BRAIN-TEASERS 05

🧠 BRAIN-TEASER 05
Boatmen, Staircase & Number Puzzles

🚤 Boatmen River Crossing: Two boatmen start simultaneously from opposite shores of a river and cross each other after 45 minutes from starting. They row till they reach the opposite shore and return immediately. When will they cross each other again?

✅ Answer: 90 minutes (1 hour 30 minutes from start)

📐 Step-by-Step Explanation

When two boats start from opposite shores, their first meeting happens after 45 minutes. At this moment, together they have covered the width of the river once. After meeting, they continue to opposite shores and turn back. To meet again, they must together cover twice the river width (each goes to the opposite end and returns partway). Since their combined speed is constant, the time for the second meeting is exactly double the first meeting time: 2 × 45 = 90 minutes from the start.

⏱️ Second crossing happens at 90 minutes (1 hour 30 minutes)

🪜 Staircase Footmarks Puzzle

Problem: Three girls climb down a staircase. Girl A climbs 2 steps at a time, Girl B climbs 3 steps, Girl C climbs 4 steps. They start together from the top, leave footmarks, and all reach the bottom in complete steps. On which steps would there be exactly one pair of footmarks? Which steps have no footmarks?

📌 Steps with exactly one pair of footmarks: 2, 3, 9, 10
❌ Steps with no footmarks: 1, 5, 7, 11

Detailed Explanation: The total number of steps is the LCM of 2, 3, and 4 = 12 steps.
• Girl A (2 steps/stride) lands on: 2, 4, 6, 8, 10, 12
• Girl B (3 steps/stride) lands on: 3, 6, 9, 12
• Girl C (4 steps/stride) lands on: 4, 8, 12
Counting footmarks per step:
Step 1: none | Step 2: A only | Step 3: B only | Step 4: A and C (two) | Step 5: none | Step 6: A and B (two)
Step 7: none | Step 8: A and C (two) | Step 9: B only | Step 10: A only | Step 11: none | Step 12: all three
Therefore, steps 2,3,9,10 have exactly one footmark. Steps 1,5,7,11 have none.

🎖️ Soldiers in Rows – Chinese Remainder Theorem

Problem: A group of soldiers: when arranged in rows of 3, there is 1 extra. In rows of 5, there are 2 extra. In rows of 7, there are 3 extra. Find the minimum number of soldiers.

👥 Answer: 52 soldiers

Step-by-step solution using the Chinese Remainder Theorem:
We need N such that:
N ≡ 1 (mod 3) → N leaves remainder 1 when divided by 3
N ≡ 2 (mod 5) → N leaves remainder 2 when divided by 5
N ≡ 3 (mod 7) → N leaves remainder 3 when divided by 7

Let's test numbers that satisfy the first two conditions:
Numbers ≡ 1 mod 3: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52...
Among these, which leave remainder 2 when divided by 5? Check: 7(remainder2✓), 22(remainder2✓), 37(remainder2✓), 52(remainder2✓)
Now check remainder when divided by 7:
7 ÷ 7 = 1 remainder 0 ❌ | 22 ÷ 7 = 3 remainder 1 ❌ | 37 ÷ 7 = 5 remainder 2 ❌ | 52 ÷ 7 = 7 remainder 3 ✅
Therefore, the minimum number is 52 soldiers.
💡 This is a classic application of the Chinese Remainder Theorem, used in cryptography and computer science!

🔢 Four 9's to Make 100

Challenge: Use exactly four 9's and any mathematical operations (+, -, ×, ÷, √, !, etc.) to make 100.

✅ 99 + 9/9 = 100

More solutions:
• 99 + (9 ÷ 9) = 99 + 1 = 100
• 9 × 9 + 9 + 9 = 81 + 18 = 99 (not 100, close!)
• (9 + 9/9) × 9 + 9? Let's check: (9+1)×9+9 = 10×9+9 = 99
• 999/9.99? That uses more than four 9's.
The simplest and most elegant is 99 + 9/9 = 100.
💡 This puzzle teaches creative thinking and that there can be multiple valid solutions to the same problem!

📊 Number of Digits in 2³⁰

Question: How many digits are in the product 2 × 2 × 2 × ... × 2 (30 times)? That is, 2³⁰.

🔢 Answer: 10 digits

Method 1 – Direct Calculation:
2³⁰ = 2¹⁰ × 2¹⁰ × 2¹⁰ = 1024 × 1024 × 1024 = 1,073,741,824 → This number has 10 digits.

Method 2 – Using Logarithms (for larger exponents):
Number of digits = floor(30 × log₁₀2) + 1
log₁₀2 ≈ 0.30102999566
30 × 0.30102999566 = 9.0308998698
floor(9.0308998698) = 9
9 + 1 = 10 digits

💡 Fun fact: 2³⁰ = 1,073,741,824 is exactly 1 Gibibyte (1 GiB) in computer memory! This is why computers have RAM sizes like 1GB, 2GB, 4GB, 8GB, 16GB, 32GB — they are powers of two.

🌟 Why These Puzzles Matter 🌟

• The boatmen puzzle teaches relative speed and the concept of combined distance.
• The staircase puzzle introduces LCM and set theory in a fun way.
• The soldiers puzzle is a real application of the Chinese Remainder Theorem used in cryptography!
• Four 9's builds creative mathematical thinking.
• 2³⁰ digits connects math to computer science.

Every puzzle you solve builds a stronger, more flexible mind. Keep going!

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