Figure It Out: Advanced Patterns

4. Calendar 2x2 Square

Let numbers be:

Diagonals:

1. $x(x+8) = x^2 + 8x$
2. $(x+1)(x+7) = x^2 + 7x + x + 7 = x^2 + 8x + 7$

Observation: The product of the secondary diagonal (top-right to bottom-left) is always 7 more than the main diagonal. $55 - 48 = 7$. Confirmed.

6. Remainders with 7

Let $N_1 = 7a + 3$ and $N_2 = 7b + 5$.

• Sum: $(7a+3) + (7b+5) = 7(a+b) + 8 = 7(a+b+1) + 1$.
  • Remainder: 1.
• Difference: $(7a+3) - (7b+5)$.
  • Wait, use modular arithmetic logic. $3 - 5 = -2 \equiv 5 \pmod 7$.
  • Remainder: 5.
• Product: $(3)(5) = 15 = 14 + 1$.
  • Remainder: 1.

10. Dhauli Park Area

Total Area = $(g+g+w) \times (g+w) = (2g+w)(g+w)$. Green Area = $2g^2$. Tiled Area = Total - Green. $(2g^2 + 2gw + gw + w^2) - 2g^2$ $= 3gw + w^2$ $= w(3g + w)$

11. Pattern Sequence

(ii) Step 10:

• Step 1: 4 blocks.
• Step 2: 7 blocks.
• Step 3: 10 blocks.
• Pattern: Starts at 4, adds 3 each time.
• Step 10: $4 + 9(3) = 31$.

(iii) Step $y$: Formula: $3y + 1$. (Check: $y=1 \rightarrow 4$. $y=2 \rightarrow 7$. Correct).