1. Problem 23: Is it possible to walk 2595 miles in 6 months? 2. To find out, we calculate the average miles per day needed. 3. There are about 30 days in a month, so 6 months is $6 \times 30 = 180$ days. 4. Average miles per day = $\frac{2595}{180} \approx 14.42$ miles/day. 5. Walking 14.42 miles every day is challenging but possible for a fit person. 6. Problem 24: Are there more or less than 50,000 seconds in a day? 7. Seconds in a day = $24 \times 60 \times 60 = 86400$ seconds. 8. Since 86400 > 50000, there are more than 50,000 seconds in a day. 9. Problem 25: Area comparison of original and doubled photograph. 10. Original area = $4 \times 5 = 20$ square inches. 11. New dimensions: width = 8, length = 10. 12. New area = $8 \times 10 = 80$ square inches. 13. Ratio of new area to original area = $\frac{80}{20} = 4$. 14. The new photograph's area is 4 times larger. 15. Problem 26: Price changes of a watch. 16. Regular price = 50. 17. Sale price after 10% reduction = $50 - 0.10 \times 50 = 50 - 5 = 45$. 18. Price after 10% increase on sale price = $45 + 0.10 \times 45 = 45 + 4.5 = 49.5$. 19. Current price 49.5 is less than regular price 50. 20. Problem 27a: Find 3-digit palindromic squares. 21. A 3-digit palindrome has form $aba$ (e.g., 121, 131). 22. Check squares of natural numbers from 10 to 31 (since $31^2=961$). 23. Palindromic squares found: 121 ($11^2$), 484 ($22^2$), 676 ($26^2$). 24. Problem 27b: Find 4-digit palindromic cubes. 25. 4-digit numbers range from 1000 to 9999. 26. Cube roots approx from $\sqrt[3]{1000}=10$ to $\sqrt[3]{9999} \approx 21.5$. 27. Check cubes of 10 to 21: - $10^3=1000$ (not palindrome) - $11^3=1331$ (palindrome) - $12^3=1728$ (no) - $13^3=2197$ (no) - $14^3=2744$ (no) - $15^3=3375$ (no) - $16^3=4096$ (no) - $17^3=4913$ (no) - $18^3=5832$ (no) - $19^3=6859$ (no) - $20^3=8000$ (no) - $21^3=9261$ (no) 28. Only 1331 is a 4-digit palindromic cube. 29. Problem 28: Speed of car between palindromic odometer readings. 30. Initial reading: 15951. 31. Next palindromic number after 15951 is 16061. 32. Distance traveled = $16061 - 15951 = 110$ miles. 33. Time = 2 hours. 34. Speed = $\frac{110}{2} = 55$ miles per hour. Final answers: - 23: Possible with ~14.42 miles/day. - 24: More than 50,000 seconds. - 25: New area 4 times larger. - 26: Current price less than regular price. - 27a: Palindromic squares: 121, 484, 676. - 27b: Palindromic cube: 1331. - 28: Speed = 55 mph.