Solutions: Sec 3.2 Supercells

Page 57: Figure It Out

Q1. Colour the supercells. Row: 6828, 670, 9435, 3780, 3708, 7308, 8000, 5583, 52

• 6828: > 670 (Yes)
• 9435: > 670 AND > 3780 (Yes)
• 7308: > 3708 AND < 8000 (No)
• 8000: > 7308 AND > 5583 (Yes)
• 52: End. Neighbor 5583 is larger. (No) Supercells: 6828, 9435, 8000.

Q5. Pattern for maximum supercells.

• For a line of $N$ cells.
• To maximize supercells, we want a pattern like High, Low, High, Low...
• Even N (e.g., 4): H, L, H, L $\rightarrow$ 2 supercells ($N/2$).
• Odd N (e.g., 5): H, L, H, L, H $\rightarrow$ 3 supercells ($(N+1)/2$).

Q6. Can you fill a table without repeating numbers such that there are no supercells? Answer: No. Reasoning: In any finite set of distinct numbers, there is always a largest number. That largest number will definitely be larger than all its neighbors. Therefore, the cell containing the largest number must be a supercell.

Q7. Will the cell having the largest number always be a supercell? Answer: Yes (as explained above). Will the cell having the smallest number be a supercell? Answer: No. It is smaller than all its neighbors, so it cannot be a supercell.

Page 58: Table 2 Puzzle

Task: Fill with 5-digit numbers using digits 1, 0, 6, 3, 9. (Table 2 in PDF). Constraint: Only colored cells are supercells. Solution Logic: The colored cells must contain large permutations (e.g., 96310). The non-colored cells neighbors to supercells must be smaller permutations (e.g., 10369).

Biggest number in table: 96,310. Smallest even number: 10,396 (ends in 6). Smallest > 50,000: 60,193.