Solved Examples

Example 1: The Stones that Shine (Page 23)

Problem: 3 daughters receive 3 baskets each. Each basket has 3 keys. Each key opens 3 rooms. Each room has 3 tables. Each table has 3 necklaces. Each necklace has 3 diamonds. How many diamonds are there?

Solution: This is a classic powers of 3 problem.

• Daughters: 3 ($3^1$)
• Baskets: $3 \times 3 = 9$ ($3^2$)
• Keys: $9 \times 3 = 27$ ($3^3$)
• Rooms: $27 \times 3 = 81$ ($3^4$)
• Tables: $81 \times 3 = 243$ ($3^5$)
• Necklaces: $243 \times 3 = 729$ ($3^6$)
• Diamonds: $729 \times 3 = 2187$ ($3^7$)

Answer: There are $3^7 = 2187$ diamonds.

Example 2: The Magical Pond (Page 25)

Problem: A lotus doubles every day. The pond is full on Day 30. On which day was it half full?

Solution: Since the lotus population doubles every day, going backwards means halving every day.

• Day 30: Full (100%)
• Day 29: Half Full (50%)
• Day 28: Quarter Full (25%)

Answer: Day 29.

Example 3: Password Combinations (Page 26)

Problem: A safe has a 5-digit password (digits 0-9). How many combinations?

Solution:

• Digit 1: 10 options (0-9)
• Digit 2: 10 options
• Digit 3: 10 options
• Digit 4: 10 options
• Digit 5: 10 options

Total combinations = $10 \times 10 \times 10 \times 10 \times 10 = 10^5 = 1,00,000$.

Example 4: Comparing Large Numbers (Page 29)

Problem: Which is larger: $10000000000000$ or $999999 \times 999999$?

Solution:

1. Roxie’s number: $10,000,000,000,000 = 10^{13}$.
2. Estu’s number: $999,999 \approx 10^6$. So $(10^6) \times (10^6) = 10^{12}$. Even precisely: $(10^6 - 1) \times (10^6 - 1) \approx 10^{12}$.

Since $10^{13} > 10^{12}$, Roxie’s number is larger.