Solutions: Sec 3.5 & 3.6 Palindromes & Kaprekar

Page 61: Digit Detectives

Q: How many times does ‘7’ occur in 1-100?

• Units place: 7, 17, 27, 37, 47, 57, 67, 77, 87, 97 (10 times).
• Tens place: 70, 71, 72, 73, 74, 75, 76, 77, 78, 79 (10 times).
• Total: 20 times.

Q: How many times in 1-1000?

• 1-100: 20 times.
• There are 10 hundreds (0-99, 100-199…). So $20 \times 10 = 200$ from last two digits.
• Plus the hundreds place (700-799): 100 times.
• Total: 300 times.

Page 62: Puzzle Time

Riddle: 5-digit palindrome, odd. $t = 2u$, $h = 2t$. Solution: 12421. (Detailed in Topic 05).

Page 63: Kaprekar Constant

Q: Try with 3-digit numbers.

• Let’s try 321.
  • $321 - 123 = 198$
  • $981 - 189 = 792$
  • $972 - 279 = 693$
  • $963 - 369 = 594$
  • $954 - 459 = 495$
  • $954 - 459 = 495$ (Repeats). Result: For 3-digit numbers, the constant is 495.