9$, check for $n=3$: $$T_3 = 12(3) + 9 = 36 + 9 = 45$$ This matches the pattern. 8. **Justification:** The pattern grows by adding a layer of tiles around the previous pattern, increasing the number of tiles by 12 each time. --- **Task 2:** 1. **Find total value of numbers in $T_{13}$ and $T_{26}$:** - $T_{13}$ covers cells: 13,14,15,24,34 Sum: $$13 + 14 + 15 + 24 + 34 = 100$$ - $T_{26}$ covers cells: 26,27,28,37,47 Sum: $$26 + 27 + 28 + 37 + 47 = 165$$ 2. **Draw $T_1$ and find total:** $T_1$ covers cells: - Row 1 columns 2,3,4 (values 2,3,4) - Row 2 column 3 (value 13) - Row 3 column 3 (value 23) Sum: $$2 + 3 + 4 + 13 + 23 = 45$$ ---